Abstract

The local density of states and response to an incident plane wave of a finite sized photonic crystal (PC) with nonlinear material (NLM) is analyzed. Of particular interest is the excitation of surface wave modes at the truncated surface of the PC, which is collocated with the NLM material. We compute the 2D Green function of the PC with linear material and then include the Kerr NLM in a self-consistent manner. The 2D PC consists of a square array of circular rods where one row of the rods is semi-circular in order to move the surface wave defect mode frequency into the band gap. Since the surface modes are resonant at the interface, the NLM should experience at least an order of magnitude increase in field intensity. This is a possible means of increasing the efficiency of the PC as a frequency conversion device.

© 2004 Optical Society of America

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References

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Appl. Phys. Lett. (1)

Y. Wang, B. Cheng, and D. Zhang, �??Distribution of density of photonic states in amorphous photonic materials,�?? Appl. Phys. Lett. 83, 2100 (2003) ; Y.W. Wang, X. Hu, X. Xu, B. Cheng, and D. Zhang, �??Localized modes in defect-free dodecagonal quasiperiodic photonic crystals,�?? Phys. Rev. B 68, 165106 (2003).
[CrossRef]

J. Appl. Phys. (2)

R. Hillebrand, St. Senz, W. Hergert, and U. Gösele, �??Macroporous-silicon-based three-dimensional photonic crystal with a large complete band gap,�?? J. Appl. Phys. 94, 2758 (2003).
[CrossRef]

B. Temelkuran, M. Bayindir, E. Ozbay, R. Biswas, M. M. Sigalas, G. Tuttle, and K.M. Ho, �??Photonic crystalbased resonant antenna with a very high directivity�??, J. Appl. Phys. 87, 603 (2000).
[CrossRef]

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

J. Opt. Soc. Am. B (2)

NATO ASI Series E: Applied Sciences (1)

P. Tran, �??Photonic band structure calculation of system possessing Kerr nonlinearity,�??�?? in �??Photonic Band Gap Materials�?? NATO ASI Series E: Applied Sciences 315, 555, Ed. C. M. Soukoulis (Kluwer Academic Publishers, Boston, 1996).

Opt. Lett. (3)

Phys. Rev. B (1)

P. Tran, �??Photonic-band-structure calculation of material possessing Kerr nonlinearity,�?? Phys. Rev. B 52, 10673 (1995) ;
[CrossRef]

Phys. Rev. E (6)

V. Lousse and P. Vigneron, �??Self-consistent photonic band structure of dielectric superlattices containing nonlinear optical materials,�?? Phys. Rev. E 63 027602 (2001)
[CrossRef]

O. J. Martin and N. B. Piller, �??Electromagnetic scattering in polarizable backgrounds,�?? Phys. Rev. E 58, 3903 (1998)
[CrossRef]

I. V. Konoplev, A. D. R. Phelps, A.W. Cross, and K. Ronald, �??Experimental studies of the influence of distributed power losses on the transparency of two-dimensional surface photonic band-gap structures,�?? Phys. Rev. E 68, 066613 (2003).
[CrossRef]

D. P. Fussell, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan,�??Three-dimensional local density of states in a finite-sized two-dimensional photonic crystal composed of cylinders,�?? Phys. Rev. E 67, 045601 (2003)
[CrossRef]

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, C. M. de Sterke, and N. A. Nicorovici, �??Twodimensional Green�??s function and local density of states in photonic crystals consisting of a finite number of cylinders of infinite length,�?? Phys. Rev. E 63, 046612 (2001)
[CrossRef]

M. Bahl, N. Panoiu, and R. Osgood, �??Nonlinear optical effects in a two-dimensional photonic crystal containing one-dimensional Kerr defects,�?? Phys. Rev. E 67 056604 (2003)
[CrossRef]

Phys. Rev. Lett. (2)

E. Yablonovitch, �??Inhibited spontaneous emission in solid-state physics and electronics,�?? Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef] [PubMed]

O. J. Martin, C. Girard, D. R. Smith, S. Schultz, �??Iterative scheme for computing exactly the total field propagating in dielectric structures of arbitrary shape,�?? Phys. Rev. Lett. 82 315 (1999)
[CrossRef]

Proc. IEEE (1)

A. D. Yaghjian, �??Electric dyadic Green�??s functions in the source region,�?? Proc. IEEE 68, 248 (1980).
[CrossRef]

Other (1)

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton NJ, 1995), See page 75 for a diagram of the surface mode dispersion.

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Figures (5)

Fig. 1.
Fig. 1.

Schematic of photonic crystal (PC) cross-section. The PC consists of infinitely long rods in a 2D array of 7 rods × 6 rods. Note that the rods in the topmost layer have been truncated along their diameter into a semi-circular shape. This truncation will allow localized surface modes to exist along this side of the PC within band gap frequencies. The lines AC and BD are guides for the eye in relating to numerical results below. The arrow and angle θ denote an incident plane wave. The PC has period a, the cylinders have radius r = 0.2a with permittivity εc = 8.9, and the background medium is vacuum. Dispersion characteristics of this PC are given in Fig. 2.

Fig. 2.
Fig. 2.

(a) Band diagram of the bulk PC for an infinite 2D array of round rods with the electric vector parallel to the rods (E) (blue) and the magnetic vector parallel to the rods (H) (red). The vertical axis is normalized frequency f = (ω/c)(a/2π). As shown by the blue curves, an infinite PC of all round rods with the parameters given in Fig. 1 has a complete band gap in the frequency range 0.32 < f < 0.44 for (E) modes. For (H) modes, there is no complete bandgap. (b) These data are a subset of the Γ-X portion of (a) where the green curve shows the dispersion of E surface modes that propagate with wavevector k parallel to the semi-circular cylinder interface at frequencies within the bandgap. The black line is the light line and the gray lines show the band gap limits. Of particular interest in the analysis to follow is the green curve in (b) for the localized surface wave mode. We have chosen to truncate the top row of cylinders along their diameter, however varying the amount of truncation could be done and this would yield different dispersion curves for the surface mode. The data in (a) and (b) are reproduced from [11].

Fig. 3.
Fig. 3.

(a) Normalized LDOS(x) vs position x within the PC for various frequencies with linear material and (b) Normalized LDOS(x) at resonance frequency compared with nonlinear material. In (a), and referring to Fig. 1, the LDOS(x) is shown on a linear scale and is calculated along line AC (between rods) and BD (through rods) where the BD data has been displaced one unit upward for clarity. For reference, the large black circles in the lower part of the figures indicate the positions of the rods in relation to the LDOS. In (a), the dimensionless frequency parameter f = (ω/c)(a/2π) is varied for ω within the band gap from 0.34 to 0.36 with a peak at f = 0.344. Along line BD, there is clear indication of localized states concentrated at the semi-circular rod interface (x ≈ 130Δ, where Δ is the spatial resolution) whereas no localization at the opposite circular rod interface (x ≈ 10Δ). Along line AC there is no comparable indication of localized states. It follows that the localized states exclusively reside within the semi-circular rods. This indicates the importance of defect parameters, such as altering the rod cross-section, in the surface mode dispersion. In (b), the resonant LDOS(x) is plotted on a log scale along lines AC and BD for f = 0.344 with χ = 0 and 0.01. The χ = 0 curves are repeated from (a). To activate the NLM, the PC is illuminated by a plane wave of unit amplitude incident at θ = 45°. Along line AC, the green and black curves are superimposed. Along line BD, the red and blue curves are nearly superimposed with the blue curve slightly larger in value that indicates a slight reduction in LDOS with non-zero χ. This reduction is not very clear on this log scale, however the log scale does show more clearly the LDOS(x) throughout the PC. This indicates that the presence of non-linear material in the semi-circular rods does not significantly alter the LDOS. Finally, these numerical data are completely independent of an incident field.

Fig. 4.
Fig. 4.

Plane wave at normalized frequency f = 0.344 incident on PC at angles of (a) 45° and (b) 135° (see Fig. 1) for both linear and non-linear material. The x-coordinate is in Δ units and the electric field is plotted on a log scale as |E|3 since this is relevant to a Kerr nonlinearity. For both angles of incidence, there is significant excitation of the surface defect mode as noted by the peaks around x ≈ 130Δ and lines BD. In (a), there are incident and reflected fields in the x ≤ 0 PC interface region. In (b), there are again the incident and reflected fields for x ≥ 130Δ in addition to the surface mode excitation field. Note also in (b) that the low field intensity for x ≈ 0 region is because this is a near-field shadow region. It is seen that for a unit amplitude incident wave, the localized |E|3 field is well in excess of an order of magnitude larger because of the surface mode excitation and the localization is precisely at the sites of the nonlinear material.

Fig. 5.
Fig. 5.

Contour plots of a plane wave at frequency f = 0.344 incident on PC at angles of (a) 45° and (b) 135° for linear material. The angles of incidence are indicated by the arrows. These plots show the electric field intensity throughout the PC and within 2 periods a outside the PC., The line plots of Fig. 4 are a subset of these data. The x and y-coordinates are in λ 0 = a/0.35 units. Surface defect mode excitation is clearly seen along the truncated layer of rods. Note that the outline of the PC rod structure is superimposed on this contour plot.

Equations (11)

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G z z ρ ρ′ = G z z 0 ρ ρ′ + k 0 2 A d 2 ρ′′ G z z 0 ρ ρ′′ ε ̂ ( ρ′′ ) G z z ρ′′ ρ′
E z ( ρ ) = E z 0 ( ρ ) + k 0 2 A d 2 ρ′ G z z ρ ρ′ ε ̂ ( ρ′ ) E z 0 ( ρ′ )
E z ( ρ ) = E z 0 ( ρ ) + k 0 2 A d 2 ρ′ G z z 0 ρ ρ′ ε ̂ ( ρ′ ) E z ( ρ′ )
ε ̂ ( ρ ) = ε c 1
G k n j n = ( I M k n n 1 ε ̂ k n ) 1 G k n j n 1 i = k n , j k n
G i j n = G i j n 1 + A ̂ G i k n n 1 ε ̂ k n G k n j n i k n , j k n
G i k n n = ( I M k n n ε ̂ k n ) 1 G i k n n 1 i k n , j = k n
M k 1 0 = k 0 2 A G k 1 k 1 0 d A = i π k 0 R H 1 ( k 0 R ) 2 1
E i = E i 0 + A ̂ n = 1 N G i k n N ε ̂ k n E k n 0
LDOS ( ρ ) = [ G ρ ρ ] [ G 0 ρ ρ ] ,
ε ̂ ( ρ ) = { ε c 1 + χ E c ( ρ ) 2 : ρ within any semicircular cylinder ε c 1 : ρ within any round cylinder 0 : ρ otherwise

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