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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 19 — Sep. 20, 2004
  • pp: 4608–4613
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Finite difference time domain study of high efficiency photonic crystal superprisms

Toshihiko Baba, Takashi Matsumoto, and Manabu Echizen  »View Author Affiliations


Optics Express, Vol. 12, Issue 19, pp. 4608-4613 (2004)
http://dx.doi.org/10.1364/OPEX.12.004608


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Abstract

Input and output boundaries of a photonic crystal (PC) are optimized so that the superprism exhibits low insertion loss. It is shown that projected-airhole and half-circular airhole interfaces achieve transmission loss of 0.3 dB and 1.0 dB, respectively, for small and large incident angles of light against normal to boundaries. The finite-difference time-domain simulation shows that a low loss is essentially realized by a periodic phase modulation of the incident beam by the interfaces. It also demonstrates the clear steering of collimated light beam with varying wavelength. The enhancement of angular dispersion is also demonstrated by a PC composed of a dispersive medium.

© 2004 Optical Society of America

Fig. 1 Contour plots of eigen frequency (thin black curve), incident angle (thick gray curve) and resolution parameter (thick black curve) of 2D PC in a triangular lattice rotated by 45°. (a) S-vector prism. Resolution parameter of > 30 is omitted. (b) k-vector prism with 45° output end. Black region indicates air lightcone when the structure is airbridge PC slab.

Fig. 2 Light transmission characteristics of S-vector prism. (a) Light intensity distribution for PC without special interfaces. (b) That with projected airhole interfaces. (c) Dependence of insertion loss on angle of projections. (d) Dependence on normalized frequency a/λ.

Figure 2(a) shows the light intensity distribution obtained by the FDTD simulation, when input and output ends have no special interfaces and the Gaussian beam is excited outside of the PC. The 1/e 2 width of the beam is fixed to 20a, which satisfies the above angular divergence condition. At the input end, light is reflected almost perfectly, and very weak intensity enters the PC, exhibiting the negative refraction of the S-vector prism. Due to additional reflection at the 0° output end, light output is hardly observed. In many calculations, we noticed that the result is strongly dependent on θin. For example, for θin = 25°, the insertion loss decreases up to 2.4 dB. However, light exhibits the positive refraction in the PC, which is not the operation expected as an S-vector prism. This implies that the interface must be optimized individually for the target incident angle and Bloch wave. In principle, however, the interface should act as a phase modulator which compensates the field mismatch. Based on this concept, we considered a projected airhole interface for θin = 10°, as shown in Fig. 2(b). Here, projections are oriented in the opposite direction against the incident beam, i.e. θpr = 20°, and their length b is set to be 1.97a. These are critical conditions under which the incident beam uniformly suffers the phase shift from each projection without duplication. Figure 3 shows magnified field profiles at input interfaces with different θpr. It is clearly observed that the optimum θpr gives an adequate modulation to the phase front so that the internal light is smoothly excited. Consequently, the insertion loss is drastically reduced and the clear negative refraction is observed. The lowest loss is 0.5 dB (0.25 dB/interface) when θpr = 20°, as shown in Fig. 2(c). This loss value is attractive even when the prism is considered as a simple diffraction grating. A loss of < 1 dB is maintained for b/a = 1.8 - 2.3 and in a wide spectral range of a/λ = 0.286 - 0.301, as shown in Fig. 2(d). Figure 4(a) demonstrates the beam steering of the S-vector prism with the optimized interface. The beam divergence at a/λ = 0.275 is caused by a small bend of the frequency contour. Except for this, clear steering of the collimated beam is observed. In the range of a/λ = 0.272 - 0.299, the beam steering angle Δθc is 16°, as shown by the solid line in Fig. 4(b). This value almost agrees with that expected from the wavelength sensitivity parameter ∂θc /∂(a/λ) in the dispersion analysis [7

7. T. Baba and T. Matsumoto, “Resolution of photonic crystal superprism,” Appl. Phys. Lett. 81, 2325–2327 (2002). [CrossRef]

].

As a k-vector prism, a structure and light output are considered, as illustrated in Fig. 5(a). A 45° output end is placed on the rear side of the PC so that the internal light dominated by the S-vector prism reaches to this end. The FDTD simulation previously showed that light output occurs in a direction determined by the second Brillouin zone [9

9. T. Matsumoto and T. Baba, “Design and FDTD simulation of photonic crystal k-vector superprism,” IEICE Trans. Electron. E87-C, 393–397 (2004).

]. Such an arrangement of the output end and a direction of light output correspond to assumptions for Fig. 1. The output end is largely inclined against the internal light, so the output beam is expanded to 75a – 150a in width and well collimated. However, the large inclination severely restricts the optimization of the interface. For example, the total insertion loss cannot be lower than 6 dB by the projected airholes, since projections cannot be freely optimized without overlapping. Through trial calculations of many interfaces, we found that half airholes oriented by 60° against normal to the output end gives a low insertion loss of 1.3 dB (~1 dB at the output end). A < 1.5 dB loss is maintained for a/λ = 0.279 - 0.289, and the steering of the collimated beam is observed outside of the PC, as shown in Fig. 5(a). The beam steering angle θout is smaller than that of the S-vector prism, and is comparable to that of the first order grating. However, the expansion of the output beam improves the beam collimation and realizes the higher resolution.

Fig. 3 Shaded drawings of field intensity distribution (H field in the direction perpendicular to the 2D plane) of light propagation at input interfaces of PCs. Dark lines denote the shape of airholes. (a) θpr = -20°. (b) θpr = 20°.
Fig. 4 Beam steering characteristics of S-vector prism. (a) Light intensity distributions for various a/λ. (b) Beam steering angle as a function of a/λ. Solid and dashed lines denote background material of PC without and with material dispersion.
Fig. 5 Beam steering characteristics of k-vector prism. (a) Calculation model and light intensity distributions for various a/λ. (b) Beam steering angle as a function of a/λ. Solid and dashed lines denote background material of PC without and with material dispersion.

In conclusion, projected airhole and half airhole interfaces were assumed, respectively, at input and output ends of a square lattice 2D PC, and the steering of collimated light beam by superprisms was successfully demonstrated in the FDTD simulation. Practical low insertion loss of 0.5 dB and 1.3 dB was estimated for the S- and k-vector prisms, respectively. Beam steering angles demonstrated here were not so large, since the prisms were optimized for high resolution. It was also shown that the material dispersion near the electronic band-edge can enhance the steering angle. These results strongly encourage future experimental studies. This work was supported by IT Program and 21st COE Program of MEXT, and by CREST of JST.

References and links

1.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals: toward microscale lightwave circuits,” J. Lightwave Technol. 17, 2032–2034 (1999). [CrossRef]

2.

T. Baba and M. Nakamura, “Photonic crystal light deflection devices using the superprism effect,” IEEE J. Quantum Electron. 38, 909–914 (2002). [CrossRef]

3.

L. J. Wu, M. Mazilu, T. Karle, and T. F. Krauss, “Superprism phenomena in planar photonic crystals,” IEEE J. Quantum Electron. 38, 915–918 (2002). [CrossRef]

4.

K. B. Chung and S. W. Hong, “Wavelength demultiplexers based on the superprism phenomena in photonic crystals,” Appl. Phys. Lett. 81, 1549–1551 (2002). [CrossRef]

5.

T. Prasad, V. Colvin, and D. Mittleman, “Superprism phenomenon in three-dimensional macroporous polymer photonic crystals,” Phys. Rev. B 67, 165103–165109 (2003) [CrossRef]

6.

D. Scrymgeour, N. Malkova, S. Kim, and V. Gopalan, “Electro-optic control of the superprism effect in photonic crystals,” Appl. Phys. Lett. 82, 3176–3178 (2003). [CrossRef]

7.

T. Baba and T. Matsumoto, “Resolution of photonic crystal superprism,” Appl. Phys. Lett. 81, 2325–2327 (2002). [CrossRef]

8.

T. Matsumoto and T. Baba, “Photonic crystal k-vector superprism,” J. Lightwave Technol. 22, 917–922 (2004). [CrossRef]

9.

T. Matsumoto and T. Baba, “Design and FDTD simulation of photonic crystal k-vector superprism,” IEICE Trans. Electron. E87-C, 393–397 (2004).

10.

T. Baba and D. Ohsaki, “Interfaces of photonic crystals for high efficiency light transmission,” Jpn. J. Appl. Phys. 40, 5920–5924 (2001). [CrossRef]

11.

J. Ushida, M. Tokushima, M. Shirane, A. Gomyo, and H. Yamada, “Immittance matching for multidimensional open-system photonic crystals,” Phys. Rev. B 68, 155115–155121 (2003). [CrossRef]

12.

Y. Suematsu, Ed., Semiconductor Lasers and Integrated Optics, Ohmsha, Tokyo, (1984).

OCIS Codes
(130.3120) Integrated optics : Integrated optics devices
(230.1950) Optical devices : Diffraction gratings
(230.3120) Optical devices : Integrated optics devices
(230.3990) Optical devices : Micro-optical devices

ToC Category:
Research Papers

History
Original Manuscript: August 5, 2004
Revised Manuscript: September 11, 2004
Published: September 20, 2004

Citation
Toshihiko Baba, Takashi Matsumoto, and Manabu Echizen, "Finite difference time domain study of high efficiency photonic crystal superprisms," Opt. Express 12, 4608-4613 (2004)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-19-4608


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References

  1. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, �??Superprism phenomena in photonic crystals: toward microscale lightwave circuits,�?? J. Lightwave Technol. 17, 2032-2034 (1999). [CrossRef]
  2. T. Baba and M. Nakamura, �??Photonic crystal light deflection devices using the superprism effect,�?? IEEE J. Quantum Electron. 38, 909-914 (2002). [CrossRef]
  3. L. J. Wu, M. Mazilu, T. Karle, and T. F. Krauss, �??Superprism phenomena in planar photonic crystals,�?? IEEE J. Quantum Electron. 38, 915-918 (2002). [CrossRef]
  4. K. B. Chung, and S. W. Hong, �??Wavelength demultiplexers based on the superprism phenomena in photonic crystals,�?? Appl. Phys. Lett. 81, 1549-1551 (2002). [CrossRef]
  5. T. Prasad, V. Colvin, and D. Mittleman, �??Superprism phenomenon in three-dimensional macroporous polymer photonic crystals,�?? Phys. Rev. B 67, 165103-165109 (2003) [CrossRef]
  6. D. Scrymgeour, N. Malkova, S. Kim, and V. Gopalan, �??Electro-optic control of the superprism effect in photonic crystals,�?? Appl. Phys. Lett. 82, 3176-3178 (2003). [CrossRef]
  7. T. Baba and T. Matsumoto, �??Resolution of photonic crystal superprism,�?? Appl. Phys. Lett. 81, 2325-2327 (2002). [CrossRef]
  8. T. Matsumoto and T. Baba, �??Photonic crystal k-vector superprism,�?? J. Lightwave Technol. 22, 917-922 (2004). [CrossRef]
  9. T. Matsumoto and T. Baba, �??Design and FDTD simulation of photonic crystal k-vector superprism,�?? IEICE Trans. Electron. E87-C, 393-397 (2004).
  10. T. Baba and D. Ohsaki, �??Interfaces of photonic crystals for high efficiency light transmission,�?? Jpn. J. Appl. Phys. 40, 5920-5924 (2001). [CrossRef]
  11. J. Ushida, M. Tokushima, M. Shirane, A. Gomyo, and H. Yamada, �??Immittance matching for multidimensional open-system photonic crystals,�?? Phys. Rev. B 68, 155115-155121 (2003). [CrossRef]
  12. Y. Suematsu, Ed., Semiconductor Lasers and Integrated Optics, Ohmsha, Tokyo, (1984).

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