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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 15 — Jul. 23, 2007
  • pp: 9287–9292
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Optical switches and logic gates based on self-collimated beams in two-dimensional photonic crystals

Yuanliang Zhang, Yao Zhang, and Baojun Li  »View Author Affiliations


Optics Express, Vol. 15, Issue 15, pp. 9287-9292 (2007)
http://dx.doi.org/10.1364/OE.15.009287


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Abstract

A device for optical switches and logic gates is proposed in two-dimensional photonic crystals based on self-collimated beams. The main structure of the device is a line-defect-induced 3 dB splitter. Operating principle, as revealed by both theoretical calculation and finite-difference time-domain simulation, is based on the interference of reflected and transmitted self-collimated beams. This device is potentially applicable for photonic integrated circuits.

© 2007 Optical Society of America

1. Introduction

Photonic crystals (PCs) have inspired great interest since their first introduction [1

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

, 2

2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987). [CrossRef] [PubMed]

]. The complex spatial dispersion properties in PCs provide various mechanisms to control light propagation, such as negative refraction [3

3. P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, “Photonic crystals: Imaging by flat lens using negative refraction,” Nature (London) , 426, 404 (2003). [CrossRef]

], superprism effect [4

4. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B , 58, R10096–R10099 (1998). [CrossRef]

], and self-collimation [5

5. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999). [CrossRef]

-12

12. D. Zhao, J. Zhang, P. Yao, X. Jiang, and X. Chen, “Photonic crystal Mach-Zehnder interferometer based on self-collimation,” Appl. Phys. Lett. 90, 231114-1 (2007). [CrossRef]

]. Recently, self-collimation effect, by which an incident light can propagate with almost no diffraction along a definite direction in a perfectly periodic PC, has attracted particular attention because of its potential for photonic integrated circuits (PICs). Several works on bending and splitting of incident self-collimated beams have been theoretically suggested [7

7. X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. 83, 3251–3253 (2003). [CrossRef]

, 8

8. S.-G. Lee, S. S. Oh, J.-E. Kim, H. Y. Park, and C.-S. Kee, “Line-defect-induced bending and splitting of self-collimated beams in two-dimensional photonic crystals,” Appl. Phys. Lett. 87, 1811061-3 (2005).

] and experimentally demonstrated [9

9. M.-W. Kim, S.-G. Lee, T.-T. Kim, J.-E. Kim, H. Y. Park, and C.-S. Kee, “Experimental demonstration of bending and splitting of self-collimated beams in two-dimensional photonic crystals,” Appl. Phys. Lett. 90, 1131211-3 (2007).

, 10

10. C. Chen, A. Sharkawy, D. M. Pustai, S. Shi, and D. W. Prather, “Optimizing bending efficiency of self-collimated beams in non-channel planar photonic crystal waveguides,” Opt. Express 11, 3153–3159 (2003). [CrossRef] [PubMed]

] for the purpose of using these phenomena as a basis for PICs. Meanwhile, optical switches and logic gates are considered as key components in future PICs. Therefore, such PC-based optical devices have attracted significant research efforts in recent years. However, most of the reported works were based on nonlinear optics [13

13. M. F. Yanki, S. Fan, M. Soljačić, and J. D. Joannopoulos, “All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry,” Opt. Lett. 28, 2506–2508 (2003). [CrossRef]

-15

15. Z.-H. Zhu, W.-M. Ye, J.-R. Ji, X.-D. Yuan, and C. Zen, “High-contrast light-by-light switching and AND gate based on nonlinear photonic crystals,” Opt. Express 14, 1783–1788 (2006). [CrossRef] [PubMed]

], which suffered from certain fundamental limitations, such as power consumption and narrow operating frequency range. By contrast, self-collimation phenomenon is independent of light intensity [5

5. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999). [CrossRef]

] and frequency range in which the phenomenon occurs is sufficient for operating in PICs [7

7. X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. 83, 3251–3253 (2003). [CrossRef]

]. Moreover, compared with conventional dielectric waveguides and PC waveguides working within the photonic band gap, self-collimation does not require a lateral confinement to prevent either the beam divergence or diffraction broadening [5

5. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999). [CrossRef]

]. It can release strict alignment for coupling light into narrow waveguides [10

10. C. Chen, A. Sharkawy, D. M. Pustai, S. Shi, and D. W. Prather, “Optimizing bending efficiency of self-collimated beams in non-channel planar photonic crystal waveguides,” Opt. Express 11, 3153–3159 (2003). [CrossRef] [PubMed]

, 11

11. B. Miao, C. Chen, S. Shi, and D. W. Prather, “A high-efficiency in-plane splitting coupler for planar photonic crystal self-collimation devices,” IEEE Photon. Technol. Lett. 17, 61–63 (2005). [CrossRef]

]. Therefore, devices based on self-collimation have intriguing potentials for high-density PICs. To take the advantages of the self-collimation, in this paper, we propose and discuss a device for optical switches and logic gates using line-defect-induced 3dB splitter in two-dimensional (2D) PCs based on the interference between incident self-collimated beams.

2. Operation principle and structure analysis

Self-collimated beams can be totally reflected at a PC-air interface with an incident angle θ>θc=arcsin (1/nH) because of the conservation of momentum components parallel to the interface, where a PC and air corresponds to an optically denser medium (high refractive index, nH) and an optically thinner medium, respectively [7

7. X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. 83, 3251–3253 (2003). [CrossRef]

, 8

8. S.-G. Lee, S. S. Oh, J.-E. Kim, H. Y. Park, and C.-S. Kee, “Line-defect-induced bending and splitting of self-collimated beams in two-dimensional photonic crystals,” Appl. Phys. Lett. 87, 1811061-3 (2005).

]. What is more, by reducing the radii of the PC rods (line defect) instead of eliminating them totally (air), the total reflection becomes frustrated and the self-collimated beam is reflected partially due to the tunneling of the evanescent wave [8

8. S.-G. Lee, S. S. Oh, J.-E. Kim, H. Y. Park, and C.-S. Kee, “Line-defect-induced bending and splitting of self-collimated beams in two-dimensional photonic crystals,” Appl. Phys. Lett. 87, 1811061-3 (2005).

]. It is expected that there should be a phase shift between the reflected beam and the tunneling beam (transmitted beam). Hence, if another self-collimated beam with an appropriate initial phase is introduced, the reflected and transmitted beams may interfere constructively or destructively to realize the switching and logical functions.

To verify our conjecture, as shown in Fig. 1, we consider a 12√2a×12√2a of 2D square lattice PC composed of silicon (Si) rods in air. The radius and dielectric constant of the host Si rods are r=0.35a and ε=12.0, respectively, where a is the lattice constant (the parameters refer to Ref. [8

8. S.-G. Lee, S. S. Oh, J.-E. Kim, H. Y. Park, and C.-S. Kee, “Line-defect-induced bending and splitting of self-collimated beams in two-dimensional photonic crystals,” Appl. Phys. Lett. 87, 1811061-3 (2005).

]). A line defect is created by reducing the radii rd=0.274a of 25 rods aligned in the Γ-X direction different from those of the host rods, as denoted by the green area in Fig. 1. The band diagram and the equifrequency contours (EFCs) of the first band for the E-polarized mode (electric-field is parallel to the rod axes) are shown in Fig. 2. In the EFCs [see Fig. 2(b)], the curves of the frequencies around 0.194(a/λ), where λ is the wavelength of light in free space, can be identified as squares with round corners centered at the M point. It is known that the light propagation direction in the PC is identical to the direction of group velocity given by vg=kω(k), where ω is the optical frequency at the wave vector k [16

16. R. S. Chu and T. Tamir, “Group velocity in space-time periodic media,” Electron. Lett. 7, 410–412 (1971). [CrossRef]

]. It means that the group velocity is perpendicular to the EFCs. So the self-collimation phenomenon occurs when the E-polarized light of the frequencies around 0.194(a/λ) propagate along the Γ-M direction. There are four faces in the device structure, two adjoining faces of them function as input faces (I1, I2), and the other two as output faces (O1, O2), as shown in Fig. 1. Before launching the self-collimated beams to investigate the switching and logical functions, the phase shift between the reflected and transmitted beams should be known.

Fig. 1. Schematic diagram of the proposed optical switches and logic gates. The defect region (green area, the reducing Si rods) and the surrounding region (cyan area, the host rods) corresponds to an optically thinner medium (low refractive index, nL) and an optically denser medium (high refractive index, nH), respectively.
Fig. 2. (a). Band diagram of the PC structure for E-polarized mode. The inset shows the PC structure, which consists of square lattice of Si rods in air. The frequency 0.194(a/λ) is marked by the orange line. (b) Equifrequency contours of the first band.

From Fig. 1, it is clear that the average index in the defect region (green area, the reducing Si rods) is lower than that in the surrounding region (cyan area, the host rods). So the proposed device can be regarded as two identical high refractive index (nH) PC prisms separated by a low refractive index (nL) gap with a spatially symmetric structure. Note that the dielectric constituents are nonabsorbing (the imaginary parts of the dielectric constants are set to zero), which means that the proposed device is a lossless (conservation of energy) system. According to Refs. [12

12. D. Zhao, J. Zhang, P. Yao, X. Jiang, and X. Chen, “Photonic crystal Mach-Zehnder interferometer based on self-collimation,” Appl. Phys. Lett. 90, 231114-1 (2007). [CrossRef]

, 17

17. Z. Y. Ou and L. Mandel, “Derivation of reciprocity relations for a beam splitter from energy balance,” Am. J. Phys. 57, 66–67 (1989). [CrossRef]

, 18

18. R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective (Morgan Kaufmann, San Francisco, 1998), chap. 3.1.

], there is a π/2 phase difference between the reflected and transmitted beams. This is a general result for a spatially symmetric, lossless beam splitting system. Especially for the splitter, by utilizing the line defect in PC to split the self-collimated beams, if the rod radii of the line defect are smaller than that of the host rods, there will be a π/2 phase lag of reflected beam compared to the transmitted beam [12

12. D. Zhao, J. Zhang, P. Yao, X. Jiang, and X. Chen, “Photonic crystal Mach-Zehnder interferometer based on self-collimation,” Appl. Phys. Lett. 90, 231114-1 (2007). [CrossRef]

]. Therefore, for the line defect created by reducing the radii of some Si rods in this work, the reflected beam has a π/2 phase lag compared to the transmitted beam after the self-collimated beam goes through the line defect.

3. Calculations of the structure

Because of the symmetry of the PC device (see Fig. 1), it can be supposed that the low index gap has a transmission amplitude te, so the reflection amplitude is rei(φ+π/2) for both the incident beams, where φ, t, and r are real, and

t2+r2=1
(1)

According to Ref. [8

8. S.-G. Lee, S. S. Oh, J.-E. Kim, H. Y. Park, and C.-S. Kee, “Line-defect-induced bending and splitting of self-collimated beams in two-dimensional photonic crystals,” Appl. Phys. Lett. 87, 1811061-3 (2005).

], the line defect just makes the PC device to be a 3 dB splitter of the self-collimated beams, as shown in Fig. 3. It means that the transmission and reflection amplitudes are equal. Considering their relationship indicated by Eq. (1), it easily gets t=r=1/√2. So the transmission and the reflection amplitudes are e/√2 and ei(φ+π/2)/√2, respectively. The computational simulation is carried out by using a finite-difference time-domain (FDTD) code with perfectly matched layer boundary condition and an E-polarized Gaussian wave with the full width at half maximum 5a is used. A monochromatic wave of the frequency 0.194(a/λ) is launched into the PC device along the Γ-M direction, and then a self-collimated beam will be excited.

Fig. 3. Simulated steady-state field distribution of the E-polarized mode at 0.194(a/λ) when incident beams propagate along the Γ-M direction. (a) and (b) for the case that the incident beam is only launched into face I1 and I2, respectively.

Further suppose that the input face I1 and I2 sees a beam field (Bloch wave) of E1=uEeiφ1 and E2=uEeiφ2, respectively, where φ1, and φ2 are real, E represents a plane wave and u is a function with the same periodicity as the PC. There is only a phase difference between the two incident beams. Then the reflected and transmitted beams can be expressed as:

TI1=E1·teiφ=uEei(φ1φ)2RI1=E1·rei(φ+π2)=uEei(φ1φπ2)2TI2=E2·teiφ=uEei(φ2φ)2RI2=E2·rei(φ+π2)=uEei(φ2φπ2)2}
(2)

The beams exiting output face O1 and O2 can be written as a linear combination (interference) of the reflected and the transmitted beams:

O1=RI1+TI2=uEei(φ1φπ2)2+uEei(φ2φ)2=2uEcos(φ1φ22+π4)ei(φ1+φ22φπ4)O2=RI2+TI1=uEei(φ2φπ2)2+uEei(φ1φ)2=2uEcos(φ1φ22+π4)ei(φ1+φ22φπ4)}
(3)

From Eq. (3) we can find the corresponding intensities,

IO1=O12=2uE2cos2(φ1φ22+π4)=uE2[1+sin(φ1φ2)]IO2=O22=2uE2cos2(φ1φ22+π4)=uE2[1sin(φ1φ2)]}
(4)

Aside from the trivial case where u or E is zero (when the phase difference becomes meaningless quantity), we can see that Io1=2|uE|2, Io2=0 when φ1-φ2=2kπ+π/2 and Io1=0, I02=2|uE|2 when φ12=2kπ-π/2, where k is an integer.

4. Results and discussions of switches and logic gates

From the discussion in section 3, it is clear that the switching function can be realized by introducing a certain phase difference between the two beams (incident on the input faces I1 and I2), as shown schematically in Fig. 4(a). A phase modulator was introduced at input branch I2, the beam coupled into which we regard as the control beam. Figures 4(b) and 4(c) show the simulated steady-state field distributions of the switch. It shows that once the control beam is introduced, the output state of the device can be controlled. When the phase difference φ1−φ2 sets as 2kπ+π/2, where k is an integer, the input lights will be output from the face O1, and there is no output light in the face O2, as shown in Fig. 4(b). If the phase difference is changed to 2kπ-π/2, the light power will be output from the face O2, and the output face O1 will be cut off, as shown in Fig. 4(c). So the switching is achieved.

Fig. 4. (a). Schematic diagram of the switch. Two beams with different phases are incident on the input faces I1 and I2. (b) and (c) Simulated steady-state field distribution of the E-polarized mode at 0.194(a/λ) when incident beams propagate along the Γ-M direction. The phase difference φ12 of the two incident beams particularly sets as π/2 and -π/2, respectively.

Table 1. The total device functions.

table-icon
View This Table

To evaluate the characters of the device, considering the symmetry, the normalized output intensity spectra from the faces O1 and O2 for Fig. 3(a) and Fig. 4(b) are shown in Fig. 5. From Fig. 5(a), it is clear that, at frequency 0.194(a/λ), the incident power is split equally into the two output faces. In the frequency range 0.188–0.199(a/λ), the fluctuation of the output intensities are within 20% referred to the intensity at frequency 0.194(a/λ), and the sum of the transmitted and reflected intensity is larger than 93%. Moreover, from Fig. 5(b), it can be found that, in the frequency range 0.188–0.199(a/λ), there is nearly no fluctuation of the output intensities from the two output faces and the extinction ratio, defined as 10log(IO1/IO2), is larger than 17 dB (maximum 20.1 dB). So we conclude that the switching and logical function of the structure is applicable in the frequency range 0.188–0.199(a/λ).

Fig. 5. (a). The normalized intensity spectra of faces O1 (green line) and O2 (violet line) for Fig. 3(a). The red line represents the total output efficiency. (b) The normalized intensity and extinction ratio spectra for Fig. 4(b).

5. Conclusion

A device for optical switches and logic gates, based on line-defect-induced 3dB splitter of self-collimated beams in a 2D PC composed of Si rods in air, is proposed and demonstrated. The switching and logical function, as revealed by both theoretical calculation and FDTD simulations, is based on the interference of the reflected and transmitted self-collimated beams and is applicable in frequency range 0.188–0.199(a/λ). The extinction ratio for the switch within the applicable frequencies is larger than 17 dB (maximum 20.1 dB). The device has simple geometric structure and clear operating principle, which shows that this device could be a strong candidate for future PICs.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant Nos. 60625404, 60577001, 90401008) and the Research Fund for the Doctoral Program of Higher Education (Grant No. 20040558009).

References and links

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.

P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, “Photonic crystals: Imaging by flat lens using negative refraction,” Nature (London) , 426, 404 (2003). [CrossRef]

4.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B , 58, R10096–R10099 (1998). [CrossRef]

5.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999). [CrossRef]

6.

J. Witzens, M. Lončar, and A. Scherer, “Self-collimation in planar photonic crystals,” IEEE J. Sel. Top. Quantum Electron. 8, 1246–1257 (2002). [CrossRef]

7.

X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. 83, 3251–3253 (2003). [CrossRef]

8.

S.-G. Lee, S. S. Oh, J.-E. Kim, H. Y. Park, and C.-S. Kee, “Line-defect-induced bending and splitting of self-collimated beams in two-dimensional photonic crystals,” Appl. Phys. Lett. 87, 1811061-3 (2005).

9.

M.-W. Kim, S.-G. Lee, T.-T. Kim, J.-E. Kim, H. Y. Park, and C.-S. Kee, “Experimental demonstration of bending and splitting of self-collimated beams in two-dimensional photonic crystals,” Appl. Phys. Lett. 90, 1131211-3 (2007).

10.

C. Chen, A. Sharkawy, D. M. Pustai, S. Shi, and D. W. Prather, “Optimizing bending efficiency of self-collimated beams in non-channel planar photonic crystal waveguides,” Opt. Express 11, 3153–3159 (2003). [CrossRef] [PubMed]

11.

B. Miao, C. Chen, S. Shi, and D. W. Prather, “A high-efficiency in-plane splitting coupler for planar photonic crystal self-collimation devices,” IEEE Photon. Technol. Lett. 17, 61–63 (2005). [CrossRef]

12.

D. Zhao, J. Zhang, P. Yao, X. Jiang, and X. Chen, “Photonic crystal Mach-Zehnder interferometer based on self-collimation,” Appl. Phys. Lett. 90, 231114-1 (2007). [CrossRef]

13.

M. F. Yanki, S. Fan, M. Soljačić, and J. D. Joannopoulos, “All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry,” Opt. Lett. 28, 2506–2508 (2003). [CrossRef]

14.

N. Moll, R. Harbers, R. F. Mahrt, and G.-L. Bona, “Integrated all-optical switch in a cross-waveguide geometry,” Appl. Phys. Lett. 88, 1711041-3 (2006). [CrossRef]

15.

Z.-H. Zhu, W.-M. Ye, J.-R. Ji, X.-D. Yuan, and C. Zen, “High-contrast light-by-light switching and AND gate based on nonlinear photonic crystals,” Opt. Express 14, 1783–1788 (2006). [CrossRef] [PubMed]

16.

R. S. Chu and T. Tamir, “Group velocity in space-time periodic media,” Electron. Lett. 7, 410–412 (1971). [CrossRef]

17.

Z. Y. Ou and L. Mandel, “Derivation of reciprocity relations for a beam splitter from energy balance,” Am. J. Phys. 57, 66–67 (1989). [CrossRef]

18.

R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective (Morgan Kaufmann, San Francisco, 1998), chap. 3.1.

OCIS Codes
(230.1150) Optical devices : All-optical devices
(250.5300) Optoelectronics : Photonic integrated circuits

ToC Category:
Photonic Crystals

History
Original Manuscript: May 11, 2007
Revised Manuscript: June 27, 2007
Manuscript Accepted: July 5, 2007
Published: July 13, 2007

Citation
Yuanliang Zhang, Yao Zhang, and Baojun Li, "Optical switches and logic gates based on self-collimated beams in two-dimensional photonic crystals," Opt. Express 15, 9287-9292 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-15-9287


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References

  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. P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, "Photonic crystals: Imaging by flat lens using negative refraction," Nature (London),  426, 404 (2003). [CrossRef]
  4. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B,  58, R10096-R10099 (1998). [CrossRef]
  5. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett. 74, 1212-1214 (1999). [CrossRef]
  6. J. Witzens, M. Lončar, and A. Scherer, "Self-collimation in planar photonic crystals," IEEE J. Sel. Top. Quantum Electron. 8, 1246-1257 (2002). [CrossRef]
  7. X. Yu and S. Fan, "Bends and splitters for self-collimated beams in photonic crystals," Appl. Phys. Lett. 83, 3251-3253 (2003). [CrossRef]
  8. S.-G. Lee, S. S. Oh, J.-E. Kim, H. Y. Park, and C.-S. Kee, "Line-defect-induced bending and splitting of self-collimated beams in two-dimensional photonic crystals," Appl. Phys. Lett. 87, 1811061-3 (2005).
  9. M.-W. Kim, S.-G. Lee, T.-T. Kim, J.-E. Kim, H. Y. Park, and C.-S. Kee, "Experimental demonstration of bending and splitting of self-collimated beams in two-dimensional photonic crystals," Appl. Phys. Lett. 90, 1131211-3 (2007).
  10. C. Chen, A. Sharkawy, D. M. Pustai, S. Shi, and D. W. Prather, "Optimizing bending efficiency of self-collimated beams in non-channel planar photonic crystal waveguides," Opt. Express 11, 3153-3159 (2003). [CrossRef] [PubMed]
  11. B. Miao, C. Chen, S. Shi, and D. W. Prather, "A high-efficiency in-plane splitting coupler for planar photonic crystal self-collimation devices," IEEE Photon. Technol. Lett. 17, 61-63 (2005). [CrossRef]
  12. D. Zhao, J. Zhang, P. Yao, X. Jiang, and X. Chen, "Photonic crystal Mach-Zehnder interferometer based on self-collimation," Appl. Phys. Lett. 90, 231114-1 (2007). [CrossRef]
  13. M. F. Yanki, S. Fan, M. Soljačić, and J. D. Joannopoulos, "All-optical transistor action with bistable switching in a photonic crystal cross-waveguide geometry," Opt. Lett. 28, 2506-2508 (2003). [CrossRef]
  14. N. Moll, R. Harbers, R. F. Mahrt, and G.-L. Bona, "Integrated all-optical switch in a cross-waveguide geometry," Appl. Phys. Lett. 88, 1711041-3 (2006). [CrossRef]
  15. Z.-H. Zhu, W.-M. Ye, J.-R. Ji, X.-D. Yuan, and C. Zen, "High-contrast light-by-light switching and AND gate based on nonlinear photonic crystals," Opt. Express 14, 1783-1788 (2006). [CrossRef] [PubMed]
  16. R. S. Chu and T. Tamir, "Group velocity in space-time periodic media," Electron. Lett. 7, 410-412 (1971). [CrossRef]
  17. Z. Y. Ou and L. Mandel, "Derivation of reciprocity relations for a beam splitter from energy balance," Am. J. Phys. 57, 66-67 (1989). [CrossRef]
  18. R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Perspective (Morgan Kaufmann, San Francisco, 1998), chap. 3.1.

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