Terahertz photonic crystal switch in silicon based on self-imaging principle

doi:10.1364/OE.14.003887

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Terahertz photonic crystal switch in silicon based on self-imaging principle

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

Optics Express, Vol. 14, Issue 9, pp. 3887-3892 (2006)
http://dx.doi.org/10.1364/OE.14.003887

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Abstract

A compact and high quality terahertz photonic crystal switch is proposed in silicon. The switch operates based on the self-imaging principle in multi-mode photonic crystal waveguide of triangular lattice. The finite-difference time-domain method and the plane wave expansion method are used to verify and analyze the characteristics of the proposed switch. Numerical simulation results agree well with the theoretical expectation. This kind of device is potentially important for terahertz application and might be a new breakthrough to design other kinds of photonic crystal switches.

1. Introduction

Terahertz (THz) radiation, which bridges the gap between optical region and microwave, offers significant scientific and technological potential in many fields. It has attracted much attention in last decade. With the realization of THz generator and detector [1

I. V. Altukhov, E. G. Chirkova, V. P. Sinis, M. S. Kagan, Y. P. Gousev, S. G. Thomas, K. L. Wang, M. A. Odnoblyudov, and I. N. Yassievich, “Towards Si1-xGex quantum-well resonant-state terahertz laser,” Appl. Phys. Lett. 79, 3909–3911 (2001). [CrossRef]

R. Kohler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature 417, 156–159 (2002). [CrossRef] [PubMed]

], THz technology develops rapidly and now involves from semiconductors, medical images, military detection, to high speed communication, chemical and biological sensing [3–6

P. H. Siegel, “Terahertz technology,” IEEE Trans. Microwave Theory Tech. 50, 910–928 (2002). [CrossRef]

]. However, if we still utilize existing waveguide technology to the THz radiation wave, the size of the device would be extremely huge. Owing to silicon (Si) is transparent to waves below 10 THz [7

D. Dragoman and M. Dragoman, “Terahertz fields and applications,” Prog. Quantum Electron. 28, 1–66 (2004). [CrossRef]

], an integrated and compact Si-based platform for THz signal propagation, guidance, manipulation and readout would be preferable.

Fortunately, photonic crystal (PC) opens a new opportunity for this critical issue because of its capability to control the electromagnetic wave in a small region of photonic band gap (PBG). Therefore, many researchers seek their new ways in the THz regime from photonic crystal technology [8–12

N. Jukam and M. S. Sherwin, “Two-dimensional terahertz photonic crystals fabricated by deep reactive ion etching in Si,” Appl. Phys. Lett. 83, 21–23 (2003). [CrossRef]

]. Recently, well-established fabrication technologies and advanced numerical methods have enabled the sophisticated engineering of PC dispersion characteristics and PC devices, like power splitter and all-optical switch [13–16

I. Park, H. S. Lee, H. J. Kim, K. M. Moon, S. G. Lee, B. H. O, S. G. Park, and E. H. Lee, “Photonic crystal power-splitter based on directional coupling,” Opt. Express 12, 3599–3604 (2004). [CrossRef] [PubMed]

]. However, to gain robust performance of devices, most works are assumed to operate under a single mode condition. In fact, multi-mode interference (MMI) devices are important components for photonic integrated circuits due to their simple structure, low loss and large optical bandwidth [17

Z. Li, Z. Chen, and B. Li, “Optical pulse controlled all-optical logic gates in SiGe/Si multi-mode interference,” Opt. Express 13, 1033–1038 (2005). [CrossRef] [PubMed]

,18

S. Nagai, G. Morishima, H. Inayoshi, and K. Utaka, “Multi-mode interference photonic switches (MIPS),” J. Lightwave Technol. 20, 675–681 (2002). [CrossRef]

]. The operation of the MMI devices is based on the self-imaging principle, a property of multi-mode waveguide, that an input field profile is reproduced at regular intervals along the propagation direction. Kim et al. reported that the self-imaging principle is available in the multi-mode photonic crystal waveguide (PCW) of square lattice [19

H. J. Kim, I. Park, B. H. O, S. G. Park, E. H. Lee, and S. G. Lee, “Self-imaging phenomena in multi-mode photonic crystal line-defect waveguides: application to wavelength de-multiplexing,” Opt. Express 12, 5625–5633 (2004). [CrossRef] [PubMed]

]. But they do not tell us whether it still holds true in a triangular lattice multi-mode PCW.

In this paper, we show that, in the triangular lattice PCW structure, the self-imaging principle is still valid. Numerical computation with the finite-difference time-domain (FDTD) method and the plane wave expansion (PWE) method [20

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001). [CrossRef] [PubMed]

] are used to simulate and analyze such phenomenon in the multi-mode PCW. Simulation results agree well with the theoretical expectation. As an application, a THz PC switch is proposed using two-dimensional (2D) Si PC of triangular lattice based on the self-imaging and wave interference principles.

2. Guided mode property

Figure 1 shows the structure of the Si-based THz PC switch with a MMI region. The structure is a high-resistivity Si slab with air holes forming a triangular lattice. The thickness of the Si slab is 525 μm. The high-resistivity Si is transparent at THz frequencies with a refractive index of 3.42 [8

N. Jukam and M. S. Sherwin, “Two-dimensional terahertz photonic crystals fabricated by deep reactive ion etching in Si,” Appl. Phys. Lett. 83, 21–23 (2003). [CrossRef]

]. In Fig. 1, the PC lattice constant is a = 65.7 μm while the air hole radius is r = 21.0 μm. In this configuration, the ratio of the air hole radius to the lattice constant is r/a = 0.32. The structure consists of two types of PCWs, i.e. PCW1 and PCW2, created by removing one row and four rows of the air holes along the Γ-K direction of the crystal, respectively. PCW1 functions as input waveguides (W1, W2) and output waveguides (W3, W4), while PCW2 serves as the MMI region. To calculate the dispersion properties of the structure, an effective refractive index of 2.8 is obtained according to the method reported in reference [21

M. Qu, “Effective index method for heterostructure-slab-waveguide-based two-dimensional photonic crystals,” Appl. Phys. Lett. 81, 1163–1165 (2002). [CrossRef]

]. Based on the band gap diagram analysis of the THz PC structure by using the 2D PWE method, there is no band gap for TM mode. For TE mode, a band gap is found between normalized frequencies of 0.26(a/λ) and 0.34(a/λ) as shown in Fig. 2. The corresponding wavelength ranges from 193.2 μm (1.553 THz) to 252.7 μm (1.187 THz). It should be noticed that all the THz waves discussed in this paper are TE mode.

Fig. 1. Multi-mode photonic crystal waveguide structure for terahertz waves.

Fig. 2. Band diagram of the THz PC switch for TE mode.

Fig. 3. (a) Dispersion curves for the input/output waveguides (PCW1) and the computational super cell (on the right side), (b) Dispersion curves for the MMI region (PCW2) and the computational super cell (on the right side).

Before launching an input THz field into the multi-mode PCW2, the properties of guided modes in the PCWs are calculated, which strongly depend on the number of modes, propagation constants, and field patterns. Figure 3 shows the calculated dispersion curves of guided modes in the input/output waveguides (PCW1) and the MMI region (PCW2), respectively. The computational super-cells for the PCW1 and the PCW2 are on the right side of Figs. 3(a) and (b), respectively. Further investigation shows that the propagation constant increases when the air holes become smaller.

Figure 3(a) indicates that the PCW1 supports one guided mode when the frequency is located between 0.305(a/λ) and the top of the band gap, while Fig. 3(b) indicates that the PCW2 supports two guided modes from 0.311(a/λ) to 0.323(a/λ). In order to have a strong confinement of light in our structure, we choose an operating frequency of 0.317(a/λ), corresponding to a wavelength of 207.2 μm (1.448 THz). Table 1 lists the mode properties of the PCW2 at a specified frequency.

Table 1. Mode properties of the PCW2 in Fig. 3(b) at a specified frequency

Frequency (a/λ)	0.317
Mode Number	Zero order	First order
Wave Vector (βa/2π)	0.25997	0.19300
Propagation Constant	β₀ = 0.51994;π/a	β₁ = 0.38600;π/a
Parity	Even	Odd

3. Self-imaging phenomenon in the PCW2

According to the self-imaging principle in conventional multi-mode waveguides [22

L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13, 615–627 (1995). [CrossRef]

], the input single mode field will mirror itself at the length of L = p(3L_π ), where p (= 0, 1, 2, …) denotes the periodic nature of imaging along the multi-mode waveguide, L_π is defined as the beat length of the two lowest-order modes and is expressed as:

L_{π} = \frac{π}{(β_{0} - β_{1})}

(1)

where β ₀ and β ₁ are the propagation constants of the two lowest order modes. By substituting β ₀ and β ₁ from Table 1 into the Eq. (1) with p = 1, we can get L = 3L_π = 22.5a.

It should be emphasized that for the square lattice PCW and the conventional waveguide [20

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001). [CrossRef] [PubMed]

, 23], the boundaries of the multi-mode area are perpendicular to the propagating direction. But in our case, there is a 60° acute angle between the boundary and the propagating direction. In other word, the PCW2 is an inclined MMI region and its boundary lies at an angle of 60° to the propagating direction. Therefore, we define L _tr as the length of the PCW2 as shown in Fig. 4. For L _tr = L = 22.5a, a length of L _z = 26a in z direction can be easily derived from plane geometry.

Fig. 4. Inclined MMI region.

Another point we concerned is the phase relationship between the input and the output PCWs. By transferring the configuration of Fig. 1 directly to the FDTD computational domain, the phase shift of the THz wave can be obtained after propagating one repeat distance L _tr = 22.5a (L _z = 26a). We assume that the original phases of the THz waves from the W1 and W2 are φ _1-in = φ _2-in = 0, the phases in the W3 and W4, induced by the THz wave coupled from the W1, are obtained to be φ ₁₃ = π/2 and φ ₁₄ = π. Similarly, the phases in the W3 and W4, induced by the THz wave coupled from the W2, are obtained to be φ ₂₃ = π and φ ₂₄ = π/2.

4. Switch design

In this section, a THz PC switch in the 2D Si is proposed based on the self-imaging principle. Since the THz wave is an electromagnetic wave, interference phenomenon of waves is still valid in the THz regime. Therefore, when the two THz waves with same polarization (TE mode) and amplitude (A ₀) interfere, the total wave amplitude (A) can be expressed as:

A = {2 A}_{0} ∣ \cos (\frac{Δ φ}{2}) ∣

(2)

where Δφ is the phase difference. From Eq. (2) we can see that A = 0 when Δφ = mπ (m = 1, 3, 5 …) and A = 2A ₀ when Δφ = nπ (n = 0, 2, 4 …).

Figure 5 shows the schematic diagram of the proposed THz PC switch based on the structure of Fig. 1. The length of the MMI region is 26a and the shape of the MMI region is trapezium. This is to ensure a symmetric field distribution when the THz wave is coupled into any one or two of the W1 and W2. In our simulation, we suppose that all the incident waves (signal wave and control wave) are with same wavelength (0317 a/λ) and polarization (TE mode). As an example, we regard the THz waves coupled into the W1 and W2 as the signal and control waves, respectively. In order to realize the switching function by wave interference, a certain phase difference Δφ _in must be introduced between the signal wave and the control wave. If we set Δφ _in appropriately to satisfy Δφ = mπ (m = 1, 3, 5 …) or Δφ = nπ (n = 0, 2, 4 …), the switch can shift the input signal coupled into the W1 to the output waveguide W3 (BAR state) or W4 (CROSS state). The desired Δφ _in is listed in Table 2. In Table 2, A ₃ and A ₄ are the wave amplitudes in the W3 and W4, respectively.

Fig. 5. Schematic diagram of the THz PC switch.

Table 2. Relation of the input phase differences (Δφ _in) and the output wave amplitudes (A ₃ and A ₄)

Δφ _in	φ ₁₃	φ ₂₃	φ ₁₃ – φ ₂₃	A₃	φ ₁₄	φ ₂₄	φ ₁₄ – φ ₂₄	A₄
π/2	π/2	π/2	0	2A₀	π	0	π	0
3π/2	π/2	-π/2	π	0	π	-π	2π	2A₀

5. Simulation results and discussion

We use the 2D FDTD method to prove and evaluate the properties of the THz PC switch. Since a finite structure is considered, the whole computational domain is surrounded by perfectly matched layers to absorb the outgoing waves. The Gaussian modulated pulse is launched at the input waveguides with a half width of 0.5a.

First, we transfer the configuration of Fig. 4 to the FDTD computational domain for numerical calculation. When the Gaussian modulated pulse with the frequency of 0317(a/λ) is launched into the W1, the distribution of time-averaged Poynting vector can be obtained after sufficient time steps and a single image is reproduced at the length of 26a in the propagation direction, which validates our assumption. Second, we transfer the THz PC switch structure of Fig. 5 to the FDTD computational domain for property evaluation. Figure 6 shows the steady-state magnetic field distributions of the switch. We can see that once the control wave is introduced, the field distribution in the MMI region is changed and hence the output state of the device can be controlled. When the phase difference between the signal wave and the control wave is π/2, the signal wave will be exported to the W3 (BAR state) as shown in Fig. 6(a). If the phase difference is changed to 3π/2, the signal wave will be outputted to the W4 (CROSS state) as shown in Fig. 6(b).

Fig. 6. Steady-state magnetic field distribution of the switch: (a) BAR state, (b) CROSS state.

To evaluate the characteristics of the switch, normalized output powers in the W3 and W4 are calculated and listed in Table 3. P _3-B and P _4-B indicate the normalized output powers in the W3 and W4 when the switch is at the BAR state, respectively while P _3-C and P _4-C are the normalized output powers in the W3 and W4 corresponding to CROSS state, respectively. Theoretical extinction ratio for the BAR state, defined as 10log(P _3-B/P _3-C), is 16.5 dB and for the CROSS state, defined as 10log(P _4-C/P _4-B), is 14.7 dB. The crosstalk at the BAR state, defined as 10log(P _4-B/P _3-B), and at the CROSS state, defined as 10log(P _3-C/P _4-C), are -14.6 dB and -16.6 dB, respectively.

Table 3. Normalized output powers in W3 and W4

Switch state	Output power in W3	Output power in W4
BAR state	P_3-B = 93.5%	P_4-B = 3.2%
CROSS state	P_3-C = 2.1%	P_4-C = 95.2%

6. Conclusion

The applicability of the self-imaging principle in the multi-mode PCW consists of triangular lattice has been validated. Numerical simulation results agree well with the theoretical expectation. As an example of application, a THz PC switch based on the self-imaging principle is proposed and characterized. Theoretical calculation shows that the transmission is up to 95.2% with an average crosstalk of -15.6 dB. This kind of device is potentially important for THz application and might be a new way to design other kinds of PC switches.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Nos. 90401008, 60577001), the Research Fund for the Doctoral Program of Higher Education (No. 20040558009), and the Program for New Century Excellent Talents in University (No. NECT-04-0796).

References and links

1.	I. V. Altukhov, E. G. Chirkova, V. P. Sinis, M. S. Kagan, Y. P. Gousev, S. G. Thomas, K. L. Wang, M. A. Odnoblyudov, and I. N. Yassievich, “Towards Si1-xGex quantum-well resonant-state terahertz laser,” Appl. Phys. Lett. 79, 3909–3911 (2001). [CrossRef]
2.	R. Kohler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, “Terahertz semiconductor-heterostructure laser,” Nature 417, 156–159 (2002). [CrossRef] [PubMed]
3.	P. H. Siegel, “Terahertz technology,” IEEE Trans. Microwave Theory Tech. 50, 910–928 (2002). [CrossRef]
4.	H. Kurt and D. S. Citrin, “Photonic crystals for biochemical sensing in the terahertz region,” Appl. Phys. Lett. 87, 041108 (2005). [CrossRef]
5.	K. L. Nguyen, M. L. Johns, L. Gladden, C. H. Worrall, P. Alexander, H. E. Beere, M. Pepper, D. A. Ritchie, J. Alton, S. Barbieri, and E. H. Linfield, “Three-dimensional imaging with a terahertz quantum cascade laser,” Opt. Express 14, 2123–2129 (2006). [CrossRef] [PubMed]
6.	B. Ferguson, S. Wang, D. Gray, D. Abbot, and X. C. Zhang, “T-ray computed tomography,” Opt. Lett. 27, 1312–1314 (2002). [CrossRef]
7.	D. Dragoman and M. Dragoman, “Terahertz fields and applications,” Prog. Quantum Electron. 28, 1–66 (2004). [CrossRef]
8.	N. Jukam and M. S. Sherwin, “Two-dimensional terahertz photonic crystals fabricated by deep reactive ion etching in Si,” Appl. Phys. Lett. 83, 21–23 (2003). [CrossRef]
9.	T. D. Drysdale, R. J. Blaikie, and D. R. S. Cumming, “Calculated and measured transmittance of a tunable metallic photonic crystal filter for terahertz frequencies,” Appl. Phys. Lett. 83, 5362–5364 (2003). [CrossRef]
10.	C. Lin, C. Chen, G. J. Schneider, P. Yao, S. Shi, A. Sharkawy, and D. W. Prather, “Wavelength scale terahertz two-dimensional photonic crystal waveguides,” Opt. Express 12, 5723–5728 (2004). [CrossRef] [PubMed]
11.	A. Bingham, Y. Zhao, and D. Grischkowsky, “THz parallel plate photonic waveguides,” Appl. Phys. Lett. 87, 051101 (2005). [CrossRef]
12.	Y. Zhang, Z. Li, and B. Li, “Multimode interference effect and self-imaging principle in two-dimensional silicon photonic crystal waveguides for terahertz waves,” Opt. Express 14, 2679–2689 (2006). [CrossRef] [PubMed]
13.	I. Park, H. S. Lee, H. J. Kim, K. M. Moon, S. G. Lee, B. H. O, S. G. Park, and E. H. Lee, “Photonic crystal power-splitter based on directional coupling,” Opt. Express 12, 3599–3604 (2004). [CrossRef] [PubMed]
14.	T. Liu, A. R. Zakharian, M. Fallahi, J. V. Moloney, and M. Mansuripur, “Multi-mode interference-based photonic crystal waveguide power splitter,” J. Lightwave Technol. 22, 2842–2846 (2004). [CrossRef]
15.	F. Cuesta-Soto, A. Martinez, J. Garcia, F. Ramos, P. Sanchis, J. Blasco, and J. Martí, “All-optical switching structure based on photonic crystal directional coupler,” Opt. Express 12, 161–167 (2003). [CrossRef]
16.	A. Locatelli, D. Modotto, D. Paloschi, and C. D. Angelis, “All optical switching in ultrashort photonic crystal couplers,” Opt. Commun. 237, 97–102 (2004). [CrossRef]
17.	Z. Li, Z. Chen, and B. Li, “Optical pulse controlled all-optical logic gates in SiGe/Si multi-mode interference,” Opt. Express 13, 1033–1038 (2005). [CrossRef] [PubMed]
18.	S. Nagai, G. Morishima, H. Inayoshi, and K. Utaka, “Multi-mode interference photonic switches (MIPS),” J. Lightwave Technol. 20, 675–681 (2002). [CrossRef]
19.	H. J. Kim, I. Park, B. H. O, S. G. Park, E. H. Lee, and S. G. Lee, “Self-imaging phenomena in multi-mode photonic crystal line-defect waveguides: application to wavelength de-multiplexing,” Opt. Express 12, 5625–5633 (2004). [CrossRef] [PubMed]
20.	S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001). [CrossRef] [PubMed]
21.	M. Qu, “Effective index method for heterostructure-slab-waveguide-based two-dimensional photonic crystals,” Appl. Phys. Lett. 81, 1163–1165 (2002). [CrossRef]
22.	L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13, 615–627 (1995). [CrossRef]

OCIS Codes
(130.1750) Integrated optics : Components
(130.3120) Integrated optics : Integrated optics devices

ToC Category:
Integrated Optics

History
Original Manuscript: February 15, 2006
Revised Manuscript: April 25, 2006
Manuscript Accepted: April 25, 2006
Published: May 1, 2006

Citation
Zhangjian Li, Yao Zhang, and Baojun Li, "Terahertz photonic crystal switch in silicon based on self-imaging principle," Opt. Express 14, 3887-3892 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-9-3887

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References

I. V. Altukhov, E. G. Chirkova, V. P. Sinis, M. S. Kagan, Y. P. Gousev, S. G. Thomas, K. L. Wang, M. A. Odnoblyudov, and I. N. Yassievich, "Towards Si1-xGex quantum-well resonant-state terahertz laser," Appl. Phys. Lett. 79, 3909-3911 (2001). [CrossRef]
R. Kohler, A. Tredicucci, F. Beltram, H. E. Beere, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, "Terahertz semiconductor-heterostructure laser," Nature 417, 156-159 (2002). [CrossRef] [PubMed]
P. H. Siegel, "Terahertz technology," IEEE Trans. Microwave Theory Tech. 50, 910-928 (2002). [CrossRef]
H. Kurt and D. S. Citrin, "Photonic crystals for biochemical sensing in the terahertz region," Appl. Phys. Lett. 87, 041108 (2005). [CrossRef]
K. L. Nguyen, M. L. Johns, L. Gladden, C. H. Worrall, P. Alexander, H. E. Beere, M. Pepper, D. A. Ritchie, J. Alton, S. Barbieri, and E. H. Linfield, "Three-dimensional imaging with a terahertz quantum cascade laser," Opt. Express 14, 2123-2129 (2006). [CrossRef] [PubMed]
B. Ferguson, S. Wang, D. Gray, D. Abbot, and X. C. Zhang, "T-ray computed tomography," Opt. Lett. 27, 1312-1314 (2002). [CrossRef]
D. Dragoman and M. Dragoman, "Terahertz fields and applications," Prog. Quantum Electron. 28, 1-66 (2004). [CrossRef]
N. Jukam and M. S. Sherwin, "Two-dimensional terahertz photonic crystals fabricated by deep reactive ion etching in Si," Appl. Phys. Lett. 83, 21-23 (2003). [CrossRef]
T. D. Drysdale, R. J. Blaikie, and D. R. S. Cumming, "Calculated and measured transmittance of a tunable metallic photonic crystal filter for terahertz frequencies," Appl. Phys. Lett. 83, 5362-5364 (2003). [CrossRef]
C. Lin, C. Chen, G. J. Schneider, P. Yao, S. Shi, A. Sharkawy, and D. W. Prather, "Wavelength scale terahertz two-dimensional photonic crystal waveguides," Opt. Express 12, 5723-5728 (2004). [CrossRef] [PubMed]
A. Bingham, Y. Zhao, and D. Grischkowsky, "THz parallel plate photonic waveguides," Appl. Phys. Lett. 87, 051101 (2005). [CrossRef]
Y. Zhang, Z. Li, B. Li, "Multimode interference effect and self-imaging principle in two-dimensional silicon photonic crystal waveguides for terahertz waves," Opt. Express 14, 2679-2689 (2006). [CrossRef] [PubMed]
I. Park, H. S. Lee, H. J. Kim, K. M. Moon, S. G. Lee, B. H. O, S. G. Park, and E. H. Lee, "Photonic crystal power-splitter based on directional coupling," Opt. Express 12, 3599-3604 (2004). [CrossRef] [PubMed]
T. Liu, A. R. Zakharian, M. Fallahi, J. V. Moloney, and M. Mansuripur, "Multi-mode interference-based photonic crystal waveguide power splitter," J. Lightwave Technol. 22, 2842-2846 (2004). [CrossRef]
F. Cuesta-Soto, A. Martinez, J. Garcia, F. Ramos, P. Sanchis, J. Blasco, and J. Martí, "All-optical switching structure based on photonic crystal directional coupler," Opt. Express 12, 161-167 (2003). [CrossRef]
A. Locatelli, D. Modotto, D. Paloschi, and C. D. Angelis, "All optical switching in ultrashort photonic crystal couplers," Opt. Commun. 237, 97-102 (2004). [CrossRef]
Z. Li, Z. Chen, and B. Li, "Optical pulse controlled all-optical logic gates in SiGe/Si multi-mode interference," Opt. Express 13, 1033-1038 (2005). [CrossRef] [PubMed]
S. Nagai, G. Morishima, H. Inayoshi, and K. Utaka, "Multi-mode interference photonic switches (MIPS)," J. Lightwave Technol. 20, 675-681 (2002). [CrossRef]
H. J. Kim, I. Park, B. H. O, S. G. Park, E. H. Lee, and S. G. Lee, "Self-imaging phenomena in multi-mode photonic crystal line-defect waveguides: application to wavelength de-multiplexing," Opt. Express 12, 5625-5633 (2004). [CrossRef] [PubMed]
S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis," Opt. Express 8, 173-190 (2001). [CrossRef] [PubMed]
M. Qu, "Effective index method for heterostructure-slab-waveguide-based two-dimensional photonic crystals," Appl. Phys. Lett. 81, 1163-1165 (2002). [CrossRef]
L. B. Soldano and E. C. M. Pennings, "Optical multi-mode interference devices based on self-imaging: principles and applications," J. Lightwave Technol. 13, 615-627 (1995). [CrossRef]

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