Abstract

A guided-mode scattering matrix approach to photonic crystal integrated devices, based on the expansion of the electromagnetic field in Wannier functions is presented and its applicability to large-scale photonic circuits is demonstrated. In particular, we design two components typically used in wavelength division multi/demultiplexing applications, namely, a directional coupler and a Mach–Zehnder interferometer, and we analyze the transmission spectra as a function of the coupler length and/or delay line length, respectively. These examples demonstrate that by cascading basic functional elements, large-scale circuits can be accurately described and efficiently designed with minimal numerical effort.

© 2008 Optical Society of America

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  1. K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, "Periodic nanostructures for photonics," Phys. Rep. 444, 101-202 (2007).
    [CrossRef]
  2. A. Mekis, J. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
    [CrossRef] [PubMed]
  3. A. A. Green, E. Istrate, and E. H. Sargent, "Efficient design and optimization of photonic crystal waveguides and couplers: the interface diffraction method," Opt. Express 13, 7304-7318 (2005).
    [CrossRef] [PubMed]
  4. A. Chutinan, M. Okano, and S. Noda, "Wider bandwidth with high transmission through waveguide bends in two-dimensional photonic crystal slabs," Appl. Phys. Lett. 80, 1698-1700 (2002).
    [CrossRef]
  5. S. F. Mingaleev, M. Schillinger, D. Hermann, and K. Busch, "Tunable photonic crystal circuits: concepts and designs based on single-pore infiltration," Opt. Lett. 29, 2858-2860 (2004).
    [CrossRef]
  6. M. Koshiba, "Wavelength division multiplexing and demultiplexing with photonic crystal waveguide couplers," J. Lightwave Technol. 19, 1970-1975 (2001).
    [CrossRef]
  7. S.-H. Jeong, N. Yamamoto, J.-I. Sugisaka, M. Okano, and K. Komori, "GaAs-based two-dimensional photonic crystal slab ring resonator consisting of a directional coupler and bent waveguides," J. Opt. Soc. Am. B 24, 1951-1959 (2007).
    [CrossRef]
  8. M. Soljacic, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen, and J. D. Joannopoulos, "Photonic-crystal slow-light enhancement of nonlinear phase sensitivity," J. Opt. Soc. Am. B 19, 2052-2059 (2002).
    [CrossRef]
  9. K. Guven and E. Ozbay, "Coupling and phase analysis of cavity structures in two-dimensional photonic crystals," Phys. Rev. B 71, 085108 (2005).
    [CrossRef]
  10. K. Busch and S. John, "Liquid crystal photonic band gap materials: the tunable electromagnetic vacuum," Phys. Rev. Lett. 83, 967-970 (1999).
    [CrossRef]
  11. K. Yoshino, Y. Shimoda, Y. Kawagishi, K. Nakayama, and M. Ozaki, "Temperature tuning of the stop band in transmission spectra of liquid-crystal infiltrated synthetic opal as tunable photonic crystal," Appl. Phys. Lett. 75, 932-934 (1999).
    [CrossRef]
  12. H. Takeda and K. Yoshino, "Tunable light propagation in Y-shaped waveguides in two-dimensional photonic crystals utilizing liquid crystals as linear defects," Phys. Rev. B 67, 073106 (2003).
    [CrossRef]
  13. T. Yasuda, Y. Tsuji, and M. Koshiba, "Tunable light propagation in photonic crystal coupler filled with liquid crystal," IEEE Photon. Technol. Lett. 17, 55-57 (2005).
    [CrossRef]
  14. S. F. Mingaleev and K. Busch, "Scattering matrix approach to large-scale photonic crystal circuits," Opt. Lett. 28, 619-621 (2003).
    [CrossRef] [PubMed]
  15. K. Busch, S. F. Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann, "The Wannier function approach to photonic crystal circuits," J. Phys. Condens. Matter 15, R1233-1256 (2003).
    [CrossRef]
  16. A. Birner, R. Wehrspohn, U. Gösele, and K. Busch, "Silicon-based photonic crystals," Adv. Mater. (Weinheim, Ger.) 13, 377-388 (2001).
    [CrossRef]
  17. F. Intonti, S. Vignolini, V. Türck, M. Colocci, P. Bettotti, L. Pavesi, S. L. Schweizer, R. Wehrspohn, and D. Wiersma, "Rewritable photonic circuits," Appl. Phys. Lett. 89, 211117 (2006).
    [CrossRef]
  18. K. Busch and S. John, "Photonic band gap formation in certain self-organizing systems," Phys. Rev. E 58, 3896-3908 (1998).
    [CrossRef]
  19. H. Brand, Schaltungslehre Linearer Mikrowellennetze (Hirzel Verlag, 1970).
  20. M. Schillinger, "Maximally localized photonic Wannier functions for the highly efficient description of integrated photonic crystal circuits," Ph.D. dissertation (University of Karlsruhe, 2006).
  21. Y. Jiao, S. Fan, and D. Miller, "Demonstration of systematic photonic crystal device design and optimization by low-rank adjustments: an extremely compact mode separator," Opt. Lett. 30, 141-143 (2005).
    [CrossRef] [PubMed]
  22. Y. Jiao, S. F. Mingaleev, M. Schillinger, D. Miller, S. Fan, and K. Busch, "Wannier basis design and optimization of a photonic crystal waveguide crossing," IEEE Photon. Technol. Lett. 17, 1875-1877 (2005).
    [CrossRef]
  23. K. Jinguji, N. Takato, Y. Hida, T. Kitoh, and M. Kawachi, "Two-port optical wavelength circuits composed of cascaded Mach-Zehnder interferometers with point-symmetrical configurations," J. Lightwave Technol. 14, 2301-2310 (1996).
    [CrossRef]
  24. M. Cherchi, "Design scheme for Mach-Zehnder interferometric coarse wavelength division multiplexing splitters and combiners," J. Opt. Soc. Am. B 23, 1752-1756 (2006).
    [CrossRef]

2007

2006

M. Cherchi, "Design scheme for Mach-Zehnder interferometric coarse wavelength division multiplexing splitters and combiners," J. Opt. Soc. Am. B 23, 1752-1756 (2006).
[CrossRef]

F. Intonti, S. Vignolini, V. Türck, M. Colocci, P. Bettotti, L. Pavesi, S. L. Schweizer, R. Wehrspohn, and D. Wiersma, "Rewritable photonic circuits," Appl. Phys. Lett. 89, 211117 (2006).
[CrossRef]

2005

Y. Jiao, S. F. Mingaleev, M. Schillinger, D. Miller, S. Fan, and K. Busch, "Wannier basis design and optimization of a photonic crystal waveguide crossing," IEEE Photon. Technol. Lett. 17, 1875-1877 (2005).
[CrossRef]

K. Guven and E. Ozbay, "Coupling and phase analysis of cavity structures in two-dimensional photonic crystals," Phys. Rev. B 71, 085108 (2005).
[CrossRef]

T. Yasuda, Y. Tsuji, and M. Koshiba, "Tunable light propagation in photonic crystal coupler filled with liquid crystal," IEEE Photon. Technol. Lett. 17, 55-57 (2005).
[CrossRef]

Y. Jiao, S. Fan, and D. Miller, "Demonstration of systematic photonic crystal device design and optimization by low-rank adjustments: an extremely compact mode separator," Opt. Lett. 30, 141-143 (2005).
[CrossRef] [PubMed]

A. A. Green, E. Istrate, and E. H. Sargent, "Efficient design and optimization of photonic crystal waveguides and couplers: the interface diffraction method," Opt. Express 13, 7304-7318 (2005).
[CrossRef] [PubMed]

2004

2003

S. F. Mingaleev and K. Busch, "Scattering matrix approach to large-scale photonic crystal circuits," Opt. Lett. 28, 619-621 (2003).
[CrossRef] [PubMed]

K. Busch, S. F. Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann, "The Wannier function approach to photonic crystal circuits," J. Phys. Condens. Matter 15, R1233-1256 (2003).
[CrossRef]

H. Takeda and K. Yoshino, "Tunable light propagation in Y-shaped waveguides in two-dimensional photonic crystals utilizing liquid crystals as linear defects," Phys. Rev. B 67, 073106 (2003).
[CrossRef]

2002

A. Chutinan, M. Okano, and S. Noda, "Wider bandwidth with high transmission through waveguide bends in two-dimensional photonic crystal slabs," Appl. Phys. Lett. 80, 1698-1700 (2002).
[CrossRef]

M. Soljacic, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen, and J. D. Joannopoulos, "Photonic-crystal slow-light enhancement of nonlinear phase sensitivity," J. Opt. Soc. Am. B 19, 2052-2059 (2002).
[CrossRef]

2001

M. Koshiba, "Wavelength division multiplexing and demultiplexing with photonic crystal waveguide couplers," J. Lightwave Technol. 19, 1970-1975 (2001).
[CrossRef]

A. Birner, R. Wehrspohn, U. Gösele, and K. Busch, "Silicon-based photonic crystals," Adv. Mater. (Weinheim, Ger.) 13, 377-388 (2001).
[CrossRef]

1999

K. Busch and S. John, "Liquid crystal photonic band gap materials: the tunable electromagnetic vacuum," Phys. Rev. Lett. 83, 967-970 (1999).
[CrossRef]

K. Yoshino, Y. Shimoda, Y. Kawagishi, K. Nakayama, and M. Ozaki, "Temperature tuning of the stop band in transmission spectra of liquid-crystal infiltrated synthetic opal as tunable photonic crystal," Appl. Phys. Lett. 75, 932-934 (1999).
[CrossRef]

1998

K. Busch and S. John, "Photonic band gap formation in certain self-organizing systems," Phys. Rev. E 58, 3896-3908 (1998).
[CrossRef]

1996

K. Jinguji, N. Takato, Y. Hida, T. Kitoh, and M. Kawachi, "Two-port optical wavelength circuits composed of cascaded Mach-Zehnder interferometers with point-symmetrical configurations," J. Lightwave Technol. 14, 2301-2310 (1996).
[CrossRef]

A. Mekis, J. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Adv. Mater. (Weinheim, Ger.)

A. Birner, R. Wehrspohn, U. Gösele, and K. Busch, "Silicon-based photonic crystals," Adv. Mater. (Weinheim, Ger.) 13, 377-388 (2001).
[CrossRef]

Appl. Phys. Lett.

F. Intonti, S. Vignolini, V. Türck, M. Colocci, P. Bettotti, L. Pavesi, S. L. Schweizer, R. Wehrspohn, and D. Wiersma, "Rewritable photonic circuits," Appl. Phys. Lett. 89, 211117 (2006).
[CrossRef]

A. Chutinan, M. Okano, and S. Noda, "Wider bandwidth with high transmission through waveguide bends in two-dimensional photonic crystal slabs," Appl. Phys. Lett. 80, 1698-1700 (2002).
[CrossRef]

K. Yoshino, Y. Shimoda, Y. Kawagishi, K. Nakayama, and M. Ozaki, "Temperature tuning of the stop band in transmission spectra of liquid-crystal infiltrated synthetic opal as tunable photonic crystal," Appl. Phys. Lett. 75, 932-934 (1999).
[CrossRef]

IEEE Photon. Technol. Lett.

T. Yasuda, Y. Tsuji, and M. Koshiba, "Tunable light propagation in photonic crystal coupler filled with liquid crystal," IEEE Photon. Technol. Lett. 17, 55-57 (2005).
[CrossRef]

Y. Jiao, S. F. Mingaleev, M. Schillinger, D. Miller, S. Fan, and K. Busch, "Wannier basis design and optimization of a photonic crystal waveguide crossing," IEEE Photon. Technol. Lett. 17, 1875-1877 (2005).
[CrossRef]

J. Lightwave Technol.

K. Jinguji, N. Takato, Y. Hida, T. Kitoh, and M. Kawachi, "Two-port optical wavelength circuits composed of cascaded Mach-Zehnder interferometers with point-symmetrical configurations," J. Lightwave Technol. 14, 2301-2310 (1996).
[CrossRef]

M. Koshiba, "Wavelength division multiplexing and demultiplexing with photonic crystal waveguide couplers," J. Lightwave Technol. 19, 1970-1975 (2001).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. Condens. Matter

K. Busch, S. F. Mingaleev, A. Garcia-Martin, M. Schillinger, and D. Hermann, "The Wannier function approach to photonic crystal circuits," J. Phys. Condens. Matter 15, R1233-1256 (2003).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rep.

K. Busch, G. von Freymann, S. Linden, S. F. Mingaleev, L. Tkeshelashvili, and M. Wegener, "Periodic nanostructures for photonics," Phys. Rep. 444, 101-202 (2007).
[CrossRef]

Phys. Rev. B

K. Guven and E. Ozbay, "Coupling and phase analysis of cavity structures in two-dimensional photonic crystals," Phys. Rev. B 71, 085108 (2005).
[CrossRef]

H. Takeda and K. Yoshino, "Tunable light propagation in Y-shaped waveguides in two-dimensional photonic crystals utilizing liquid crystals as linear defects," Phys. Rev. B 67, 073106 (2003).
[CrossRef]

Phys. Rev. E

K. Busch and S. John, "Photonic band gap formation in certain self-organizing systems," Phys. Rev. E 58, 3896-3908 (1998).
[CrossRef]

Phys. Rev. Lett.

K. Busch and S. John, "Liquid crystal photonic band gap materials: the tunable electromagnetic vacuum," Phys. Rev. Lett. 83, 967-970 (1999).
[CrossRef]

A. Mekis, J. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Other

H. Brand, Schaltungslehre Linearer Mikrowellennetze (Hirzel Verlag, 1970).

M. Schillinger, "Maximally localized photonic Wannier functions for the highly efficient description of integrated photonic crystal circuits," Ph.D. dissertation (University of Karlsruhe, 2006).

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Figures (13)

Fig. 1
Fig. 1

Model system: 2D macroporous silicon PhC with pores of r = 0.45 a arranged on a triangular lattice. A large bandgap ( Δ ω ω c = 49 % ) exists for H-polarized light. Within the gap, the waveguide dispersion is shown for a line of pores filled with a low-index material of refractive index n. The dispersion calculated with the Wannier-function technique (WF; symbols) is compared with results from plane-wave based supercell calculations (PWE; curves). We are mostly interested in the single-mode frequency range below the cutoff frequency of the almost flat dispersion branch, i.e., a λ = [ 0.357 , 0.407 ] for n = 1.7 . The corresponding propagating waveguide mode exhibits even symmetry (lower right panel).

Fig. 2
Fig. 2

Principle of the S-matrix formalism: a large-scale circuit is divided into basic FEs that are each described by their individual S-matrices. Typically, the same FEs appear multiple times in different orientations, and their S-matrices can be reused. The complex MZI circuit depicted above is composed of only two basic FEs.

Fig. 3
Fig. 3

Maximally localized photonic Wannier functions related to the 26 lowest bands of the model system. Each Wannier function is labeled by its band index n. The Wannier functions associated with bands 4–9, 10–15, and 18–23 can, respectively, be obtained from the three representative functions shown in the center of the figure through five successive 60° rotations. For instance, the set of Wannier functions 4–9 forms a sixfold star, where each spike of the star is identical to the (suitably rotated) Wannier function shown. For the parameters of the model system, we refer to Fig. 1.

Fig. 4
Fig. 4

Division of a given FE into two regions, Ω D and Ω W . In the waveguiding region, Ω W (shaded), the electromagnetic field can be described by incoming and outgoing guided modes and, therefore, the Wannier coefficients H β can be replaced by guided mode amplitudes, a l (incoming) and b l (outgoing), in that region.

Fig. 5
Fig. 5

Illustration of how to connect two individual FEs with S-matrices S and S to form a more complex FE. The interior ports 4 and 2 are eliminated, while the exterior ports 1, 2, 3, 1 , and 3 form the new ports of the combined FE. The length of the waveguide connecting the two FEs appears as a phase factor in the switching matrix T. In addition, the switching matrix accounts for the potential mode mismatch in the case of when waveguides at ports 4 and 2 are different.

Fig. 6
Fig. 6

Reflectance of waveguide bends in the model system. The FE was optimized with respect to minimal total reflectance in the frequency range a λ = [ 0.370 , 0.390 ] . A set of 3 × 3 pores was chosen, into which different low-index materials with refractive indices of n = 1.55 , 1.60, 1.65, and 1.70 were allowed to be infiltrated, indicated by numbers from 1 to 4 in the lower right panel, which depicts the optimized design. Only symmetric configurations with respect to a mirror reflection on the plane bisecting the angle enclosed by the bend were scanned. Consequently, the reflectance spectra of ( 1 + 4 ) 6 = 15,625 configurations were computed. The left panel shows the reflectance of the simple (dashed curve) and the optimized design (solid curve). The two vertical dashed lines mark the frequencies of minimal reflectance of the optimized design ( a λ = 0.373 and 0.384). These lines are repeated in all the subsequent spectra in this paper.

Fig. 7
Fig. 7

Coupler end point design (right panel) and its spectral performance (left panel). The FE consists of two single-mode waveguide ports (1 and 2) on the left and one dual-mode waveguide port (3) on the right. Undesirable reflectances have been reduced by utilizing the optimized design of the waveguide bend (see Fig. 6).

Fig. 8
Fig. 8

Directional coupler built into a PhC. It is decomposed into two coupler end points with a variable length waveguide between them (shaded region). Only the S-matrix of a single coupler end point and the (simple) switching matrix of the connecting dual-mode waveguide have to be computed for all frequencies of interest to allow design studies with couplers of any length.

Fig. 9
Fig. 9

Length dependence of the reflectances and transmittances of the DC shown in Fig. 8 for a fixed frequency a λ = 0.373 .

Fig. 10
Fig. 10

Reflectance/transmittance spectrum of a fixed-length DC as shown in Fig. 8 with L = 97 a . For this length, the DC is in the bar state for a λ = 0.373 and in the cross state for a λ = 0.384 .

Fig. 11
Fig. 11

MZI consisting of two DCs of lengths L 1 and L 2 connected via two separate arms. In one arm a delay line of length L D is realized by filling the pores with a different polymer with n = 1.65 .

Fig. 12
Fig. 12

Spectrum of the MZI depicted in Fig. 11 with L 1 = L 2 = 49 a without a delay line (both arms have the same optical length). This balanced MZI acts as a DC of length L 1 + L 2 = 98 a . Compare with Fig. 10.

Fig. 13
Fig. 13

Spectrum of the MZI depicted in Fig. 11 with L 1 = L 2 = 49 a and a delay line of variable length L D = 17 a (solid curves), 10 a (dashed curves) and 6 a (dotted curves). For L D = 17 a , the relative phase shift approximately corresponds to Δ ϕ = 0 at a λ = 0.373 and Δ ϕ = π at a λ = 0.384 . For shorter delay lines, it is still zero at a λ = 0.373 , but takes some value 0 < Δ ϕ < π at a λ = 0.384 . Compare with the balanced MZI case ( Δ ϕ = 0 at all frequencies) in Fig. 12.

Equations (4)

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[ 2 1 ϵ ( r ) 2 + ω 2 c 2 ] H z ( r ) = 0 ,
β M α β ( ω ) H β = 0 .
β Ω D M α β H β + β Ω W M α β l = 1 N l G β l out b l = β Ω W M α β l = 1 N l G β l in a l .
S combined = S e,e + S e,i ( T S i,i ) 1 S i,e .

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