Abstract

General properties of N × M self-images in a strongly confined rectangular waveguide are given. Analytical formulas are derived for the positions, amplitudes, and phases of the N × M images at the end of multimode interference section. The formulas are verified with numerical simulation of a three-dimensional semivectorial beam propagation method.

© 2003 Optical Society of America

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References

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  1. L. B. Soldano, E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13, 615–627 (1995).
    [CrossRef]
  2. L. B. Soldano, F. B. Veerman, M. K. Smit, B. H. Verbeek, A. H. Dubost, E. C. M. Pennings, “Planar monomode optical couplers based on multimode interference effects,” J. Lightwave Technol. 10, 1843–1850 (1992).
    [CrossRef]
  3. M. Bachmann, P. A. Besse, H. Melchior, “General self-imaging properties in N × N multimode interference couplers,” Appl. Opt. 33, 3905–3911 (1994).
    [CrossRef] [PubMed]
  4. M. Bachmann, P. A. Besse, H. Melchior, “Overlapping-image multimode interference couplers with a reduced number of self-images for uniform and nonuniform power splitting,” Appl. Opt. 34, 6898–6910 (1995).
    [CrossRef] [PubMed]
  5. E. Voges, R. Ulrich, “Self-imaging by phase coincidences in rectangular dielectric waveguides,” presented at the 6th European Microwave Conference, Rome, Italy, 14–17 September 1976.
  6. S. L. Tsao, P. C. Peng, “Design of two-dimensional 1 × 16 and 1 × 32 array waveguide optical power splitters,” in Optoelectronic Materials and Devices II, Y.-K. Su, P. Bhattacharya, eds., Proc. SPIE4078, 373–382 (2000).
    [CrossRef]
  7. S. M. Garner, S. S. Lee, V. Chuyanov, A. Chen, A. Yacoubian, W. H. Steier, L. R. Dalton, “Three-dimensional integrated optics using polymers,” IEEE J. Quantum Electron. 35, 1146–1155 (1999).
    [CrossRef]
  8. Y. L. Hsueh, M. C. Yang, H. C. Chang, “Three-dimensional noniterative full-vectorial beam propagation method based on alternating direction implicit method,” J. Lightwave Technol. 17, 2389–2397 (1999).
    [CrossRef]
  9. W. P. Huang, C. L. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett. 8, 649–651 (1996).
    [CrossRef]

1999 (2)

S. M. Garner, S. S. Lee, V. Chuyanov, A. Chen, A. Yacoubian, W. H. Steier, L. R. Dalton, “Three-dimensional integrated optics using polymers,” IEEE J. Quantum Electron. 35, 1146–1155 (1999).
[CrossRef]

Y. L. Hsueh, M. C. Yang, H. C. Chang, “Three-dimensional noniterative full-vectorial beam propagation method based on alternating direction implicit method,” J. Lightwave Technol. 17, 2389–2397 (1999).
[CrossRef]

1996 (1)

W. P. Huang, C. L. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett. 8, 649–651 (1996).
[CrossRef]

1995 (2)

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

M. Bachmann, P. A. Besse, H. Melchior, “Overlapping-image multimode interference couplers with a reduced number of self-images for uniform and nonuniform power splitting,” Appl. Opt. 34, 6898–6910 (1995).
[CrossRef] [PubMed]

1994 (1)

1992 (1)

L. B. Soldano, F. B. Veerman, M. K. Smit, B. H. Verbeek, A. H. Dubost, E. C. M. Pennings, “Planar monomode optical couplers based on multimode interference effects,” J. Lightwave Technol. 10, 1843–1850 (1992).
[CrossRef]

Bachmann, M.

Besse, P. A.

Chang, H. C.

Chen, A.

S. M. Garner, S. S. Lee, V. Chuyanov, A. Chen, A. Yacoubian, W. H. Steier, L. R. Dalton, “Three-dimensional integrated optics using polymers,” IEEE J. Quantum Electron. 35, 1146–1155 (1999).
[CrossRef]

Chuyanov, V.

S. M. Garner, S. S. Lee, V. Chuyanov, A. Chen, A. Yacoubian, W. H. Steier, L. R. Dalton, “Three-dimensional integrated optics using polymers,” IEEE J. Quantum Electron. 35, 1146–1155 (1999).
[CrossRef]

Dalton, L. R.

S. M. Garner, S. S. Lee, V. Chuyanov, A. Chen, A. Yacoubian, W. H. Steier, L. R. Dalton, “Three-dimensional integrated optics using polymers,” IEEE J. Quantum Electron. 35, 1146–1155 (1999).
[CrossRef]

Dubost, A. H.

L. B. Soldano, F. B. Veerman, M. K. Smit, B. H. Verbeek, A. H. Dubost, E. C. M. Pennings, “Planar monomode optical couplers based on multimode interference effects,” J. Lightwave Technol. 10, 1843–1850 (1992).
[CrossRef]

Garner, S. M.

S. M. Garner, S. S. Lee, V. Chuyanov, A. Chen, A. Yacoubian, W. H. Steier, L. R. Dalton, “Three-dimensional integrated optics using polymers,” IEEE J. Quantum Electron. 35, 1146–1155 (1999).
[CrossRef]

Hsueh, Y. L.

Huang, W. P.

W. P. Huang, C. L. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett. 8, 649–651 (1996).
[CrossRef]

Lee, S. S.

S. M. Garner, S. S. Lee, V. Chuyanov, A. Chen, A. Yacoubian, W. H. Steier, L. R. Dalton, “Three-dimensional integrated optics using polymers,” IEEE J. Quantum Electron. 35, 1146–1155 (1999).
[CrossRef]

Lui, W.

W. P. Huang, C. L. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett. 8, 649–651 (1996).
[CrossRef]

Melchior, H.

Peng, P. C.

S. L. Tsao, P. C. Peng, “Design of two-dimensional 1 × 16 and 1 × 32 array waveguide optical power splitters,” in Optoelectronic Materials and Devices II, Y.-K. Su, P. Bhattacharya, eds., Proc. SPIE4078, 373–382 (2000).
[CrossRef]

Pennings, E. C. M.

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

L. B. Soldano, F. B. Veerman, M. K. Smit, B. H. Verbeek, A. H. Dubost, E. C. M. Pennings, “Planar monomode optical couplers based on multimode interference effects,” J. Lightwave Technol. 10, 1843–1850 (1992).
[CrossRef]

Smit, M. K.

L. B. Soldano, F. B. Veerman, M. K. Smit, B. H. Verbeek, A. H. Dubost, E. C. M. Pennings, “Planar monomode optical couplers based on multimode interference effects,” J. Lightwave Technol. 10, 1843–1850 (1992).
[CrossRef]

Soldano, L. B.

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

L. B. Soldano, F. B. Veerman, M. K. Smit, B. H. Verbeek, A. H. Dubost, E. C. M. Pennings, “Planar monomode optical couplers based on multimode interference effects,” J. Lightwave Technol. 10, 1843–1850 (1992).
[CrossRef]

Steier, W. H.

S. M. Garner, S. S. Lee, V. Chuyanov, A. Chen, A. Yacoubian, W. H. Steier, L. R. Dalton, “Three-dimensional integrated optics using polymers,” IEEE J. Quantum Electron. 35, 1146–1155 (1999).
[CrossRef]

Tsao, S. L.

S. L. Tsao, P. C. Peng, “Design of two-dimensional 1 × 16 and 1 × 32 array waveguide optical power splitters,” in Optoelectronic Materials and Devices II, Y.-K. Su, P. Bhattacharya, eds., Proc. SPIE4078, 373–382 (2000).
[CrossRef]

Ulrich, R.

E. Voges, R. Ulrich, “Self-imaging by phase coincidences in rectangular dielectric waveguides,” presented at the 6th European Microwave Conference, Rome, Italy, 14–17 September 1976.

Veerman, F. B.

L. B. Soldano, F. B. Veerman, M. K. Smit, B. H. Verbeek, A. H. Dubost, E. C. M. Pennings, “Planar monomode optical couplers based on multimode interference effects,” J. Lightwave Technol. 10, 1843–1850 (1992).
[CrossRef]

Verbeek, B. H.

L. B. Soldano, F. B. Veerman, M. K. Smit, B. H. Verbeek, A. H. Dubost, E. C. M. Pennings, “Planar monomode optical couplers based on multimode interference effects,” J. Lightwave Technol. 10, 1843–1850 (1992).
[CrossRef]

Voges, E.

E. Voges, R. Ulrich, “Self-imaging by phase coincidences in rectangular dielectric waveguides,” presented at the 6th European Microwave Conference, Rome, Italy, 14–17 September 1976.

Xu, C. L.

W. P. Huang, C. L. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett. 8, 649–651 (1996).
[CrossRef]

Yacoubian, A.

S. M. Garner, S. S. Lee, V. Chuyanov, A. Chen, A. Yacoubian, W. H. Steier, L. R. Dalton, “Three-dimensional integrated optics using polymers,” IEEE J. Quantum Electron. 35, 1146–1155 (1999).
[CrossRef]

Yang, M. C.

Yokoyama, K.

W. P. Huang, C. L. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett. 8, 649–651 (1996).
[CrossRef]

Appl. Opt. (2)

IEEE J. Quantum Electron. (1)

S. M. Garner, S. S. Lee, V. Chuyanov, A. Chen, A. Yacoubian, W. H. Steier, L. R. Dalton, “Three-dimensional integrated optics using polymers,” IEEE J. Quantum Electron. 35, 1146–1155 (1999).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

W. P. Huang, C. L. Xu, W. Lui, K. Yokoyama, “The perfectly matched layer (PML) boundary condition for the beam propagation method,” IEEE Photon. Technol. Lett. 8, 649–651 (1996).
[CrossRef]

J. Lightwave Technol. (3)

Y. L. Hsueh, M. C. Yang, H. C. Chang, “Three-dimensional noniterative full-vectorial beam propagation method based on alternating direction implicit method,” J. Lightwave Technol. 17, 2389–2397 (1999).
[CrossRef]

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

L. B. Soldano, F. B. Veerman, M. K. Smit, B. H. Verbeek, A. H. Dubost, E. C. M. Pennings, “Planar monomode optical couplers based on multimode interference effects,” J. Lightwave Technol. 10, 1843–1850 (1992).
[CrossRef]

Other (2)

E. Voges, R. Ulrich, “Self-imaging by phase coincidences in rectangular dielectric waveguides,” presented at the 6th European Microwave Conference, Rome, Italy, 14–17 September 1976.

S. L. Tsao, P. C. Peng, “Design of two-dimensional 1 × 16 and 1 × 32 array waveguide optical power splitters,” in Optoelectronic Materials and Devices II, Y.-K. Su, P. Bhattacharya, eds., Proc. SPIE4078, 373–382 (2000).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Schematic diagram of a rectangular waveguide. The refractive indices of the core and the cladding are nf and nc, respectively.

Fig. 2
Fig. 2

Field distribution at the end of the 3 × 2 MMI section calculated by a BPM simulation with a zero boundary condition. (a) three-dimensional plot, (b) contour plot. The plus signs indicate the self-imaging centers predicted by our formulas.

Fig. 3
Fig. 3

Field distribution at the end of the designed silicon MMI section (with 3 × 2 self-images) calculated by a BPM simulation with a PML boundary treatment.

Tables (1)

Tables Icon

Table 1 Relative Phases Predicted by Our Analytical Formulas and the Corresponding Values Calculated by the BPM with Zero Boundary Condition

Equations (20)

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Uk,lx, y=U0 sinπkxWx+θxsinπlyWy+θy,
2+k02nf2ϕ=0 ϕx, y, 0=fx, yfor 0<x<Wx, 0<y<Wy,
βk,l=k02nf2-πkWx2-πlWy21/2k0nf-πλ04nfkWx2+lWy2.
ϕx, y, 0=k,l=1 ck,l sinπkxWxsinπlyWy=k,l=- ck,l expjπkxWx+lyWy,
-ck,l=ck,-l=c-k,l=-c-k,-l=14 ck,l.
ϕx, y, 0=fx, y-f-x, y-fx, y+f-x,-y.
ϕx, y, z=k,l=- ck,l expjπkxWx+lyWyexp-jβk,lz.
ϕx, y, L=k,l=-ck,l expjπkxWx+lyWy×expj πλ04nfk2-1Wx2+l2-1Wy2L=k,l=-ck,l expjπkxWx+lyWy×exp jπk2-1LLx+l2-1LLy,
Lx=4nfλ0 Wx2, Ly=4nfλ0 Wy2.
LLx=nN  and  LLy=mM,
ϕx, y, L=k,l=-ck,l expjπkxWx+lyWy×expjπ nNk2-1×expjπ mMl2-1.
expjπ nNk2-1=1C1q=0N-1exp-jπ kxqWx-αqexpjπ mMl2-1=1C2p=0M-1exp-jπ lypWy-γp,
xq=2q-NnN Wx,
yp=2p-MmM Wy,
αq=qN-qnN π,
γp=pM-pmM π,
ϕx, y, L=1C1C2k,l=-ck,l expjπkxWx+lyWy×q=0N-1p=0M-1exp-jπkxqWx+lypWyexpjαq+γp=1C1C2q=0N-1p=0M-1expjαq+γp×k,l=- ck,l expjπx-xqWx+l y-ypWy=1C1C2q=0N-1p=0M-1expjαq+γp×ϕx-xq, y-yp, 0.
LxLy=mNMn,
L=nN Lx.
WξeffWξ+λπncnf2σnf2-nc2,

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