Abstract

The self-imaging property in multimode waveguides is related to the waveguide widths and lengths. By engineering the diffraction properties of multimode waveguides, we propose a scheme to design devices with reduced self-imaging lengths at a fixed width. Using computer-generated planar holograms, the coupling coefficients between the guided modes are adjusted to generate the desired diffraction properties. Calculations based on the coupled-mode theory are presented. Devices are designed based on a silicon-on-insulator (SOI) platform. Beam propagation simulations are used to verify the coupled-mode theory analysis.

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References

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2009 (1)

2008 (2)

2007 (1)

2006 (1)

2005 (2)

2004 (1)

2003 (1)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[CrossRef] [PubMed]

2001 (1)

1999 (1)

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical bloch oscillations,” Phys. Rev. Lett. 83(23), 4756–4759 (1999).
[CrossRef]

1997 (1)

D. S. Levy, Y. M. Li, R. Scarmozzino, and R. M. Osgood, Jr., “A multimode interference-based variable power splitter in GaAs-AlGaAs,” IEEE Photon. Technol. Lett. 9(10), 1373–1375 (1997).
[CrossRef]

1996 (1)

P. A. Besse, E. Gini, M. Bachmann, and H. Melchior, “New 2×2 and 1×3 multimode interference couplers with free selection of power splitting ratios,” J. Lightwave Technol. 14(10), 2286–2293 (1996).
[CrossRef]

1995 (2)

L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 (1995).
[CrossRef]

M. Bachmann, P. A. Besse, and H. Melchior, “Overlapping-image multimode interference couplers with a reduced number of self-images for uniform and nonuniform power splitting,” Appl. Opt. 34(30), 6898–6910 (1995).
[CrossRef] [PubMed]

1994 (1)

P. A. Besse, M. Bachmann, H. Melchior, L. B. Soldano, and M. K. Smit, “Optical bandwidth and fabrication tolerances of multimode interference couplers,” J. Lightwave Technol. 12(6), 1004–1009 (1994).
[CrossRef]

1993 (1)

1992 (1)

1975 (1)

R. Ulrich, “Image formation by phase coincidence in optical waveguides,” Opt. Commun. 13(3), 259–264 (1975).
[CrossRef]

1973 (2)

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9(9), 919–933 (1973).
[CrossRef]

O. Bryngdahl, “Image formation using self-imaging techniques,” J. Opt. Soc. Am. 63(4), 416–419 (1973).
[CrossRef]

Aitchison, J. S.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical bloch oscillations,” Phys. Rev. Lett. 83(23), 4756–4759 (1999).
[CrossRef]

Bachmann, M.

P. A. Besse, E. Gini, M. Bachmann, and H. Melchior, “New 2×2 and 1×3 multimode interference couplers with free selection of power splitting ratios,” J. Lightwave Technol. 14(10), 2286–2293 (1996).
[CrossRef]

M. Bachmann, P. A. Besse, and H. Melchior, “Overlapping-image multimode interference couplers with a reduced number of self-images for uniform and nonuniform power splitting,” Appl. Opt. 34(30), 6898–6910 (1995).
[CrossRef] [PubMed]

P. A. Besse, M. Bachmann, H. Melchior, L. B. Soldano, and M. K. Smit, “Optical bandwidth and fabrication tolerances of multimode interference couplers,” J. Lightwave Technol. 12(6), 1004–1009 (1994).
[CrossRef]

Besse, P. A.

P. A. Besse, E. Gini, M. Bachmann, and H. Melchior, “New 2×2 and 1×3 multimode interference couplers with free selection of power splitting ratios,” J. Lightwave Technol. 14(10), 2286–2293 (1996).
[CrossRef]

M. Bachmann, P. A. Besse, and H. Melchior, “Overlapping-image multimode interference couplers with a reduced number of self-images for uniform and nonuniform power splitting,” Appl. Opt. 34(30), 6898–6910 (1995).
[CrossRef] [PubMed]

P. A. Besse, M. Bachmann, H. Melchior, L. B. Soldano, and M. K. Smit, “Optical bandwidth and fabrication tolerances of multimode interference couplers,” J. Lightwave Technol. 12(6), 1004–1009 (1994).
[CrossRef]

Bryngdahl, O.

Choi, S.

Christodoulides, D. N.

N. K. Efremidis and D. N. Christodoulides, “Revivals in engineered waveguide arrays,” Opt. Commun. 246(4-6), 345–356 (2005).
[CrossRef]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[CrossRef] [PubMed]

Efremidis, N. K.

N. K. Efremidis and D. N. Christodoulides, “Revivals in engineered waveguide arrays,” Opt. Commun. 246(4-6), 345–356 (2005).
[CrossRef]

Eisenberg, H. S.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical bloch oscillations,” Phys. Rev. Lett. 83(23), 4756–4759 (1999).
[CrossRef]

Fallahkhair, A. B.

Feng, D. J. Y.

Fuentes-Hernandez, C.

Gini, E.

P. A. Besse, E. Gini, M. Bachmann, and H. Melchior, “New 2×2 and 1×3 multimode interference couplers with free selection of power splitting ratios,” J. Lightwave Technol. 14(10), 2286–2293 (1996).
[CrossRef]

Goldhar, J.

Gordon, R.

Hadley, G. R.

Hashimoto, T.

Kim, Y.

Kippelen, B.

Kohtoku, M.

Lay, T. S.

Lederer, F.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[CrossRef] [PubMed]

Levy, D. S.

D. S. Levy, Y. M. Li, R. Scarmozzino, and R. M. Osgood, Jr., “A multimode interference-based variable power splitter in GaAs-AlGaAs,” IEEE Photon. Technol. Lett. 9(10), 1373–1375 (1997).
[CrossRef]

Li, K. S.

Li, Y. M.

D. S. Levy, Y. M. Li, R. Scarmozzino, and R. M. Osgood, Jr., “A multimode interference-based variable power splitter in GaAs-AlGaAs,” IEEE Photon. Technol. Lett. 9(10), 1373–1375 (1997).
[CrossRef]

Melchior, H.

P. A. Besse, E. Gini, M. Bachmann, and H. Melchior, “New 2×2 and 1×3 multimode interference couplers with free selection of power splitting ratios,” J. Lightwave Technol. 14(10), 2286–2293 (1996).
[CrossRef]

M. Bachmann, P. A. Besse, and H. Melchior, “Overlapping-image multimode interference couplers with a reduced number of self-images for uniform and nonuniform power splitting,” Appl. Opt. 34(30), 6898–6910 (1995).
[CrossRef] [PubMed]

P. A. Besse, M. Bachmann, H. Melchior, L. B. Soldano, and M. K. Smit, “Optical bandwidth and fabrication tolerances of multimode interference couplers,” J. Lightwave Technol. 12(6), 1004–1009 (1994).
[CrossRef]

Mendlovic, D.

Morandotti, R.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical bloch oscillations,” Phys. Rev. Lett. 83(23), 4756–4759 (1999).
[CrossRef]

Mossberg, T. W.

Murphy, T. E.

Ogawa, I.

Osgood, Jr., R. M.

D. S. Levy, Y. M. Li, R. Scarmozzino, and R. M. Osgood, Jr., “A multimode interference-based variable power splitter in GaAs-AlGaAs,” IEEE Photon. Technol. Lett. 9(10), 1373–1375 (1997).
[CrossRef]

Owens, D.

Ozaktas, H. M.

Pennings, E. C. M.

L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 (1995).
[CrossRef]

Peschel, U.

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical bloch oscillations,” Phys. Rev. Lett. 83(23), 4756–4759 (1999).
[CrossRef]

Richardson, C. J. K.

Saida, T.

Scarmozzino, R.

D. S. Levy, Y. M. Li, R. Scarmozzino, and R. M. Osgood, Jr., “A multimode interference-based variable power splitter in GaAs-AlGaAs,” IEEE Photon. Technol. Lett. 9(10), 1373–1375 (1997).
[CrossRef]

Shibata, T.

Silberberg, Y.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[CrossRef] [PubMed]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical bloch oscillations,” Phys. Rev. Lett. 83(23), 4756–4759 (1999).
[CrossRef]

Smit, M. K.

P. A. Besse, M. Bachmann, H. Melchior, L. B. Soldano, and M. K. Smit, “Optical bandwidth and fabrication tolerances of multimode interference couplers,” J. Lightwave Technol. 12(6), 1004–1009 (1994).
[CrossRef]

Soldano, L. B.

L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13(4), 615–627 (1995).
[CrossRef]

P. A. Besse, M. Bachmann, H. Melchior, L. B. Soldano, and M. K. Smit, “Optical bandwidth and fabrication tolerances of multimode interference couplers,” J. Lightwave Technol. 12(6), 1004–1009 (1994).
[CrossRef]

Takahashi, H.

Tseng, S.-Y.

Ulrich, R.

R. Ulrich, “Image formation by phase coincidence in optical waveguides,” Opt. Commun. 13(3), 259–264 (1975).
[CrossRef]

Yariv, A.

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9(9), 919–933 (1973).
[CrossRef]

Appl. Opt. (2)

IEEE J. Quantum Electron. (1)

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quantum Electron. 9(9), 919–933 (1973).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

D. S. Levy, Y. M. Li, R. Scarmozzino, and R. M. Osgood, Jr., “A multimode interference-based variable power splitter in GaAs-AlGaAs,” IEEE Photon. Technol. Lett. 9(10), 1373–1375 (1997).
[CrossRef]

J. Lightwave Technol. (4)

P. A. Besse, E. Gini, M. Bachmann, and H. Melchior, “New 2×2 and 1×3 multimode interference couplers with free selection of power splitting ratios,” J. Lightwave Technol. 14(10), 2286–2293 (1996).
[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(4), 615–627 (1995).
[CrossRef]

P. A. Besse, M. Bachmann, H. Melchior, L. B. Soldano, and M. K. Smit, “Optical bandwidth and fabrication tolerances of multimode interference couplers,” J. Lightwave Technol. 12(6), 1004–1009 (1994).
[CrossRef]

A. B. Fallahkhair, K. S. Li, and T. E. Murphy, “Vector finite difference modesolver for anisotropic dielectric waveguides,” J. Lightwave Technol. 26(11), 1423–1431 (2008).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Nature (1)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
[CrossRef] [PubMed]

Opt. Commun. (2)

R. Ulrich, “Image formation by phase coincidence in optical waveguides,” Opt. Commun. 13(3), 259–264 (1975).
[CrossRef]

N. K. Efremidis and D. N. Christodoulides, “Revivals in engineered waveguide arrays,” Opt. Commun. 246(4-6), 345–356 (2005).
[CrossRef]

Opt. Express (2)

Opt. Lett. (5)

Phys. Rev. Lett. (1)

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical bloch oscillations,” Phys. Rev. Lett. 83(23), 4756–4759 (1999).
[CrossRef]

Other (3)

K. Okamoto, Fundamentals of Optical Waveguides. (Academic, Burlington, MA, 2006).

K. Kawano, and T. Kitoh, Introduction to Optical Waveguide Analysis (Wiley, New York, 2001).

M. T. Chu, and G. H. Golub, Inverse Eigenvalue Problems: Theory, Algorithms, and Applications, (Oxford, 2005).

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Figures (3)

Fig. 1
Fig. 1

Self-imaging (SI) lengths of multimode waveguides with different index modulation depths. Coupled-mode theory (CMT) calculations are obtained by solving (9) numerically. WA-BPM simulations are obtained using the finite difference algorithm in [23]. Uniform index change indicates the SI lengths when Δn is applied uniformly throughout the multimode waveguide.

Fig. 2
Fig. 2

Calculated CGPH for self-imaging (SI) length engineering. The pattern is obtained using the WA-BPM based code and the design procedure outlined in [17].

Fig. 3
Fig. 3

The intensity patterns of the light in multimode waveguides without and with the CGPH. (a) no CGPH: SI occurs at 1379μm. (b) with CGPH (index modulation depth Δn = 0.002): SI occurs at a shorter length of 1250μm.

Equations (14)

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E ( x , z ) = 1 M A m ( z ) ϕ m ( x ) exp ( j β m z ) ,
ϕ m ϕ n d x = 2 ω μ β n δ m n .
β m k 0 n r A m 2 = k 0 n r m 2 π λ 4 n r W e 2    ( m = 1 M ) ,
Δ n ( x , z ) = m n f m n ( x ) exp ( j 2 π z / Λ m n ) ,
d E m d z = j β m E m j κ m n E m n ,
κ m n = ω ε 0 n r 2 f m n ( x ) ϕ m ϕ n d x    ( m n ) .
d E d z = j K E ,
U m ( z ) = U m ( 0 ) exp ( j γ m z ) .
K U m = γ m U m .
γ m = β A m 2 .
L π = π β 1 β 2 4 n r W e 2 3 λ = π 3 A ,
L π = π γ 1 γ 2 π 3 A .
K = [ β 1 0 0 κ 1 0 β 2 κ 2 0 0 κ 2 β 3 0 κ 1 0 0 β 4 ] ,
γ 1 = β 1 + β 4 2 + 1 2 [ ( β 1 β 4 ) 2 + 4 | κ 1 | 2 ] 1 / 2 γ 2 = β 2 + β 3 2 + 1 2 [ ( β 2 β 3 ) 2 + 4 | κ 2 | 2 ] 1 / 2 γ 3 = β 2 + β 3 2 1 2 [ ( β 2 β 3 ) 2 + 4 | κ 2 | 2 ] 1 / 2 γ 4 = β 1 + β 4 2 1 2 [ ( β 1 β 4 ) 2 + 4 | κ 1 | 2 ] 1 / 2 .

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