Abstract

We report a theoretical and numerical study of self-imaging properties, including time domain and pulse spreading, caused by modal group-delay dispersion in generalized N×N multimode interference devices achieved by using a mode-propagation analysis and finite-difference time-domain method. It was found that the spatial self-imaging condition does not realize temporal self-imaging but lets waveforms separate whose shape depends on input position and input field distribution. Pulse spreading, which is sensitive to beam diameter, has a very large variation (420  fs) among input positions as well as rising to a very high 900  fs in response to a 21 fs and spatially Gaussian pulse for the conveniently smallest size with 10 channels.

© 2007 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  8. B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Rendina, and F. Coppinger, "Advances in silicon-on-insulator optoelectronics," IEEE J. Sel. Top. Quantum Electron. 4, 938-947 (1998).
    [CrossRef]
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    [CrossRef]
  11. G. W. Euliss, "Temporal characteristics and scaling considerations of multimode interference couplers," J. Lightwave Technol. 17, 1206-1210 (1999).
    [CrossRef]
  12. M. J. Yadlowsky and A. R. Mickelson, "Distributed loss and mode coupling and their effect on time-dependent propagation in multimode fibers," Appl. Opt. 32, 6664-6676 (1993).
    [CrossRef] [PubMed]
  13. L. Raddatz, I. H. White, D. G. Cunningham, and M. C. Nowell, "An experimental and theoretical study of the offset launch technique for the enhancement of the bandwidth of multimode fiber links," J. Lightwave Technol. 16, 324-331 (1998).
    [CrossRef]
  14. S. Fan and J. M. Kahn, "Principal modes in multimode waveguides," Opt. Lett. 30, 135-137 (2005).
    [CrossRef] [PubMed]
  15. R. Pimpinella and A. Brunsting, "Differential mode delay (DMD) for multimode fiber types and its relationship to measured performance," presented at the Optical Fiber Communication Conference, Anaheim, Calif., March 2005, paper NWF2.
  16. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Kluwer Academic, 2000).
  17. C. Vazquez, F. J. Mustieles, and F. Hernandez-Gil, "Three-dimensional method for simulation of multimode interference couplers," J. Lightwave Technol. 13, 2296-2299 (1995).
    [CrossRef]

2006

2005

1999

1998

L. Raddatz, I. H. White, D. G. Cunningham, and M. C. Nowell, "An experimental and theoretical study of the offset launch technique for the enhancement of the bandwidth of multimode fiber links," J. Lightwave Technol. 16, 324-331 (1998).
[CrossRef]

B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Rendina, and F. Coppinger, "Advances in silicon-on-insulator optoelectronics," IEEE J. Sel. Top. Quantum Electron. 4, 938-947 (1998).
[CrossRef]

1996

K.-C. Lin and W.-Y. Lee, "Guided-wave 1.3/1.55 µm wavelength division multiplexer based on multimode interference," Electron. Lett. 32, 1259-1261 (1996).
[CrossRef]

1995

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]

C. Vazquez, F. J. Mustieles, and F. Hernandez-Gil, "Three-dimensional method for simulation of multimode interference couplers," J. Lightwave Technol. 13, 2296-2299 (1995).
[CrossRef]

1994

1993

M. J. Yadlowsky and A. R. Mickelson, "Distributed loss and mode coupling and their effect on time-dependent propagation in multimode fibers," Appl. Opt. 32, 6664-6676 (1993).
[CrossRef] [PubMed]

A. Ferreras, F. Rodríguez, E. Gómez-Salas, J. L. de Miguel, and F. Hernández-Gil, "Useful formulas for multimode interference power splitter/combiner design," IEEE Photon. Technol. Lett. 5, 1224-1227 (1993).
[CrossRef]

1992

1981

D. C. Chang and E. F. Kuester, "A hybrid method for paraxial beam propagation in multimode optical waveguides," IEEE Trans. Microwave Theory Technol. MTT-29, 923-933 (1981).
[CrossRef]

1975

R. Ulrich, "Image formation by phase coincidences in optical waveguides," Opt. Commun. 13, 259-264 (1975).
[CrossRef]

1973

Bachmann, M.

Besse, P. A.

Brunsting, A.

R. Pimpinella and A. Brunsting, "Differential mode delay (DMD) for multimode fiber types and its relationship to measured performance," presented at the Optical Fiber Communication Conference, Anaheim, Calif., March 2005, paper NWF2.

Bryngdahl, O.

Chang, D. C.

D. C. Chang and E. F. Kuester, "A hybrid method for paraxial beam propagation in multimode optical waveguides," IEEE Trans. Microwave Theory Technol. MTT-29, 923-933 (1981).
[CrossRef]

Coppinger, F.

B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Rendina, and F. Coppinger, "Advances in silicon-on-insulator optoelectronics," IEEE J. Sel. Top. Quantum Electron. 4, 938-947 (1998).
[CrossRef]

Cunningham, D. G.

de Miguel, J. L.

A. Ferreras, F. Rodríguez, E. Gómez-Salas, J. L. de Miguel, and F. Hernández-Gil, "Useful formulas for multimode interference power splitter/combiner design," IEEE Photon. Technol. Lett. 5, 1224-1227 (1993).
[CrossRef]

Deveraux, R. W.

Euliss, G. W.

Fan, S.

Ferreras, A.

A. Ferreras, F. Rodríguez, E. Gómez-Salas, J. L. de Miguel, and F. Hernández-Gil, "Useful formulas for multimode interference power splitter/combiner design," IEEE Photon. Technol. Lett. 5, 1224-1227 (1993).
[CrossRef]

Gómez-Salas, E.

A. Ferreras, F. Rodríguez, E. Gómez-Salas, J. L. de Miguel, and F. Hernández-Gil, "Useful formulas for multimode interference power splitter/combiner design," IEEE Photon. Technol. Lett. 5, 1224-1227 (1993).
[CrossRef]

Hamada, H.

Heaton, J. M.

Hernandez-Gil, F.

C. Vazquez, F. J. Mustieles, and F. Hernandez-Gil, "Three-dimensional method for simulation of multimode interference couplers," J. Lightwave Technol. 13, 2296-2299 (1995).
[CrossRef]

Hernández-Gil, F.

A. Ferreras, F. Rodríguez, E. Gómez-Salas, J. L. de Miguel, and F. Hernández-Gil, "Useful formulas for multimode interference power splitter/combiner design," IEEE Photon. Technol. Lett. 5, 1224-1227 (1993).
[CrossRef]

Jalali, B.

B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Rendina, and F. Coppinger, "Advances in silicon-on-insulator optoelectronics," IEEE J. Sel. Top. Quantum Electron. 4, 938-947 (1998).
[CrossRef]

Jenkins, R. M.

Kahn, J. M.

Kuester, E. F.

D. C. Chang and E. F. Kuester, "A hybrid method for paraxial beam propagation in multimode optical waveguides," IEEE Trans. Microwave Theory Technol. MTT-29, 923-933 (1981).
[CrossRef]

Lee, W.-Y.

K.-C. Lin and W.-Y. Lee, "Guided-wave 1.3/1.55 µm wavelength division multiplexer based on multimode interference," Electron. Lett. 32, 1259-1261 (1996).
[CrossRef]

Lin, K.-C.

K.-C. Lin and W.-Y. Lee, "Guided-wave 1.3/1.55 µm wavelength division multiplexer based on multimode interference," Electron. Lett. 32, 1259-1261 (1996).
[CrossRef]

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Kluwer Academic, 2000).

Melchior, H.

Mickelson, A. R.

Mustieles, F. J.

C. Vazquez, F. J. Mustieles, and F. Hernandez-Gil, "Three-dimensional method for simulation of multimode interference couplers," J. Lightwave Technol. 13, 2296-2299 (1995).
[CrossRef]

Nowell, M. C.

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, 615-627 (1995).
[CrossRef]

Pimpinella, R.

R. Pimpinella and A. Brunsting, "Differential mode delay (DMD) for multimode fiber types and its relationship to measured performance," presented at the Optical Fiber Communication Conference, Anaheim, Calif., March 2005, paper NWF2.

Raddatz, L.

Rendina, I.

B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Rendina, and F. Coppinger, "Advances in silicon-on-insulator optoelectronics," IEEE J. Sel. Top. Quantum Electron. 4, 938-947 (1998).
[CrossRef]

Rodríguez, F.

A. Ferreras, F. Rodríguez, E. Gómez-Salas, J. L. de Miguel, and F. Hernández-Gil, "Useful formulas for multimode interference power splitter/combiner design," IEEE Photon. Technol. Lett. 5, 1224-1227 (1993).
[CrossRef]

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Kluwer Academic, 2000).

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, 615-627 (1995).
[CrossRef]

Ulrich, R.

R. Ulrich, "Image formation by phase coincidences in optical waveguides," Opt. Commun. 13, 259-264 (1975).
[CrossRef]

Vazquez, C.

C. Vazquez, F. J. Mustieles, and F. Hernandez-Gil, "Three-dimensional method for simulation of multimode interference couplers," J. Lightwave Technol. 13, 2296-2299 (1995).
[CrossRef]

White, I. H.

Yadlowsky, M. J.

Yegnanarayanan, S.

B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Rendina, and F. Coppinger, "Advances in silicon-on-insulator optoelectronics," IEEE J. Sel. Top. Quantum Electron. 4, 938-947 (1998).
[CrossRef]

Yoon, T.

B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Rendina, and F. Coppinger, "Advances in silicon-on-insulator optoelectronics," IEEE J. Sel. Top. Quantum Electron. 4, 938-947 (1998).
[CrossRef]

Yoshimoto, T.

B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Rendina, and F. Coppinger, "Advances in silicon-on-insulator optoelectronics," IEEE J. Sel. Top. Quantum Electron. 4, 938-947 (1998).
[CrossRef]

Appl. Opt.

Electron. Lett.

K.-C. Lin and W.-Y. Lee, "Guided-wave 1.3/1.55 µm wavelength division multiplexer based on multimode interference," Electron. Lett. 32, 1259-1261 (1996).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

B. Jalali, S. Yegnanarayanan, T. Yoon, T. Yoshimoto, I. Rendina, and F. Coppinger, "Advances in silicon-on-insulator optoelectronics," IEEE J. Sel. Top. Quantum Electron. 4, 938-947 (1998).
[CrossRef]

IEEE Photon. Technol. Lett.

A. Ferreras, F. Rodríguez, E. Gómez-Salas, J. L. de Miguel, and F. Hernández-Gil, "Useful formulas for multimode interference power splitter/combiner design," IEEE Photon. Technol. Lett. 5, 1224-1227 (1993).
[CrossRef]

IEEE Trans. Microwave Theory Technol.

D. C. Chang and E. F. Kuester, "A hybrid method for paraxial beam propagation in multimode optical waveguides," IEEE Trans. Microwave Theory Technol. MTT-29, 923-933 (1981).
[CrossRef]

J. Lightwave Technol.

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]

C. Vazquez, F. J. Mustieles, and F. Hernandez-Gil, "Three-dimensional method for simulation of multimode interference couplers," J. Lightwave Technol. 13, 2296-2299 (1995).
[CrossRef]

G. W. Euliss, "Temporal characteristics and scaling considerations of multimode interference couplers," J. Lightwave Technol. 17, 1206-1210 (1999).
[CrossRef]

L. Raddatz, I. H. White, D. G. Cunningham, and M. C. Nowell, "An experimental and theoretical study of the offset launch technique for the enhancement of the bandwidth of multimode fiber links," J. Lightwave Technol. 16, 324-331 (1998).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

Opt. Commun.

R. Ulrich, "Image formation by phase coincidences in optical waveguides," Opt. Commun. 13, 259-264 (1975).
[CrossRef]

Opt. Lett.

Other

R. Pimpinella and A. Brunsting, "Differential mode delay (DMD) for multimode fiber types and its relationship to measured performance," presented at the Optical Fiber Communication Conference, Anaheim, Calif., March 2005, paper NWF2.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Kluwer Academic, 2000).

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Figures (7)

Fig. 1
Fig. 1

(Color online) Simulation model of a two-dimensional MMWG of width W and length 3 L π that satisfies the self-imaging condition that arbitrary input light distributions can produce the first single image mirrored with respect to the x = 0 plane at the output. The input light has a spatially and temporally Gaussian distribution of input position x u , beam diameter σ, and 21 fs pulse width.

Fig. 2
Fig. 2

(a) Contour plot for lower modes than the 10th of the normalized depending part on x u / W of c p u j ϕ p ( x u ) versus normalized input position x u / W . (b) Contour plot of the normalized dependent part on σ / W of c p u j ϕ p ( x u ) versus normalized beam diameter σ / W . c p u j ϕ p ( x u ) is the strength of the excited modes in the spatial self-imaging position and has separable components depending on x u / W and σ / W , respectively.

Fig. 3
Fig. 3

Contour plot of c p u j ϕ p ( x u ) versus x u / W and mode number p at selected σ / W values: (a) 0.1, (b) 0.2, (c) 0.3, and (d) 0.4. The mode number has the same order with time because higher mode numbers have a larger group delay.

Fig. 4
Fig. 4

Plot of output waveform versus x u / W at selected σ / W values: (a) 0.1, (b) 0.2, (c) 0.3, and (d) 0.4.

Fig. 5
Fig. 5

Output waveform simulated by FDTD versus x u / W at σ / W of 0.2 and comparable theoretical results (a) excluding or (b) including the spread group delay. The input pulse applied to the MMWG is shown in the inset.

Fig. 6
Fig. 6

Plot of the peak delay time versus x u / W every σ / W estimated from the theoretical results and FDTD.

Fig. 7
Fig. 7

Plot of the pulse spreading versus x u / W every σ / W estimated from the theoretical results and FDTD.

Tables (2)

Tables Icon

Table 1 Physical Behavior of Multimode Waveguide and Input Light Model

Tables Icon

Table 2 Finite-Difference Time-Domain Simulation Conditions

Equations (198)

Equations on this page are rendered with MathJax. Learn more.

N × N
( 420   fs )
900   fs
2 × 2
1 × 2
N × N
| p
β p
Θ ( ω ) | p = β p | p ,
{ p }
q | F ( ω ) | p = i q | T + ( ω ) T ( ω ) ω | p = z β p ω δ q p z sin c ( β q β p 2 z ) × exp [ i ( β q β p ) 2 z ] q | d Θ ( ω ) d ω | p ( 1 δ q p ) ,
T = exp [ i Θ ( ω ) z ]
sin c ( x ) sin ( x ) / x
( [ F , T ] 0 )
( [ F , T ] = 0 )
exp ( i ω t )
exp ( i ω l t )
T | p , l exp ( i ω t ) = T | p exp [ i ( ω + ω l ) t ] ¯ = exp [ i ( ω t β p z ) ] | p × exp [ i ω l ( t z β p ω ) ] ¯ ¯ ,
| p , l = | p exp ( i ω l t )
exp ( i ω l t )
β p = n 0 k 0 ( p + 1 ) 2 π 3 L π , d β p d ω = 1 ω [ n 0 k 0 + ( p + 1 ) 2 π 3 L π ] ,
k 0
n 0
L π
L π = π β 0 β 1 = 2 n 0 k 0 W 0 2 3 π .
n c
x = 0
| u , j , s = p , l c p u j b l s | p , l = p , l c p u j | p b l s exp ( i ω l t ) ,
| u , j , s
ψ u j ( x )
ζ s ( t )
c p u j
c p u j = ψ u j ( x ) ϕ p ( x ) d x ϕ p ( x ) 2 d x = u , j p p p ,
ψ u j ( x )
ϕ p ( x ) = sin [ ( p + 1 ) π ( x W + 1 2 ) ] ,
b l s
b l s = ζ s ( t ) exp ( i ω l t ) d x = s l ,
ζ s ( t )
exp ( i ω l t )
T | u , j , s exp ( i ω t ) = p , l { c p u j exp [ i ( β 0 β p ) z ] ¯ × exp [ i ( ω t β 0 z ) ] | p ¯ b l s ¯ ¯ × exp [ i ω l z ( β 0 ω β p ω ) ] ¯ ¯ × exp [ i ω l ( t z β 0 ω ) ] ¯ ¯ } .
z = m 3 L π
exp [ i ( β 0 β p ) z ] = ( 1 ) m
exp [ i ω l z ( β 0 / ω β p / ω ) ] = exp [ i m p ( p + 2 ) π ω l / ω ]
δ ω
ω c
τ p
ω c
Δ τ p
m 3 L π β p ω = m 3 L π | β p ω | ω = ω c + m 3 L π δ ω | 2 β p ω 2 | ω = ω c = τ p + Δ τ p ,
T | u , j , s exp ( i ω t ) = ( 1 ) m exp [ i ( ω t β 0 m 3 L π ) ] × p , l { c p u j | p ¯ b l s ¯ ¯ × exp [ i ω l ( t τ p Δ τ p ) ] ¯ ¯ } .
c p u j
ψ u j ( x )
x u
σ j
Δ τ p
τ p
Δ τ p
τ peak
Δ τ pulse
τ p
515   μm
x = 0
c p u j ϕ p ( x u )
W x u / W
W σ / W
x u / W
c p u j ϕ p ( x u )
x u / W
x u / W = 0
x u / W = ζ / ( p + 1 ) 1 / 2
x u / W = ( 2 ζ + 1 ) / ( p + 1 ) 1 / 2
σ / W
σ / W
0.2
p c p u j ϕ p ( x u )
c p u j ϕ p ( x u )
σ / W
0.2
p c p u j ϕ p ( x u )
c p u j ϕ p ( x u )
c p u j ϕ p ( x u )
x u / W
x = 0
x u / W
σ / W
Δ τ p
Δ τ p
500   μm
11   μm
1   μm
500   μm
x = 0
Δ τ p
x u / W
σ / W
Δ τ p
x u / W
Δ τ p
Δ τ p
x u / W
x u / W = 0
x u / W 1 / 6
x u / W = 0
τ peak
Δ τ pulse
x u / W
σ / W
τ peak
Δ τ pulse
Δ τ pulse
x u / W = 0
x u / W 1 / 6
x u / W
τ peak
x u / W τ peak
σ / W
τ peak
x u / W
σ / W = 0.4
σ / W = 0.1
Δ τ pulse
x u / W
σ / W
x u / W
σ / W
σ / W
σ / W = 0.4
σ / W = 0.1
σ / W
x u / W
Δ τ pulse
σ / W
Δ τ pulse
x u / W = 0.25
σ / W
x u / W
Δ τ pulse
x u / W
σ / W
x u / W
σ / W
58 f s
σ / W = 0.4
σ / W = 0.1
Δ τ pulse
Δ τ pulse
σ / W
W / σ
Δ τ p
Δ τ pulse
x u / W
Δ τ pulse
x u / W
σ / W
σ / W
x u / W
σ / W
Δ τ pulse
x u / W = 0.1
σ / W = 0.2
Δ τ pulse
x u / W = 0.25
σ / W = 0.35
Δ τ pulse
σ / W
N × N
x u
x u
x u
420 fs
500   μm
Δ τ pulse
n core
n clad
3 L π
x = 0
x u
x u / W
c p u j ϕ p ( x u )
x u / W
σ / W
c p u j ϕ p ( x u )
σ / W
c p u j ϕ p ( x u )
x u / W
σ / W
c p u j ϕ p ( x u )
x u / W
σ / W
x u / W
σ / W
x u / W
σ / W
x u / W
σ / W
x u / W
σ / W
2.66   GHz

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