Abstract

I propose an analytical approach to design flattened wavelength splitters with cascaded Mach–Zehnder interferometers when wavelength dependence of the directional couplers cannot be neglected. I start from a geometrical representation of the action of a doubly point-symmetrical filter, assuming no wavelength dependence of the couplers. Next I derive the analytical formulas behind its working principle and extend them to the wavelength-dependent case. I also show how the geometrical representation allows one to broaden the class of working structures.

© 2006 Optical Society of America

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References

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  1. C. K. Madsen, Optical Filter Design and Analysis (Wiley, 1999).
    [CrossRef]
  2. M. Kuznetsov, "Cascaded coupler Mach-Zehnder channel dropping filter for wavelength-division-multiplexed optical systems," J. Lightwave Technol. 12, 226-230 (1994).
    [CrossRef]
  3. B. J. Offrein, R. Germann, F. Horst, H. W. M. Salemink, R. Beyeler, and G. L. Bona, "Resonant coupler-based tunable add-after-drop filter in silicon-oxynitride technology for WDM networks," IEEE J. Sel. Top. Quantum Electron. 5, 1400-1406 (1999).
    [CrossRef]
  4. 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]
  5. M. Cherchi, "Wavelength-flattened directional couplers: a geometrical approach," Appl. Opt. 42, 7141-7148 (2003).
    [CrossRef]
  6. M. Tormen and M. Cherchi, "Wavelength-flattened directional couplers for mirror-symmetric interferometers," J. Lightwave Technol. 23, 4387-4392 (2005).
    [CrossRef]
  7. R. Ulrich, "Representation of codirectional coupled waves," Opt. Lett. 1, 109-111 (1977).
    [CrossRef] [PubMed]
  8. N. Frigo, "A generalized geometrical representation of coupled mode theory," IEEE J. Quantum Electron. 22, 2131-2140 (1986).
    [CrossRef]
  9. S. K. Korotky, "Three-space representation of phase-mismatch switching in coupled two-state optical systems," IEEE J. Quantum Electron. QE-22, 952-958 (1986).
    [CrossRef]
  10. Fimmwave, fully vectorial semianalytical mode solver by PhotonDesign, http://www.photond.com.
  11. BeamPROP, vectorial beam propagator by RSoft Inc., http://www.rsoftdesign.com.

2005 (1)

2003 (1)

1999 (1)

B. J. Offrein, R. Germann, F. Horst, H. W. M. Salemink, R. Beyeler, and G. L. Bona, "Resonant coupler-based tunable add-after-drop filter in silicon-oxynitride technology for WDM networks," IEEE J. Sel. Top. Quantum Electron. 5, 1400-1406 (1999).
[CrossRef]

1996 (1)

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]

1994 (1)

M. Kuznetsov, "Cascaded coupler Mach-Zehnder channel dropping filter for wavelength-division-multiplexed optical systems," J. Lightwave Technol. 12, 226-230 (1994).
[CrossRef]

1986 (2)

N. Frigo, "A generalized geometrical representation of coupled mode theory," IEEE J. Quantum Electron. 22, 2131-2140 (1986).
[CrossRef]

S. K. Korotky, "Three-space representation of phase-mismatch switching in coupled two-state optical systems," IEEE J. Quantum Electron. QE-22, 952-958 (1986).
[CrossRef]

1977 (1)

Beyeler, R.

B. J. Offrein, R. Germann, F. Horst, H. W. M. Salemink, R. Beyeler, and G. L. Bona, "Resonant coupler-based tunable add-after-drop filter in silicon-oxynitride technology for WDM networks," IEEE J. Sel. Top. Quantum Electron. 5, 1400-1406 (1999).
[CrossRef]

Bona, G. L.

B. J. Offrein, R. Germann, F. Horst, H. W. M. Salemink, R. Beyeler, and G. L. Bona, "Resonant coupler-based tunable add-after-drop filter in silicon-oxynitride technology for WDM networks," IEEE J. Sel. Top. Quantum Electron. 5, 1400-1406 (1999).
[CrossRef]

Cherchi, M.

Frigo, N.

N. Frigo, "A generalized geometrical representation of coupled mode theory," IEEE J. Quantum Electron. 22, 2131-2140 (1986).
[CrossRef]

Germann, R.

B. J. Offrein, R. Germann, F. Horst, H. W. M. Salemink, R. Beyeler, and G. L. Bona, "Resonant coupler-based tunable add-after-drop filter in silicon-oxynitride technology for WDM networks," IEEE J. Sel. Top. Quantum Electron. 5, 1400-1406 (1999).
[CrossRef]

Hida, Y.

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]

Horst, F.

B. J. Offrein, R. Germann, F. Horst, H. W. M. Salemink, R. Beyeler, and G. L. Bona, "Resonant coupler-based tunable add-after-drop filter in silicon-oxynitride technology for WDM networks," IEEE J. Sel. Top. Quantum Electron. 5, 1400-1406 (1999).
[CrossRef]

Jinguji, K.

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]

Kawachi, M.

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]

Kitoh, T.

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]

Korotky, S. K.

S. K. Korotky, "Three-space representation of phase-mismatch switching in coupled two-state optical systems," IEEE J. Quantum Electron. QE-22, 952-958 (1986).
[CrossRef]

Kuznetsov, M.

M. Kuznetsov, "Cascaded coupler Mach-Zehnder channel dropping filter for wavelength-division-multiplexed optical systems," J. Lightwave Technol. 12, 226-230 (1994).
[CrossRef]

Madsen, C. K.

C. K. Madsen, Optical Filter Design and Analysis (Wiley, 1999).
[CrossRef]

Offrein, B. J.

B. J. Offrein, R. Germann, F. Horst, H. W. M. Salemink, R. Beyeler, and G. L. Bona, "Resonant coupler-based tunable add-after-drop filter in silicon-oxynitride technology for WDM networks," IEEE J. Sel. Top. Quantum Electron. 5, 1400-1406 (1999).
[CrossRef]

Salemink, H. W. M.

B. J. Offrein, R. Germann, F. Horst, H. W. M. Salemink, R. Beyeler, and G. L. Bona, "Resonant coupler-based tunable add-after-drop filter in silicon-oxynitride technology for WDM networks," IEEE J. Sel. Top. Quantum Electron. 5, 1400-1406 (1999).
[CrossRef]

Takato, N.

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]

Tormen, M.

Ulrich, R.

Appl. Opt. (1)

IEEE J. Quantum Electron. (2)

N. Frigo, "A generalized geometrical representation of coupled mode theory," IEEE J. Quantum Electron. 22, 2131-2140 (1986).
[CrossRef]

S. K. Korotky, "Three-space representation of phase-mismatch switching in coupled two-state optical systems," IEEE J. Quantum Electron. QE-22, 952-958 (1986).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

B. J. Offrein, R. Germann, F. Horst, H. W. M. Salemink, R. Beyeler, and G. L. Bona, "Resonant coupler-based tunable add-after-drop filter in silicon-oxynitride technology for WDM networks," IEEE J. Sel. Top. Quantum Electron. 5, 1400-1406 (1999).
[CrossRef]

J. Lightwave Technol. (3)

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. Kuznetsov, "Cascaded coupler Mach-Zehnder channel dropping filter for wavelength-division-multiplexed optical systems," J. Lightwave Technol. 12, 226-230 (1994).
[CrossRef]

M. Tormen and M. Cherchi, "Wavelength-flattened directional couplers for mirror-symmetric interferometers," J. Lightwave Technol. 23, 4387-4392 (2005).
[CrossRef]

Opt. Lett. (1)

Other (3)

C. K. Madsen, Optical Filter Design and Analysis (Wiley, 1999).
[CrossRef]

Fimmwave, fully vectorial semianalytical mode solver by PhotonDesign, http://www.photond.com.

BeamPROP, vectorial beam propagator by RSoft Inc., http://www.rsoftdesign.com.

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Figures (8)

Fig. 1
Fig. 1

Doubly point-symmetric structure. First, the building block, composed of a type A coupler and a half-type B coupler, is repeated point symmetrically, resulting in an A B A structure. This structure is repeated point symmetrically to give the desired result.

Fig. 2
Fig. 2

Generalized Poincaré sphere for the analysis of two coupled waveguides. The generic point P is represented together with its relative phase angle θ and power-splitting angle α. Physical transformations are represented by composition of rotations about the axes on the S 1 S 2 plane. The points on the rotation axis represent the eigenstates of the system. In particular a synchronous coupler with coupling angle ϕ is represented by a 2 ϕ rotation about the S 2 axis, whereas a θ phase shift is represented by a θ rotation about the S 1 axis.

Fig. 3
Fig. 3

Geometric representation on the S 1 S 3 plane of the action of a Mach–Zehnder interferometer in the two cases of (a) even and (b) odd phase shifts. A coupler with coupling angle ϕ gives rise to a 2 ϕ rotation, a 2 n π phase shift leaves the system as it is, whereas a ( 2 n + 1 ) π phase shift turns over the state with respect to the S 1 axis.

Fig. 4
Fig. 4

Simulated spectral response for the bar port (solid curve) and the cross port (dashed curve) of a splitter with t = 1 , m = 1 , k = 0 , ideally designed for separating a 1490 nm channel in the bar port from a 1550 nm channel in the cross port. The doubly point-symmetric structure ensures flatness for both ports.

Fig. 5
Fig. 5

Simulated spectral response for the bar port (solid curve) and the cross port (dashed curve) of a splitter with t = 0 , m = 0 , k = 0 , ideally designed for separating a 1490 nm channel in the cross port from a 1550 nm channel in the bar port. In this case the response is not flat for the bar port.

Fig. 6
Fig. 6

Geometrical representation on the full three-dimensional Poincaré sphere of the response in Fig. 5 for a wavelength slightly different by 1550 nm , corresponding to phase shifts slightly greater than π. The sequence A C A C moves the trajectory on the spherical surface away from the S 1 S 3 plane, since the angular excess adds up at each stage. This means that, unlike the nominal wavelength case (which trajectory remains confined in the S 1 S 3 plane), the coupler rotations are performed at an increasing distance from the S 1 S 3 plane, that is, far from the nominal ending point E 1 .

Fig. 7
Fig. 7

Same as in Fig. 6 when the sequence A A C C is chosen. In this case the trajectory remains close to the S 1 S 3 plane. The angular excess of the second (fourth) phase shifter compensates the angular excess of the first (third) phase shifter. In this way all the coupler rotations are performed close to the S 1 S 3 plane and they add up to almost zero, like at the nominal wavelength.

Fig. 8
Fig. 8

Spectral response of a t = 0 , m = 0 , k = 0 structure when the A A C C configuration is chosen. In this case, as explained in Fig. 7, the 1550 nm port is well flattened.

Equations (6)

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ϕ ( λ ) = κ ( λ ) [ L + δ L ( λ ) ] ,
{ 4 ϕ A 2 ϕ B = t π 2 + k π 4 ϕ A + 2 ϕ B = ( 1 t ) π 2 + m π } ,
{ ϕ A = π 16 + ( m + k ) π 8 ϕ B = π 8 + ( m k t ) π 4 } ,
{ m k , for t = 0 m k + 1 , for t = 1 and k 0 m k , for t = 1 and k < 0 } .
{ 4 ϕ 2 A 2 ϕ 2 B = t π 2 + k π 4 ϕ 1 A + 2 ϕ 1 B = ( 1 t ) π 2 + m π } ,
{ L A = [ ( 1 t ) + 2 m κ 1 + t + 2 k κ 2 ] π 16 δ L 2 + 3 δ L 1 4 L B = [ ( 1 t ) + 2 m κ 1 t + 2 k κ 2 ] π 8 + δ L 2 3 δ L 1 2 } .

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