Abstract

We present a Maclaurin-series method for calculating the dispersion from phase error and amplitude distributions in arrayed waveguide grating (AWG) multiplexers. By using this method, we can easily derive the intercept, the gradient, and the curvature of the dispersion in the center frequency region of a passband. A third-order Maclaurin series was calculated by using the measured phase error and amplitude distributions of AWGs having a channel frequency spacing of 12.5GHz. The calculated results are in good agreement with the dispersions measured with an optical network analyzer. We also discuss the physical effect of the phase error on dispersion by assuming certain limited cases.

© 2010 Optical Society of America

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References

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  1. M. K. Smit, “New focusing and dispersive planar components based on an optical phase array,” Electron. Lett. 24, 385–386 (1988).
    [CrossRef]
  2. H. Takahashi, K. Oda, H. Toba, Y. Inoue, “Transmission characteristics of arrayed waveguide N×N wavelength multiplexers,” J. Lightwave Technol. 13, 447–455 (1995).
    [CrossRef]
  3. H. Yamada, K. Okamoto, A. Kaneko, A. Sugita “Dispersion resulting from phase and amplitude errors in arrayed-waveguide grating multiplexers–demultiplexers,” Opt. Lett. 25, 569–571 (2000).
    [CrossRef]
  4. M. C. Parker, S. D. Walker, “Design of arrayed-waveguide gratings using hybrid Fourier-Fresnel transform techniques,” IEEE J. Sel. Top. Quantum Electron. 5, 1379–1384 (1999).
    [CrossRef]
  5. J. Gehler, W. Spahn, “Dispersion measurement of arrayed-waveguide gratings by Fourier transforms spectroscopy,” Electron. Lett. 36, 338–339 (2000).
    [CrossRef]
  6. M. E. Marhic, X. Yi, “Calculation of dispersion in arrayed waveguide grating demultiplexers by a shifting-image method,” IEEE J. Sel. Top. Quantum Electron. 8, 1149–1157 (2002).
    [CrossRef]
  7. K. Takada, H. Yamada, Y. Inoue, “Optical low coherence method for characterizing silica-based arrayed-waveguide grating multiplexers,” J. Lightwave Technol. 141677–1689 (1996).
    [CrossRef]

2002 (1)

M. E. Marhic, X. Yi, “Calculation of dispersion in arrayed waveguide grating demultiplexers by a shifting-image method,” IEEE J. Sel. Top. Quantum Electron. 8, 1149–1157 (2002).
[CrossRef]

2000 (2)

J. Gehler, W. Spahn, “Dispersion measurement of arrayed-waveguide gratings by Fourier transforms spectroscopy,” Electron. Lett. 36, 338–339 (2000).
[CrossRef]

H. Yamada, K. Okamoto, A. Kaneko, A. Sugita “Dispersion resulting from phase and amplitude errors in arrayed-waveguide grating multiplexers–demultiplexers,” Opt. Lett. 25, 569–571 (2000).
[CrossRef]

1999 (1)

M. C. Parker, S. D. Walker, “Design of arrayed-waveguide gratings using hybrid Fourier-Fresnel transform techniques,” IEEE J. Sel. Top. Quantum Electron. 5, 1379–1384 (1999).
[CrossRef]

1996 (1)

K. Takada, H. Yamada, Y. Inoue, “Optical low coherence method for characterizing silica-based arrayed-waveguide grating multiplexers,” J. Lightwave Technol. 141677–1689 (1996).
[CrossRef]

1995 (1)

H. Takahashi, K. Oda, H. Toba, Y. Inoue, “Transmission characteristics of arrayed waveguide N×N wavelength multiplexers,” J. Lightwave Technol. 13, 447–455 (1995).
[CrossRef]

1988 (1)

M. K. Smit, “New focusing and dispersive planar components based on an optical phase array,” Electron. Lett. 24, 385–386 (1988).
[CrossRef]

Gehler, J.

J. Gehler, W. Spahn, “Dispersion measurement of arrayed-waveguide gratings by Fourier transforms spectroscopy,” Electron. Lett. 36, 338–339 (2000).
[CrossRef]

Inoue, Y.

K. Takada, H. Yamada, Y. Inoue, “Optical low coherence method for characterizing silica-based arrayed-waveguide grating multiplexers,” J. Lightwave Technol. 141677–1689 (1996).
[CrossRef]

H. Takahashi, K. Oda, H. Toba, Y. Inoue, “Transmission characteristics of arrayed waveguide N×N wavelength multiplexers,” J. Lightwave Technol. 13, 447–455 (1995).
[CrossRef]

Kaneko, A.

Marhic, M. E.

M. E. Marhic, X. Yi, “Calculation of dispersion in arrayed waveguide grating demultiplexers by a shifting-image method,” IEEE J. Sel. Top. Quantum Electron. 8, 1149–1157 (2002).
[CrossRef]

Oda, K.

H. Takahashi, K. Oda, H. Toba, Y. Inoue, “Transmission characteristics of arrayed waveguide N×N wavelength multiplexers,” J. Lightwave Technol. 13, 447–455 (1995).
[CrossRef]

Okamoto, K.

Parker, M. C.

M. C. Parker, S. D. Walker, “Design of arrayed-waveguide gratings using hybrid Fourier-Fresnel transform techniques,” IEEE J. Sel. Top. Quantum Electron. 5, 1379–1384 (1999).
[CrossRef]

Smit, M. K.

M. K. Smit, “New focusing and dispersive planar components based on an optical phase array,” Electron. Lett. 24, 385–386 (1988).
[CrossRef]

Spahn, W.

J. Gehler, W. Spahn, “Dispersion measurement of arrayed-waveguide gratings by Fourier transforms spectroscopy,” Electron. Lett. 36, 338–339 (2000).
[CrossRef]

Sugita, A.

Takada, K.

K. Takada, H. Yamada, Y. Inoue, “Optical low coherence method for characterizing silica-based arrayed-waveguide grating multiplexers,” J. Lightwave Technol. 141677–1689 (1996).
[CrossRef]

Takahashi, H.

H. Takahashi, K. Oda, H. Toba, Y. Inoue, “Transmission characteristics of arrayed waveguide N×N wavelength multiplexers,” J. Lightwave Technol. 13, 447–455 (1995).
[CrossRef]

Toba, H.

H. Takahashi, K. Oda, H. Toba, Y. Inoue, “Transmission characteristics of arrayed waveguide N×N wavelength multiplexers,” J. Lightwave Technol. 13, 447–455 (1995).
[CrossRef]

Walker, S. D.

M. C. Parker, S. D. Walker, “Design of arrayed-waveguide gratings using hybrid Fourier-Fresnel transform techniques,” IEEE J. Sel. Top. Quantum Electron. 5, 1379–1384 (1999).
[CrossRef]

Yamada, H.

H. Yamada, K. Okamoto, A. Kaneko, A. Sugita “Dispersion resulting from phase and amplitude errors in arrayed-waveguide grating multiplexers–demultiplexers,” Opt. Lett. 25, 569–571 (2000).
[CrossRef]

K. Takada, H. Yamada, Y. Inoue, “Optical low coherence method for characterizing silica-based arrayed-waveguide grating multiplexers,” J. Lightwave Technol. 141677–1689 (1996).
[CrossRef]

Yi, X.

M. E. Marhic, X. Yi, “Calculation of dispersion in arrayed waveguide grating demultiplexers by a shifting-image method,” IEEE J. Sel. Top. Quantum Electron. 8, 1149–1157 (2002).
[CrossRef]

Electron. Lett. (2)

J. Gehler, W. Spahn, “Dispersion measurement of arrayed-waveguide gratings by Fourier transforms spectroscopy,” Electron. Lett. 36, 338–339 (2000).
[CrossRef]

M. K. Smit, “New focusing and dispersive planar components based on an optical phase array,” Electron. Lett. 24, 385–386 (1988).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

M. E. Marhic, X. Yi, “Calculation of dispersion in arrayed waveguide grating demultiplexers by a shifting-image method,” IEEE J. Sel. Top. Quantum Electron. 8, 1149–1157 (2002).
[CrossRef]

M. C. Parker, S. D. Walker, “Design of arrayed-waveguide gratings using hybrid Fourier-Fresnel transform techniques,” IEEE J. Sel. Top. Quantum Electron. 5, 1379–1384 (1999).
[CrossRef]

J. Lightwave Technol. (2)

K. Takada, H. Yamada, Y. Inoue, “Optical low coherence method for characterizing silica-based arrayed-waveguide grating multiplexers,” J. Lightwave Technol. 141677–1689 (1996).
[CrossRef]

H. Takahashi, K. Oda, H. Toba, Y. Inoue, “Transmission characteristics of arrayed waveguide N×N wavelength multiplexers,” J. Lightwave Technol. 13, 447–455 (1995).
[CrossRef]

Opt. Lett. (1)

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram of an arrayed waveguide grating multiplexer. The AWG consists of input–output ports, slab lenses, and arrayed waveguides with a constant length difference Δ L between neighboring waveguides. The optical length error Δ L m degrades the transmission properties in terms of loss, cross talk, and dispersion.

Fig. 2
Fig. 2

Diagram of AWG samples that are silica based planar waveguides fabricated on a Si substrate. The samples had a channel-spaced frequency of 12.5 GHz , 48 channels, and a Gaussian passband. The number of arrayed waveguides N and the FSR were 144 and 0.6 THz , respectively.

Fig. 3
Fig. 3

Experimental setup of a Fourier transform interferometer. The experimental setup is based on a fiber optic Mach–Zehnder interferometer. An edge-emitting LED with a center wavelength of 1.55 μm is used as an optical source. A movable retroreflector provides an optical path difference. A tunable LD operating at 1.3 μm generates a sinusoidal interference pattern with an optical path change. A zero-cross detector produced the sampling pulse by the sinusoidal interference pattern of the laser diode.

Fig. 4
Fig. 4

Phase error and amplitude for the AWGs under test; experimental results for (a) AWG #1 and (b) AWG #2. The circles represent the normalized amplitude and the triangles represent the phase errors measured with a Fourier transform interferometer.

Fig. 5
Fig. 5

Transmission spectrum and dispersion in AWGs under test; experimental results for (a) AWG #1 and (b) AWG #2. Circles, normalized transmission; triangles, dispersion measured with an optical network analyzer; solid curve, results obtained with the third-order Maclaurin-series method.

Fig. 6
Fig. 6

Dispersion obtained with the Maclaurin-series method from zeroth to third order; the results are for (a) AWG #1 and (b) AWG #2. Circles, experimental dispersion plots; solid, dash–dot, dashed, and dotted curves, results obtained with the 3rd-, 2nd-, 1st-, and 0th-order Maclaurin-series method, respectively.

Equations (19)

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H ( x ) = m = 0 N 1 a m exp [ i ( 2 π m x + ε m ) ] ,
D = 2 ϕ 2 π λ ν = ν 2 2 π c FSR ϕ ( 2 ) ( x ) ,
ϕ ( x ) = tan 1 [ S ( x ) C ( x ) ] ,
S ( x ) = m = 0 N 1 a m sin ( 2 π m x + ε m ) ,
C ( x ) = m = 0 N 1 a m cos ( 2 π m x + ε m ) .
ϕ ( 2 ) ( x ) = ϕ ( 2 ) ( 0 ) + ϕ ( 3 ) ( 0 ) x + ϕ ( 4 ) ( 0 ) x 2 2 + ϕ ( 5 ) ( 0 ) x 3 6 .
ϕ ( 1 ) ( 0 ) = C 0 S 0 ( 1 ) S 0 C 0 ( 1 ) C 0 2 + S 0 2 ,
ϕ ( 2 ) ( 0 ) = C 0 S 0 ( 2 ) S 0 C 0 ( 2 ) 2 ϕ ( 1 ) ( 0 ) ( C 0 C 0 ( 1 ) + S 0 S 0 ( 1 ) ) C 0 2 + S 0 2 ,
ϕ ( 3 ) ( 0 ) = 1 C 0 2 + S 0 2 { C 0 S 0 ( 3 ) S 0 C 0 ( 3 ) + C 0 ( 1 ) S 0 ( 2 ) S 0 ( 1 ) C 0 ( 2 ) 4 ϕ ( 2 ) ( 0 ) ( C 0 C 0 ( 1 ) + S 0 S 0 ( 1 ) ) 2 ϕ ( 1 ) ( 0 ) [ ( C 0 ( 1 ) ) 2 + ( S 0 ( 1 ) ) 2 + C 0 C 0 ( 2 ) + S 0 S 0 ( 2 ) ] } ,
S 0 ( i ) = { ( 1 ) i + 2 2 m = 0 N 1 ( 2 π · m ) i a m sin ( ε m ) ( i = 0 , 2 , 4 ) ( 1 ) i + 2 2 m = 0 N 1 ( 2 π · m ) i a m cos ( ε m ) ( i = 1 , 3 , 5 ) ,
C 0 ( i ) = { ( 1 ) i + 4 2 m = 0 N 1 ( 2 π m ) i a m cos ( ε m ) ( i = 0 , 2 , 4 ) ( 1 ) i + 5 2 m = 0 N 1 ( 2 π m ) i a m sin ( ε m ) ( i = 1 , 3 , 5 ) .
H ( x ) = exp ( i 2 π m c x ) exp [ ( 1 Δ N 2 i ε 0 ) m 2 i 2 π x m ] d m .
D a = π ν 2 c FSR 2 ε 0 Δ N 4 1 + ε 0 2 Δ N 4 .
ε m = ε o + ε e ,
S ( x ) = m = 0 N 1 a m ε e [ cos ( 2 π m x ) ε o sin ( 2 π m x ) ] ,
C ( x ) =     m = 0 N 1 a m [ cos ( 2 π m x ) ε o sin ( 2 π m x ) ] .
ϕ ( x ) = tan 1 { m = 0 N 1 a m ε e [ cos ( 2 π m x ) ε o sin ( 2 π m x ) ] m = 0 N 1 a m [ cos ( 2 π m x ) ε o sin ( 2 π m x ) ] } .
ϕ ( 4 ) ( 0 ) = 1 C 0 2 + S 0 2 { C 0 S 0 ( 4 ) S 0 C 0 ( 4 ) + 2 C 0 ( 1 ) S 0 ( 3 ) 2 S 0 ( 1 ) C 0 ( 3 ) 6 ϕ ( 3 ) ( 0 ) ( C 0 C 0 ( 1 ) + S 0 S 0 ( 1 ) ) 6 ϕ ( 2 ) ( 0 ) [ ( C 0 ( 1 ) ) 2 + ( S 0 ( 1 ) ) 2 + C 0 C 0 ( 2 ) + S 0 S 0 ( 2 ) ] 2 ϕ ( 1 ) ( 0 ) ( 3 C 0 ( 1 ) C 0 ( 2 ) + 3 S 0 ( 1 ) S 0 ( 2 ) + C 0 C 0 ( 3 ) + S 0 S 0 ( 3 ) ) } ,
ϕ ( 5 ) ( 0 ) = 1 C 0 2 + S 0 2 { C 0 S 0 ( 5 ) S 0 C 0 ( 5 ) + 3 C 0 ( 1 ) S 0 ( 4 ) 3 S 0 ( 1 ) C 0 ( 4 ) + 2 C 0 ( 2 ) S 0 ( 3 ) 2 S 0 ( 2 ) C 0 ( 3 ) 8 ϕ ( 4 ) ( 0 ) ( C 0 S 0 ( 1 ) + S 0 · C 0 ( 1 ) ) 12 ϕ ( 3 ) ( 0 ) [ ( C 0 ( 1 ) ) 2 + ( S 0 ( 1 ) ) 2 + C 0 C 0 ( 2 ) + S 0 S 0 ( 2 ) ] 8 ϕ ( 2 ) ( 0 ) ( 3 C 0 ( 1 ) C 0 ( 2 ) + 3 S 0 ( 1 ) S 0 ( 2 ) + C 0 C 0 ( 3 ) + S 0 S 0 ( 3 ) ) 2 ϕ ( 1 ) ( 0 ) [ 4 C 0 ( 1 ) C 0 ( 3 ) + 4 S 0 ( 1 ) S 0 ( 3 ) + C 0 ( 1 ) C 0 ( 4 ) + S 0 ( 1 ) S 0 ( 4 ) + 3 ( C 0 ( 2 ) ) 2 + 3 ( S 0 ( 2 ) ) 2 ] } ,

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