Abstract

In this paper we present an improved Fourier Optics model to calculate the transmission characteristics between any arbitrary pair of input/output ports (IOPs) of an Arrayed Waveguide Grating (AWG). In this model the input and output sections of the AWG are modeled using the same approximations, thus removing some reciprocity-related inconsistencies present in previously existing models. The expressions which summarize the model are compact and easily interpretable. Simple quasi-analytical expressions are also derived under the Gaussian approximation of the mode field profiles.

© 2004 Optical Society of America

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  8. A.A. Bernussi, L Grave de Peralta and H. Temkin, �??Electric field distribution in a grating of a folded Arrayed-Waveguide Multiplexer,�?? IEEE Photon. Technol. Lett. 16, 448-490 (2004)
    [CrossRef]

IEEE J. Sel. Top. Quantum Electron.

P. Muñoz, D. Pastor, J. Capmany and S. Sales, �??Analytical and numerical analysis of phase and amplitude errors in the performance of arrayed waveguide gratings,�?? IEEE J. Sel. Top. Quantum Electron. 8, 1130�??1141 (2002)
[CrossRef]

IEEE J. Select. Topics Quantum Electron.

Y. Chu, X. Zheng, H. Zhang, X. Liu and Y. Guo, �??The impact of phase errors on arrayed waveguide gratings,�?? IEEE J. Select. Topics Quantum Electron. 8, 1122�??1129 (2002)
[CrossRef]

M.K. Smit and C. van Dam, �??PHASAR-based WDM-devices: Principles, design and applications,�?? IEEE J. Select. Topics Quantum Electron. 2, 236�??250 (1996)
[CrossRef]

IEEE Photon. Technol. Lett.

A.A. Bernussi, L Grave de Peralta and H. Temkin, �??Electric field distribution in a grating of a folded Arrayed-Waveguide Multiplexer,�?? IEEE Photon. Technol. Lett. 16, 448-490 (2004)
[CrossRef]

S. Vallon, P. Chevallier, L. Guiziou, G. Alibert, L.S. How Kee Chun and N. Boos, �??40-band integrated static gain-flattening filter,�?? IEEE Photon. Technol. Lett. 15, 554-556 (2003)
[CrossRef]

J. Lightwave Technol.

Opt. Commun.

J. Zhou, N. Q. Ngo, K. Pita, C.H. Kam, P.V. Ramana and M.K. Iyer, �??Determining the minimum number of arrayed waveguides and the optimal orientation angle of slab for the design of arrayed waveguide gratings,�?? Opt. Commun. 226, 181-189 (2003)
[CrossRef]

Opt. Express

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Figures (4)

Fig. 1.
Fig. 1.

Layout of the AWG showing one input and one output waveguides located at arbitrary positions

Fig. 2.
Fig. 2.

Optical imaging system showing the meaning of qM(x)

Fig. 3.
Fig. 3.

Coupling between input and output ports of the system of Fig.2 versus normalized position of the output waveguide.

Fig. 4:
Fig. 4:

Illumination at the FPR2 output plane for the centered and outermost input waveguide of a 16×1 multiplexer

Equations (76)

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B g ( f ) = TF { b g ( x ) } = b g ( x ) · e j 2 πxf dx
f 0 x 0 d i ν = b i ν ( x 0 d i )
f 0 ( x 0 ) = b i ν ( x 0 d i )
f 1 ( x 1 ) = 1 α F 0 ( f ) f = x 1 α = 1 α F 0 ( x 1 α )
α = λ L f n s
f 1 d ( x 1 ) = r = N 1 2 N 1 2 a r · b g ( x 1 r · d w )
a r = f 1 ( x 1 ) · b g ( x 1 r · d w ) d x 1
f 1 d ( x 1 ) = r = g 1 ( r · d w ) · b g ( x 1 r · d w ) = ( g 1 ( x 1 ) · δ ω ( x 1 ) ) * b g ( x 1 )
g 1 ( x 1 ) = ( f 1 ( x 1 ) * b g ( x 1 ) ) · ( x 1 N · d w )
b g ( x 1 ) ( b g ( x 1 ) d x 1 ) · δ ( x 1 ) = 2 π ω g 2 4 · δ ( x 1 ) g 1 ( x 1 ) = 2 π ω g 2 4 f 1 ( x 1 ) · ( x 1 N · d w )
f 1 d ( x 1 ) = 2 π ω g 2 4 ( f 1 ( x 1 ) · ( x 1 N · d w ) · δ ω ( x 1 ) ) * b g ( x 1 )
Δ l = m λ 0 n c = mc n c ν 0
ϕ x 2 ν = exp [ j 2 πm ( ν ν 0 ) · ( x 2 d w ) ]
f 2 ( x 2 ) = ( g 1 ( x 2 ) · δ ω ( x 2 ) · ϕ x 2 ν ) * b g ( x 2 )
f 3 ( x 3 ) = 1 α F 2 ( f ) f = x 3 α = 1 α F 2 ( x 3 α )
f 3 ( x 3 ) = 1 d w · B g ( x 3 α ) · r = h ( x 3 + d w ν 0 ν d w )
h ( x 3 ) = 1 α · G 1 ( x 3 α ) = ( b i ( ( x 3 + d i ) ) · B g ( x 3 α ) ) * ( N · d w α · Sinc ( N · d w α · x 3 ) )
b i ( ( x 3 + d i ) ) · B g ( x 3 α ) b i ( ( x 3 + d i ) ) · B g ( d i α )
h ( x 3 ) = B g ( d i α ) · f M ( x 3 + d i )
f M ( x 3 ) = b i ( x 3 ) * ( N · d w α · Sinc ( N · d w α · x 3 ) )
f 3 x 3 d i ν = 1 d w · B g ( d i α ) · B g ( x 3 α ) · r = f M ( x 3 + d i + d w ν 0 ν d w )
t d i d o ν = f 3 x 3 d i ν · b o ( x 3 d o ) d x 3
t d i d o ν 1 d w · B g ( d i α ) · B g ( d o α ) · { r = f M ( x 3 + d i + d w ν 0 ν d w ) } · b o ( x 3 d o ) d x 3
t d i d o ν 1 d w · B g ( d i α ) · B g ( d o α ) · r = q M ( d i + d o + d w ν 0 ν d w )
q M ( x ) = b o ( x ) * f M ( x ) = ( b o ( x ) * b i ( x ) ) * ( N · d w α · Sinc ( N . d w α · x )
q M ( x ) = lim N q M ( x ) = b o ( x ) * b i ( x )
q M ( 0 ) = b i ( x ) b i ( x ) dx = b i 2 ( x ) dx = 1
t d i d o ν = 2 π ω g d w exp { ( π ω g α ) 2 ( d i 2 + d o 2 ) } · r = ( d i + d o + d w ν 0 ν d w )
q M ( x ) = exp ( ( x 2 ω i ) 2 ) · Re al { erf ( πN d w ω i 2 α + j x 2 ω i ) }
q M ( x ) = lim N q M ( x ) = exp ( ( x 2 ω i ) 2 )
d i + d o + α d w ( m ν 0 ν r ) = 0 d i + d o α d w + ν ν o m = r
L d d i d o | min = 1 t d i d o max = d w B g ( d i α ) B g ( d o α ) 1 r = q M ( ( r m ) α d w )
d w B g ( d i α ) · d w B g ( d o α ) · 1 q M ( 0 )
f 3 x 3 d i ν = ( 2 π ω g 2 ) ¼ d w · B g ( x 3 α ) · r = f M ( x 3 + d i + d w ν 0 ν d w )
t d i d o ν = ( 2 π ω g 2 ) ¼ d w · B g ( d o α ) · r = q M ( d i + d o + d w ν 0 ν d w )
f 1 d ( x 1 ) = r = N 1 2 N 1 2 a r · b g ( x 1 r · d w )
a r = f 1 ( x 1 ) · b g ( x 1 r · d w 1 ) d x 1
q 1 ( x ) = f 1 ( x ) * b g ( x ) = f 1 ( x 1 ) · b g ( ( x x 1 ) ) dx = f 1 ( x 1 ) · b g ( x 1 x ) ) dx
a r = q 1 ( r d w )
f 1 d ( x 1 ) = r = N 1 2 N 1 2 q 1 ( r d w ) · b g ( x 1 r · d w )
g 1 ( x 1 ) = q 1 ( x 1 ) · ( x 1 N · d w 1 )
( x 1 N · d w 1 ) = { 1 , si x 1 N · d w 1 2 0 , si x 1 > N · d w 1 2 }
δ ω ( x 1 ) = r = δ ( x 1 r · d w )
f 1 d ( x 1 ) = r = g 1 ( r · d w ) · b g ( x 1 r · d w ) = ( g 1 ( x 1 ) · δ ω ( x 1 ) ) * b g ( x 1 )
g 1 ( x 1 ) = ( f 1 ( x 1 ) * b g ( x 1 ) ) · ( x 1 N · d w )
ϕ x 2 ν = exp [ j 2 πm ( ν ν 0 ) · ( x 2 d w ) ]
f 2 ( x 2 ) = ( g 1 ( x 2 ) · δ ω ( x 2 ) · ϕ x 2 ν ) * b g ( x 2 )
f 3 ( x 3 ) = 1 α F 2 ( x 3 α )
F 2 ( f ) = B g ( f ) · ( G 1 ( f ) * Δ ω ( f ) * Φ f ν )
Δ w ( f ) = 1 d w r = δ ( f r d w )
Φ f ν = δ ( f + m d w ν 0 ν )
F 2 ( f ) = 1 d w · B g ( f ) · r = G 1 ( f + m d w ν 0 ν r d w )
f 3 ( x 3 ) = 1 α d w · B g ( x 3 α ) · r = G 1 ( x 3 α + m d w ν 0 ν r d w )
G 1 ( f ) = ( F 1 ( f ) · B g ( f ) ) * ( N · d w · Sinc ( N · d w · f ) )
F 1 ( f ) = TF { f 1 ( x 1 ) } = TF { 1 α F 0 ( x 1 α ) } = α f 0 ( αf )
f ( x ) F ( f )
f ( x α ) αF ( αf )
TF { TF ( f ( x ) ) } = f ( x )
h ( x 3 ) = 1 α · G 1 ( x 3 α )
h ( x 3 ) = ( f 0 ( x 3 ) · B g ( x 3 α ) ) * ( N · d w α · Sinc ( N · d w α ) · x 3 )
h ( x ) = h 1 ( x ) * h 2 ( x ) h ( x α ) = 1 α ( h 1 ( x α ) * h 2 ( x α ) * h 2 ( x α ) )
h ( x 3 ) = ( b i ( ( x 3 + d i ) · B g ( x 3 α ) ) * ( N · d w α · Sinc ( N · d w α ) · x 3 ) )
f 3 ( x 3 ) = 1 d w · B g ( x 3 α ) · r = h ( x 3 + d w ν 0 ν d w )
t d i d o ν 1 d w · B g ( d i α ) · B g ( d o α ) · { r = f M ( x 3 + d i + d w ν 0 ν d w ) · b o ( x 3 d o ) d x 3
t d i d o ν = 1 d w · B g ( d i α ) · B g ( d o α ) · r = f M ( x 3 + d i + d w ν 0 ν d w ) · b o ( x 3 d o ) d x 3
t d i d o ν 1 d w · B g ( d i α ) · B g ( d o α ) · r = f M ( x 3 + d i + d o + d w ν 0 ν d w ) · b o ( x 3 ) d x 3
q M ( x ) = b o ( x ) * f M ( x ) = b o ( x 3 ) f M ( x x 3 ) d x 3
t d i d o ν = 1 d w · B g ( d i α ) · B g ( d o α ) · r = q M ( d i + d o + d w ν 0 ν d w )
q M ( x ) = b o ( x ) * f M ( x ) = ( b o ( x ) * b i ( x ) ) * ( N · d w α · Sinc ( N · d w α · x ) )
b i , o , g ( x ) = 2 π ω i , o , g 2 4 exp ( ( x ω i , o , g ) 2 )
B g ( f ) = 2 π ω g 2 4 exp ( ( π ω g f ) 2 )
t d i d o ν = 2 π ω g d w exp { ( π ω g α ) 2 ( d i 2 + d o 2 ) } · r = q M ( d i + d o + d w ν 0 ν d w )
b o ( x ) * b i ( x ) = 2 ω i ω o ω s 2 exp ( ( x ω s ) 2 )
ω s = ω i 2 + ω o 2
q M ( x ) = 2 ω i ω o ω s 2 exp ( ( x ω s ) 2 ) · Re al { erf ( πN d w ω s 2 α + j x ω s ) }
q M ( x ) = exp ( ( x 2 ω i ) 2 ) · Re al { erf ( πN d w ω i 2 α + j x 2 ω i ) }

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