Abstract

A twofold generalization of the optical schemes that perform the discrete Fourier transform (DFT) is given: new passive planar architectures are presented where the 2×2 3dB couplers are replaced by M×M hybrids, reducing the number of required connections and phase shifters. Furthermore, the planar implementation of the discrete fractional Fourier transform (DFrFT) is also described, with a waveguide grating router (WGR) configuration and a properly modified slab coupler.

© 2011 Optical Society of America

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References

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2011 (1)

2010 (2)

2009 (2)

L. Zimmermann, K. Voigt, G. Winzer, K. Petermann, and C. M. Weinert, IEEE Photon. Technol. Lett. 21, 143 (2009).
[CrossRef]

W. Shieh and I. Djordjevic, OFDM for Optical Communications (Academic, 2009).

2004 (1)

2002 (1)

G. Cincotti, IEEE J. Quantum Electron. 38, 1420 (2002).
[CrossRef]

2001 (2)

A. E. Siegman, Opt. Lett. 26, 1215 (2001).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

1987 (1)

Cincotti, G.

G. Cincotti, J. Lightwave Technol. 22, 1642 (2004).
[CrossRef]

G. Cincotti, IEEE J. Quantum Electron. 38, 1420 (2002).
[CrossRef]

Djordjevic, I.

W. Shieh and I. Djordjevic, OFDM for Optical Communications (Academic, 2009).

Doerr, C. R.

Ezra, S. Ben

Freude, W.

Hillerkuss, D.

Kutay, M. A.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Leuthold, J.

Li, J.

Lowery, A. J.

Marchic, M. E.

Marculescu, A.

Narkiss, N.

Ozaktas, H. M.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Petermann, K.

L. Zimmermann, K. Voigt, G. Winzer, K. Petermann, and C. M. Weinert, IEEE Photon. Technol. Lett. 21, 143 (2009).
[CrossRef]

Shieh, W.

W. Shieh and I. Djordjevic, OFDM for Optical Communications (Academic, 2009).

Siegman, A. E.

Sigurdsson, G.

Teschke, M.

Voigt, K.

L. Zimmermann, K. Voigt, G. Winzer, K. Petermann, and C. M. Weinert, IEEE Photon. Technol. Lett. 21, 143 (2009).
[CrossRef]

Weinert, C. M.

L. Zimmermann, K. Voigt, G. Winzer, K. Petermann, and C. M. Weinert, IEEE Photon. Technol. Lett. 21, 143 (2009).
[CrossRef]

Winter, M.

Winzer, G.

L. Zimmermann, K. Voigt, G. Winzer, K. Petermann, and C. M. Weinert, IEEE Photon. Technol. Lett. 21, 143 (2009).
[CrossRef]

Winzer, P. J.

Worms, K.

Zalevsky, Z.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Zhang, L.

Zimmermann, L.

L. Zimmermann, K. Voigt, G. Winzer, K. Petermann, and C. M. Weinert, IEEE Photon. Technol. Lett. 21, 143 (2009).
[CrossRef]

IEEE J. Quantum Electron. (1)

G. Cincotti, IEEE J. Quantum Electron. 38, 1420 (2002).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

L. Zimmermann, K. Voigt, G. Winzer, K. Petermann, and C. M. Weinert, IEEE Photon. Technol. Lett. 21, 143 (2009).
[CrossRef]

J. Lightwave Technol. (2)

Opt. Express (2)

Opt. Lett. (2)

Other (2)

W. Shieh and I. Djordjevic, OFDM for Optical Communications (Academic, 2009).

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

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Figures (3)

Fig. 1
Fig. 1

N = 8 th-order DFT of (a) parallel and (b) serial input using two 90 ° hybrids and a set of asymmetric couplers (or 180 ° hybrids). Ellipses represent time delays, Y branch represents symmetric splitters, and boxes are constant phase shifts of value q 2 π / N (q is the value inside the box).

Fig. 2
Fig. 2

(a)  N = 8 th-order DFT (DFrFT) of a serial input using a 45 ° hybrid (modified slab coupler). (b) Detail of a slab coupler with N = 8 input/output ports; R is the curvature radius, l the slab length, d and d o are the input and output grating pitches. In the case R = l , the slab coupler is a ( 360 / N ) ° = 45 ° hybrid.

Fig. 3
Fig. 3

Transfer functions of the device in Fig. 2a for τ = 5 ps ( FSR = 200 GHz ) and different values of the parameter p.

Equations (10)

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b m = 1 N n = 0 N 1 a n exp ( j 2 π m n / N ) 0 m N 1 ,
b m = 1 N [ n = 0 N / 4 1 a 4 n exp ( j 8 π m n / N ) + n = 0 N / 4 1 a 4 n + 1 exp ( j 8 π m n / N ) exp ( j 2 π m / N ) + n = 0 N / 4 1 a 4 n + 2 exp ( j 8 π m n / N ) exp ( j 4 π m / N ) + n = 0 N / 4 1 a 4 n + 3 exp ( j 8 π m n / N ) exp ( j 6 π m / N ) ] .
b m = 1 N p = 0 N / P 1 [ n = 0 P 1 a N P n + p exp ( j 2 π m n / P ) ] · exp ( j 2 π m p / N )
b m = 1 M n = 0 M 1 a n exp ( j 2 π m n / M ) 0 m M 1 .
d = d o = λ R N ,
l = R 1 sin ( p π 2 ) R = R 1 tan ( p π 4 ) ,
F p [ f ( x ) ] ( X ) = 1 j tan ( p π 2 ) f ( x ) exp { j π [ 2 x X sin ( p π 2 ) x 2 + X 2 tan ( p π 2 ) ] } d x .
F p { F q [ f ( x ) ] } = F p + q [ f ( x ) ] ,
d = d o = λ R 1 N ,
b m = 1 N n = 0 N 1 a n exp [ 2 π j m n N sin ( p π 2 ) ] · exp [ π j n 2 + m 2 N tan ( p π 2 ) ] 0 m N 1 .

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