Abstract

In this paper we propose a concept of irregularly arrayed waveguide gratings (IAWGs). By replacing regularly arrayed waveguides of conventional AWGs with irregularly arrayed ones, we found theoretically that the IAWGs have arbitrary free spectral ranges and are suitable to provide a large number of channels with relatively smaller circuit regions. A Fourier optics model is presented, which is able to calculate the transmission characteristics between any arbitrary pair of input/output ports in either conventional AWGs or IAWGs. As an example, a 1080-channel IAWG with 0.199 nm spacing is designed and simulated. Results show this device can be contained in a 6-inch wafer and provide two work patterns: all 1080 channels with a relatively high crosstalk of about -20 dB, and 128 successive channels which can be arbitrarily moveable over the range of the 1080 channels with a crosstalk of about -32 dB.

© 2007 Optical Society of America

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  1. M. K. Smit and A. C. Dam, “PHASAR-based WDM-devices: principles, design and applications,” IEEE J. Select. Topics Quantum Electron. 2,236–250 (1996).
    [Crossref]
  2. Y. Hibino, “Recent advances in high-density and large-scale AWG multi/demultiplexers with higher index-contrast silica-based PLCs,” IEEE J. Select. Topics Quantum Electron. 8,1090 (2002).
    [Crossref]
  3. Y. Hida, Y. Hibino, T. Kitoh, Y. Inoue, M. Itoh, T. Shibata, A. Sugita, and A. Himeno, “400-channel arrayed-waveguide grating with 25 GHz spacing using 1.5% - Δ waveguides on 6-inch Si wafer,” Electron. Lett. 37,576–577 (2001).
    [Crossref]
  4. K. Takada, M. Abe, M. Shibata, M. Ishii, and K. Okamoto, “Low-crosstalk 10-GHz-spaced 512-channel arrayed-waveguide grating multi/demultiplexer fabricated on a 4-in wafer,” IEEE Photon. Technol. Lett. 13,1182–1184 (2001).
    [Crossref]
  5. K. Takada, M. Abe, T. Shibata, and K. Okamoto, “A 25-GHz-spaced 1080-channel tandem multi/demultiplexer covering the S-, C-, and L-bands using an arrayed-waveguide grating with Gaussian passbands as a primary filter,” IEEE Photon. Technol. Lett. 14,648–650 (2002).
    [Crossref]
  6. J. H. Abeles and R. J. Deri, “Suppression of sidelobes in the far-field radiation patterns of optical waveguide arrays,” Appl. Phys. Lett. 53,1375–1377 (1988).
    [Crossref]
  7. A. Ishimaru and H. -S. Tuan, “Theory of frequency scanning of antennas,” IEEE Trans. Antennas Propag. 10(2),144–150 (1962).
  8. F. Xiao, W. W. Hu, and A. S. Xu, “Optical phased-array beam steering controlled by wavelength,” Appl. Opt. 44,5429–5433 (2005).
    [Crossref] [PubMed]
  9. P. Munoz, D. Pastor, and J. Capmany, “Modeling and design of arrayed waveguide gratings,” J. Lightwave Tech-nol. 20,661–674 (2002).
    [Crossref]
  10. I. Molina-Fernandez and J. G. Wanguemert-Perez, “Improved AWG Fourier optics model,” Opt. Express 12,4804–4821 (2004).
    [Crossref] [PubMed]
  11. M. G. Bray, D. H. Werner, D. W. Boeringer, and D. W. Machuga, “Optimization of thinned aperiodic linear phased arrays using genetic algorithms to reduce grating lobes during scanning,” IEEE Trans. Antennas Propag. 50,1732–1742 (2002).
    [Crossref]
  12. A. Lommi, A. Massa, E. Storti, and A. Trucco, “Sidelobe reduction in sparse linear arrays by genetic algorithms,” Microw. Opt. Technol. Lett. 32,194–196 (2002).
    [Crossref]
  13. D. W. Boeringer and D. H. Werner, “Particle swarm optimization versus genetic algorithms for phased array synthesis,” IEEE Trans. Antennas Propag. 52,771–779 (2004).
    [Crossref]
  14. J. Y. Yang, X. Q. Jiang, M. H. Wang, and Y. L. Wang, “Two-dimensional wavelength demultiplexing employing multilevel arrayed waveguides,” Optics Express 12,1084–1089 (2004).
    [Crossref] [PubMed]

2005 (1)

2004 (3)

I. Molina-Fernandez and J. G. Wanguemert-Perez, “Improved AWG Fourier optics model,” Opt. Express 12,4804–4821 (2004).
[Crossref] [PubMed]

D. W. Boeringer and D. H. Werner, “Particle swarm optimization versus genetic algorithms for phased array synthesis,” IEEE Trans. Antennas Propag. 52,771–779 (2004).
[Crossref]

J. Y. Yang, X. Q. Jiang, M. H. Wang, and Y. L. Wang, “Two-dimensional wavelength demultiplexing employing multilevel arrayed waveguides,” Optics Express 12,1084–1089 (2004).
[Crossref] [PubMed]

2002 (5)

M. G. Bray, D. H. Werner, D. W. Boeringer, and D. W. Machuga, “Optimization of thinned aperiodic linear phased arrays using genetic algorithms to reduce grating lobes during scanning,” IEEE Trans. Antennas Propag. 50,1732–1742 (2002).
[Crossref]

A. Lommi, A. Massa, E. Storti, and A. Trucco, “Sidelobe reduction in sparse linear arrays by genetic algorithms,” Microw. Opt. Technol. Lett. 32,194–196 (2002).
[Crossref]

P. Munoz, D. Pastor, and J. Capmany, “Modeling and design of arrayed waveguide gratings,” J. Lightwave Tech-nol. 20,661–674 (2002).
[Crossref]

Y. Hibino, “Recent advances in high-density and large-scale AWG multi/demultiplexers with higher index-contrast silica-based PLCs,” IEEE J. Select. Topics Quantum Electron. 8,1090 (2002).
[Crossref]

K. Takada, M. Abe, T. Shibata, and K. Okamoto, “A 25-GHz-spaced 1080-channel tandem multi/demultiplexer covering the S-, C-, and L-bands using an arrayed-waveguide grating with Gaussian passbands as a primary filter,” IEEE Photon. Technol. Lett. 14,648–650 (2002).
[Crossref]

2001 (2)

Y. Hida, Y. Hibino, T. Kitoh, Y. Inoue, M. Itoh, T. Shibata, A. Sugita, and A. Himeno, “400-channel arrayed-waveguide grating with 25 GHz spacing using 1.5% - Δ waveguides on 6-inch Si wafer,” Electron. Lett. 37,576–577 (2001).
[Crossref]

K. Takada, M. Abe, M. Shibata, M. Ishii, and K. Okamoto, “Low-crosstalk 10-GHz-spaced 512-channel arrayed-waveguide grating multi/demultiplexer fabricated on a 4-in wafer,” IEEE Photon. Technol. Lett. 13,1182–1184 (2001).
[Crossref]

1996 (1)

M. K. Smit and A. C. Dam, “PHASAR-based WDM-devices: principles, design and applications,” IEEE J. Select. Topics Quantum Electron. 2,236–250 (1996).
[Crossref]

1988 (1)

J. H. Abeles and R. J. Deri, “Suppression of sidelobes in the far-field radiation patterns of optical waveguide arrays,” Appl. Phys. Lett. 53,1375–1377 (1988).
[Crossref]

1962 (1)

A. Ishimaru and H. -S. Tuan, “Theory of frequency scanning of antennas,” IEEE Trans. Antennas Propag. 10(2),144–150 (1962).

Abe, M.

K. Takada, M. Abe, T. Shibata, and K. Okamoto, “A 25-GHz-spaced 1080-channel tandem multi/demultiplexer covering the S-, C-, and L-bands using an arrayed-waveguide grating with Gaussian passbands as a primary filter,” IEEE Photon. Technol. Lett. 14,648–650 (2002).
[Crossref]

K. Takada, M. Abe, M. Shibata, M. Ishii, and K. Okamoto, “Low-crosstalk 10-GHz-spaced 512-channel arrayed-waveguide grating multi/demultiplexer fabricated on a 4-in wafer,” IEEE Photon. Technol. Lett. 13,1182–1184 (2001).
[Crossref]

Abeles, J. H.

J. H. Abeles and R. J. Deri, “Suppression of sidelobes in the far-field radiation patterns of optical waveguide arrays,” Appl. Phys. Lett. 53,1375–1377 (1988).
[Crossref]

Boeringer, D. W.

D. W. Boeringer and D. H. Werner, “Particle swarm optimization versus genetic algorithms for phased array synthesis,” IEEE Trans. Antennas Propag. 52,771–779 (2004).
[Crossref]

M. G. Bray, D. H. Werner, D. W. Boeringer, and D. W. Machuga, “Optimization of thinned aperiodic linear phased arrays using genetic algorithms to reduce grating lobes during scanning,” IEEE Trans. Antennas Propag. 50,1732–1742 (2002).
[Crossref]

Bray, M. G.

M. G. Bray, D. H. Werner, D. W. Boeringer, and D. W. Machuga, “Optimization of thinned aperiodic linear phased arrays using genetic algorithms to reduce grating lobes during scanning,” IEEE Trans. Antennas Propag. 50,1732–1742 (2002).
[Crossref]

Capmany, J.

P. Munoz, D. Pastor, and J. Capmany, “Modeling and design of arrayed waveguide gratings,” J. Lightwave Tech-nol. 20,661–674 (2002).
[Crossref]

Dam, A. C.

M. K. Smit and A. C. Dam, “PHASAR-based WDM-devices: principles, design and applications,” IEEE J. Select. Topics Quantum Electron. 2,236–250 (1996).
[Crossref]

Deri, R. J.

J. H. Abeles and R. J. Deri, “Suppression of sidelobes in the far-field radiation patterns of optical waveguide arrays,” Appl. Phys. Lett. 53,1375–1377 (1988).
[Crossref]

Hibino, Y.

Y. Hibino, “Recent advances in high-density and large-scale AWG multi/demultiplexers with higher index-contrast silica-based PLCs,” IEEE J. Select. Topics Quantum Electron. 8,1090 (2002).
[Crossref]

Y. Hida, Y. Hibino, T. Kitoh, Y. Inoue, M. Itoh, T. Shibata, A. Sugita, and A. Himeno, “400-channel arrayed-waveguide grating with 25 GHz spacing using 1.5% - Δ waveguides on 6-inch Si wafer,” Electron. Lett. 37,576–577 (2001).
[Crossref]

Hida, Y.

Y. Hida, Y. Hibino, T. Kitoh, Y. Inoue, M. Itoh, T. Shibata, A. Sugita, and A. Himeno, “400-channel arrayed-waveguide grating with 25 GHz spacing using 1.5% - Δ waveguides on 6-inch Si wafer,” Electron. Lett. 37,576–577 (2001).
[Crossref]

Himeno, A.

Y. Hida, Y. Hibino, T. Kitoh, Y. Inoue, M. Itoh, T. Shibata, A. Sugita, and A. Himeno, “400-channel arrayed-waveguide grating with 25 GHz spacing using 1.5% - Δ waveguides on 6-inch Si wafer,” Electron. Lett. 37,576–577 (2001).
[Crossref]

Hu, W. W.

Inoue, Y.

Y. Hida, Y. Hibino, T. Kitoh, Y. Inoue, M. Itoh, T. Shibata, A. Sugita, and A. Himeno, “400-channel arrayed-waveguide grating with 25 GHz spacing using 1.5% - Δ waveguides on 6-inch Si wafer,” Electron. Lett. 37,576–577 (2001).
[Crossref]

Ishii, M.

K. Takada, M. Abe, M. Shibata, M. Ishii, and K. Okamoto, “Low-crosstalk 10-GHz-spaced 512-channel arrayed-waveguide grating multi/demultiplexer fabricated on a 4-in wafer,” IEEE Photon. Technol. Lett. 13,1182–1184 (2001).
[Crossref]

Ishimaru, A.

A. Ishimaru and H. -S. Tuan, “Theory of frequency scanning of antennas,” IEEE Trans. Antennas Propag. 10(2),144–150 (1962).

Itoh, M.

Y. Hida, Y. Hibino, T. Kitoh, Y. Inoue, M. Itoh, T. Shibata, A. Sugita, and A. Himeno, “400-channel arrayed-waveguide grating with 25 GHz spacing using 1.5% - Δ waveguides on 6-inch Si wafer,” Electron. Lett. 37,576–577 (2001).
[Crossref]

Jiang, X. Q.

J. Y. Yang, X. Q. Jiang, M. H. Wang, and Y. L. Wang, “Two-dimensional wavelength demultiplexing employing multilevel arrayed waveguides,” Optics Express 12,1084–1089 (2004).
[Crossref] [PubMed]

Kitoh, T.

Y. Hida, Y. Hibino, T. Kitoh, Y. Inoue, M. Itoh, T. Shibata, A. Sugita, and A. Himeno, “400-channel arrayed-waveguide grating with 25 GHz spacing using 1.5% - Δ waveguides on 6-inch Si wafer,” Electron. Lett. 37,576–577 (2001).
[Crossref]

Lommi, A.

A. Lommi, A. Massa, E. Storti, and A. Trucco, “Sidelobe reduction in sparse linear arrays by genetic algorithms,” Microw. Opt. Technol. Lett. 32,194–196 (2002).
[Crossref]

Machuga, D. W.

M. G. Bray, D. H. Werner, D. W. Boeringer, and D. W. Machuga, “Optimization of thinned aperiodic linear phased arrays using genetic algorithms to reduce grating lobes during scanning,” IEEE Trans. Antennas Propag. 50,1732–1742 (2002).
[Crossref]

Massa, A.

A. Lommi, A. Massa, E. Storti, and A. Trucco, “Sidelobe reduction in sparse linear arrays by genetic algorithms,” Microw. Opt. Technol. Lett. 32,194–196 (2002).
[Crossref]

Molina-Fernandez, I.

Munoz, P.

P. Munoz, D. Pastor, and J. Capmany, “Modeling and design of arrayed waveguide gratings,” J. Lightwave Tech-nol. 20,661–674 (2002).
[Crossref]

Okamoto, K.

K. Takada, M. Abe, T. Shibata, and K. Okamoto, “A 25-GHz-spaced 1080-channel tandem multi/demultiplexer covering the S-, C-, and L-bands using an arrayed-waveguide grating with Gaussian passbands as a primary filter,” IEEE Photon. Technol. Lett. 14,648–650 (2002).
[Crossref]

K. Takada, M. Abe, M. Shibata, M. Ishii, and K. Okamoto, “Low-crosstalk 10-GHz-spaced 512-channel arrayed-waveguide grating multi/demultiplexer fabricated on a 4-in wafer,” IEEE Photon. Technol. Lett. 13,1182–1184 (2001).
[Crossref]

Pastor, D.

P. Munoz, D. Pastor, and J. Capmany, “Modeling and design of arrayed waveguide gratings,” J. Lightwave Tech-nol. 20,661–674 (2002).
[Crossref]

Shibata, M.

K. Takada, M. Abe, M. Shibata, M. Ishii, and K. Okamoto, “Low-crosstalk 10-GHz-spaced 512-channel arrayed-waveguide grating multi/demultiplexer fabricated on a 4-in wafer,” IEEE Photon. Technol. Lett. 13,1182–1184 (2001).
[Crossref]

Shibata, T.

K. Takada, M. Abe, T. Shibata, and K. Okamoto, “A 25-GHz-spaced 1080-channel tandem multi/demultiplexer covering the S-, C-, and L-bands using an arrayed-waveguide grating with Gaussian passbands as a primary filter,” IEEE Photon. Technol. Lett. 14,648–650 (2002).
[Crossref]

Y. Hida, Y. Hibino, T. Kitoh, Y. Inoue, M. Itoh, T. Shibata, A. Sugita, and A. Himeno, “400-channel arrayed-waveguide grating with 25 GHz spacing using 1.5% - Δ waveguides on 6-inch Si wafer,” Electron. Lett. 37,576–577 (2001).
[Crossref]

Smit, M. K.

M. K. Smit and A. C. Dam, “PHASAR-based WDM-devices: principles, design and applications,” IEEE J. Select. Topics Quantum Electron. 2,236–250 (1996).
[Crossref]

Storti, E.

A. Lommi, A. Massa, E. Storti, and A. Trucco, “Sidelobe reduction in sparse linear arrays by genetic algorithms,” Microw. Opt. Technol. Lett. 32,194–196 (2002).
[Crossref]

Sugita, A.

Y. Hida, Y. Hibino, T. Kitoh, Y. Inoue, M. Itoh, T. Shibata, A. Sugita, and A. Himeno, “400-channel arrayed-waveguide grating with 25 GHz spacing using 1.5% - Δ waveguides on 6-inch Si wafer,” Electron. Lett. 37,576–577 (2001).
[Crossref]

Takada, K.

K. Takada, M. Abe, T. Shibata, and K. Okamoto, “A 25-GHz-spaced 1080-channel tandem multi/demultiplexer covering the S-, C-, and L-bands using an arrayed-waveguide grating with Gaussian passbands as a primary filter,” IEEE Photon. Technol. Lett. 14,648–650 (2002).
[Crossref]

K. Takada, M. Abe, M. Shibata, M. Ishii, and K. Okamoto, “Low-crosstalk 10-GHz-spaced 512-channel arrayed-waveguide grating multi/demultiplexer fabricated on a 4-in wafer,” IEEE Photon. Technol. Lett. 13,1182–1184 (2001).
[Crossref]

Trucco, A.

A. Lommi, A. Massa, E. Storti, and A. Trucco, “Sidelobe reduction in sparse linear arrays by genetic algorithms,” Microw. Opt. Technol. Lett. 32,194–196 (2002).
[Crossref]

Tuan, H. -S.

A. Ishimaru and H. -S. Tuan, “Theory of frequency scanning of antennas,” IEEE Trans. Antennas Propag. 10(2),144–150 (1962).

Wang, M. H.

J. Y. Yang, X. Q. Jiang, M. H. Wang, and Y. L. Wang, “Two-dimensional wavelength demultiplexing employing multilevel arrayed waveguides,” Optics Express 12,1084–1089 (2004).
[Crossref] [PubMed]

Wang, Y. L.

J. Y. Yang, X. Q. Jiang, M. H. Wang, and Y. L. Wang, “Two-dimensional wavelength demultiplexing employing multilevel arrayed waveguides,” Optics Express 12,1084–1089 (2004).
[Crossref] [PubMed]

Wanguemert-Perez, J. G.

Werner, D. H.

D. W. Boeringer and D. H. Werner, “Particle swarm optimization versus genetic algorithms for phased array synthesis,” IEEE Trans. Antennas Propag. 52,771–779 (2004).
[Crossref]

M. G. Bray, D. H. Werner, D. W. Boeringer, and D. W. Machuga, “Optimization of thinned aperiodic linear phased arrays using genetic algorithms to reduce grating lobes during scanning,” IEEE Trans. Antennas Propag. 50,1732–1742 (2002).
[Crossref]

Xiao, F.

Xu, A. S.

Yang, J. Y.

J. Y. Yang, X. Q. Jiang, M. H. Wang, and Y. L. Wang, “Two-dimensional wavelength demultiplexing employing multilevel arrayed waveguides,” Optics Express 12,1084–1089 (2004).
[Crossref] [PubMed]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

J. H. Abeles and R. J. Deri, “Suppression of sidelobes in the far-field radiation patterns of optical waveguide arrays,” Appl. Phys. Lett. 53,1375–1377 (1988).
[Crossref]

Electron. Lett. (1)

Y. Hida, Y. Hibino, T. Kitoh, Y. Inoue, M. Itoh, T. Shibata, A. Sugita, and A. Himeno, “400-channel arrayed-waveguide grating with 25 GHz spacing using 1.5% - Δ waveguides on 6-inch Si wafer,” Electron. Lett. 37,576–577 (2001).
[Crossref]

IEEE J. Select. Topics Quantum Electron. (2)

M. K. Smit and A. C. Dam, “PHASAR-based WDM-devices: principles, design and applications,” IEEE J. Select. Topics Quantum Electron. 2,236–250 (1996).
[Crossref]

Y. Hibino, “Recent advances in high-density and large-scale AWG multi/demultiplexers with higher index-contrast silica-based PLCs,” IEEE J. Select. Topics Quantum Electron. 8,1090 (2002).
[Crossref]

IEEE Photon. Technol. Lett. (2)

K. Takada, M. Abe, M. Shibata, M. Ishii, and K. Okamoto, “Low-crosstalk 10-GHz-spaced 512-channel arrayed-waveguide grating multi/demultiplexer fabricated on a 4-in wafer,” IEEE Photon. Technol. Lett. 13,1182–1184 (2001).
[Crossref]

K. Takada, M. Abe, T. Shibata, and K. Okamoto, “A 25-GHz-spaced 1080-channel tandem multi/demultiplexer covering the S-, C-, and L-bands using an arrayed-waveguide grating with Gaussian passbands as a primary filter,” IEEE Photon. Technol. Lett. 14,648–650 (2002).
[Crossref]

IEEE Trans. Antennas Propag. (3)

A. Ishimaru and H. -S. Tuan, “Theory of frequency scanning of antennas,” IEEE Trans. Antennas Propag. 10(2),144–150 (1962).

M. G. Bray, D. H. Werner, D. W. Boeringer, and D. W. Machuga, “Optimization of thinned aperiodic linear phased arrays using genetic algorithms to reduce grating lobes during scanning,” IEEE Trans. Antennas Propag. 50,1732–1742 (2002).
[Crossref]

D. W. Boeringer and D. H. Werner, “Particle swarm optimization versus genetic algorithms for phased array synthesis,” IEEE Trans. Antennas Propag. 52,771–779 (2004).
[Crossref]

J. Lightwave Tech-nol. (1)

P. Munoz, D. Pastor, and J. Capmany, “Modeling and design of arrayed waveguide gratings,” J. Lightwave Tech-nol. 20,661–674 (2002).
[Crossref]

Microw. Opt. Technol. Lett. (1)

A. Lommi, A. Massa, E. Storti, and A. Trucco, “Sidelobe reduction in sparse linear arrays by genetic algorithms,” Microw. Opt. Technol. Lett. 32,194–196 (2002).
[Crossref]

Opt. Express (1)

Optics Express (1)

J. Y. Yang, X. Q. Jiang, M. H. Wang, and Y. L. Wang, “Two-dimensional wavelength demultiplexing employing multilevel arrayed waveguides,” Optics Express 12,1084–1089 (2004).
[Crossref] [PubMed]

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Figures (5)

Fig. 1.
Fig. 1.

Layout of IAWGs, where the arrayed waveguides is irregularly spaced.

Fig. 2.
Fig. 2.

Layout of the designed IAWGs.

Fig. 3.
Fig. 3.

Transmission spectra of all 1080 channels with the wavelengths ranging from 1435 nm to 1650 nm and the channel spacing of 0.199 nm.

Fig. 4.
Fig. 4.

Transmission spectrum of an arbitrary channel.

Fig. 5.
Fig. 5.

The new work pattern with 128 successive channels that can be arbitrarily movable among the 1080 channels.

Tables (1)

Tables Icon

Table 1. Elementary parameters for the designed IAWG

Equations (49)

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

θ max = arcsin m Δ λ max n s d ,
n c Δ l r = m r λ 0 ,
n s d r = D m r ∙Δ λ max ,
θ main = arcsin ( m r Δλ n s d r ) = arcsin ( Δλ D Δ λ max ) .
f 0 ( x 0 , d i , v ) = b i v ( x 0 d i ) ,
f 1 ( x 1 ) = 1 α TF { f 0 ( x 0 ) } f = x 1 α = 1 α F 0 ( x 1 α ) ,
α = λL f n s ,
f 1 d ( x 1 ) = n = 0 N 1 a n b g ( x 1 ( r = 0 n d r d s ) ) ,
d s = ( r = 0 N 1 d r ) 2 ,
a n = f 1 ( x 1 ) b g ( x 1 ( r = 0 n d r d s ) ) dx 1 .
g 1 ( x ) = f 1 ( x ) * b g ( x ) = f 1 ( x 1 ) b g ( x 1 x ) dx 1 ,
a n = g 1 ( r = 0 n d r d s ) .
f 1 d ( x 1 ) = n = 0 N 1 g 1 ( r = 0 n d r d s ) b g ( x 1 ( r = 0 n d r d s ) )
= ( g 1 ( x 1 ) δ w ( x 1 ) ) * b g ( x 1 ) ,
g 1 ( x 1 ) = f 1 ( x 1 ) * b g ( x 1 ) ,
δ w ( x 1 ) = n = 0 N 1 δ ( x 1 ( r = 0 n d r d s ) ) .
l n = l 0 + r = 0 n Δ l r ,
Δ l r = m r λ 0 n c ,
Δϕ n = 2 π λ n c l n = 2 π λ ( n c l 0 + ( r = 0 n m r ) λ 0 ) .
f 2 ( x 2 ) = ( g 1 ( x 2 ) δ w ( x 2 ) ϕ x 2 v ) * b g ( x 2 ) .
ϕ x 2 v = exp [ j 2 π λ ( n c l 0 + ( x 2 + d s ) n s D Δ λ max λ 0 ) ] = ψ ( v ) exp [ j 2 πn s ( v v 0 ) x 2 D Δ λ max ] ,
ψ ( v ) = exp [ j 2 πv ( n c l 0 c + n s d s ( v 0 D Δ λ max ) ) ] ,
f 3 ( x 3 ) = 1 α TF { f 2 ( x 2 ) } f = x 3 α = 1 α F 2 ( x 3 α ) .
F 2 ( f ) = B g ( f ) ( G 1 ( f ) * Δ w ( f ) * Φ f v ) ,
B f ( f ) = TF { b g ( x 2 ) } ,
G 1 ( f ) = TF { f 1 ( x 2 ) * b g ( x 2 ) } = F 1 ( f ) B g ( f ) ,
Δ w ( f ) = TF { δ w ( x 2 ) } = n = 0 N 1 exp [ j 2 πf ( r = 0 n d r d s ) ] ,
Φ ( f ) = TF { ϕ x 2 v } = ψ ( v ) δ ( f + n s v v 0 D Δ λ max ) .
f 3 ( x 3 ) = 1 α α B g ( x 3 α ) ψ ( v ) n = 0 N 1 G 1 ( x 3 α + n s v v 0 D Δ λ max ) * exp [ j 2 π ( r = 0 n d r d s ) x 3 α ] .
F 1 ( f ) = TF { f 1 ( x 1 ) } = TF { 1 α F 0 ( x 1 α ) } = α f 0 ( αf ) ,
αF ( αf ) = TF { f ( x α ) } ,
f ( x ) = TF { TF { f ( x ) } } .
G 1 ( x 3 α ) = α f 0 ( x 3 ) B g ( x 3 α ) = α b i ( ( x 3 + d i ) ) B g ( x 3 α ) .
G 1 ( x 3 α ) α b i ( ( x 3 + d i ) ) B g ( d i α ) .
f 3 ( x 3 , d i , v ) = 1 α B g ( d i α ) B g ( x 3 α ) ψ ( v ) n = 0 N 1 h n ( x 3 + n s v 0 D Δ λ max + d i ) ,
h n ( x 3 ) = b i ( x 3 ) * exp [ j 2 π ( r = 0 n d r d s ) x 3 α ] .
t ( d i , d o , v ) = f 3 ( x 3 , d i , v ) b 0 ( x 3 d o ) dx 3 .
t ( d i , d o , v ) 1 α B g ( d i α ) B g ( d 0 α ) ψ ( v ) n = 0 N 1 h n ( x 3 + n s v 0 D Δλ max + d i ) b o ( x 3 d 0 ) dx 3 .
h n ( x 3 + n s v 0 D Δλ max + d i ) b o ( x 3 d o ) dx 3 = h n ( x 3 + n s v 0 D Δλ max + d i + d o ) b o ( x 3 ) dx 3 ,
t ( d i , d o , v ) = 1 α B g ( d i α ) B g ( d o α ) ψ ( v ) n = 0 N 1 q n ( n s v 0 D Δλ max + d i + d o ) ,
q n ( x ) = b o ( x ) * h n ( x ) = b o ( x 3 ) h n ( x x 3 ) dx 3 .
q n ( x ) = b i ( x ) * exp [ j 2 π ( r = 0 n d r d s ) x α ] * b o ( x ) .
t ( d i , d o , v ) = 1 α B g ( d i α ) B g ( d o α ) ψ ( v ) q ( n s v 0 D Δ λ max + d i + d o ) ,
q ( x ) = n = 0 N 1 q n ( x ) = b i ( x ) * n = 0 N 1 exp [ j 2 π ( r = 0 n d r d s ) x α ] * b o ( x ) .
b i , o , g = 2 π ω i , o , g 2 4 exp [ ( x ω i , o , g ) 2 ] ,
t ( d i , d o , v ) = 2 π ω g α exp [ ( π ω g α ) 2 ( d i 2 + d o 2 ) ] ψ ( v ) n = 0 N 1 q n ( λ 0 L f D Δ λ max + d i + d o ) ,
q n ( x ) = 2 π ω i ω o exp [ π 2 ( r = 0 n d r d s ) 2 n s 2 ( ω i 2 + ω o 2 ) λ 2 L f 2 ] × exp [ 2 πn s ( r = 0 n d t d s ) x λL f ] .
L u ( dB ) = 20 log 10 ( B g ( 0 ) B g ( ( C N 1 ) d out ( 2 α ) ) ) ,
θ max = ( C N 1 ) d out 2 L f .

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