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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 14 — Jul. 4, 2011
  • pp: 13590–13597
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System tolerance of all-optical sampling OFDM using AWG discrete Fourier transform

Seong-Jin Lim and June-Koo Kevin Rhee  »View Author Affiliations


Optics Express, Vol. 19, Issue 14, pp. 13590-13597 (2011)
http://dx.doi.org/10.1364/OE.19.013590


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Abstract

The fundamental-mode arrayed waveguide grating (AWG) for all-optical discrete Fourier transformer (DFT) shows significant feasibility in the system tolerance of all-optical sampling orthogonal frequency division multiplexing (AOS-OFDM) systems. We discuss the system tolerance of AWG-based DFT designs for 100/160Gbps OFDM transmission system in comparison with coupler-based DFT designs.

© 2011 OSA

1. Introduction

2. Phase errors in all-optical DFT processors

An AOS-OFDM multiplexer and demultiplexer consist of two functions: DFT and optical delay array [4

4. K. Lee, C. T. D. Thai, and J.-K. K. Rhee, “All optical discrete Fourier transform processor for 100 Gbps OFDM transmission,” Opt. Express 16(6), 4023–4028 (2008). [CrossRef] [PubMed]

]. The DFT of time-domain sequence εn with a sampling size of N is defined by Ek=1Nn=0N1εne2πikn/N,   0k<N, and the corresponding IDFT of frequency-domain sequence Ek is given byεn=1Nk=0N1Eke2πikn/N, 0n<N, where k and n are integer. Considering the mathematical expression of DFT, we can implement that a DFT processor with N sampling ports introduces fixed phase changes of εn,k=1NEkei2πkn/Nand Ek,n=1Nεnei2πkn/N for inverse and forward DFT, respectively. Here, εn,k is the n-th time domain sample for the k-th inputEkfor an inverse DFT. Similarly, Ek,nis the k-th frequency domain output for the n-th inputεnfor a forward DFT. In order to realize the time domain representation as a waveform, we require an array of time delays, that is, nΔTfor the n-th time-domain port. Here, ΔT=T/N where T is the symbol period.

For a realization of an all-optical DFT, one can employ an optical DFT circuit design introduced by Marhic, which consists of 2x2 couplers and phase shifters [12

12. M. E. Marhic, “Discrete Fourier transforms by single-mode star networks,” Opt. Lett. 12(1), 63–65 (1987). [CrossRef] [PubMed]

14

14. Y.-K. Huang, R. Saperstein, and T. Wang, “All-optical OFDM transmission with coupler-based IFFT/FFT and pulse interleaving,” in proceedings of IEEE conference on Lasers and Electro-Optics Society (Institute of Electrical and Electronics Engineers, Acapulco, 2008), pp.408–409.

]. Figure 1(a)
Fig. 1 Schematic illustration of phase errors of IDFTs and DFTs in coupler-based Marhic circuits (a) and fundamental-mode AWGs (b).
shows 4x4 coupler-based IDFT and DFT circuits as an example for multiplexing and demultiplexing 4-subcarrier OFDM symbols [6

6. W. Li, X. Liang, W. Ma, T. Zhou, B. Huang, and D. Liu, “A planar waveguide optical discrete Fourier transformer design for 160 Gb/s all-optical OFDM systems,” Opt. Fiber Technol. 16(1), 5–11 (2010). [CrossRef]

]. This design allows for the simplest optical circuit design with O(NlogN) scalability, where N is the number of optical OFDM subcarriers. However, in manufacturing, every waveguide between couplers always has a certain phase error, which is indicated with a yellow marker in Fig. 1(a), caused by little difference of waveguide width and length. As the coupler-based Marhic DFT circuits utilize complex interferences in a network of couplers, phase error can break the inter-subcarrier orthogonality in an AOS-OFDM system. In the case of an NxN coupler-based DFT, the number of phase error impairments grows with 2NlogN. As a result, performance evaluation shows serious penalties due to phase errors in the Marhic DFT [7

7. S. Lim and J.-K. K. Rhee, “System performance of 2x2 coupler-based all-optical OFDM System,” in Photonics in Switching, OSA Technical Digest (CD) (Optical Society of America, 2010), paper PWF2. http://www.opticsinfobase.org/abstract.cfm?URI=PS-2010-PWF2

]. A practical AOS-OFDM system may require an active phase error compensation technique [7

7. S. Lim and J.-K. K. Rhee, “System performance of 2x2 coupler-based all-optical OFDM System,” in Photonics in Switching, OSA Technical Digest (CD) (Optical Society of America, 2010), paper PWF2. http://www.opticsinfobase.org/abstract.cfm?URI=PS-2010-PWF2

]. Precise insertion loss control of each waveguide is also important to realize the orthogonality because the inter-subcarrier crosstalk cancelling by destructive interference among multiple light circuit paths becomes incomplete for uneven loss. However, the phase error gives more critical penalty than loss uniformity error [7

7. S. Lim and J.-K. K. Rhee, “System performance of 2x2 coupler-based all-optical OFDM System,” in Photonics in Switching, OSA Technical Digest (CD) (Optical Society of America, 2010), paper PWF2. http://www.opticsinfobase.org/abstract.cfm?URI=PS-2010-PWF2

]. In this paper, we focus on the manufacturability study of optical DFT devices from the point of view of waveguide phase tolerance.

A fundamental mode AWG (FM-AWG) employs N arrayed waveguides for N subcarrier channels between two slab waveguides which are two-dimensional free propagation regions (FPR), as shown in Fig. 1(b). A careful design of waveguide ports on a slab waveguide can achieve the DFT phase relations [15

15. G. Cincotti, N. Wad, and K. Kitayama, “Characterization of a full encoder/decoder in the AWG configuration for code-based photonic routers,” J. Lightwave Technol. 24(1), 103–112 (2006). [CrossRef]

17

17. Z. Wang, K. S. Kravtsov, Y.-K. Huang, and P. R. Prucnal, “Optical FFT/IFFT circuit realization using arrayed waveguide gratings and the applications in all-optical OFDM system,” Opt. Express 19(5), 4501–4512 (2011). [CrossRef] [PubMed]

]. The length of waveguides vary to produce propagation time delays, nΔT, n = 1… N, to multiplex and demultiplex an OFDM symbol [18

18. K. Takiguchi, T. Kitoh, A. Mori, M. Oguma, and H. Takahashi, “Optical orthogonal frequency division multiplexing demultiplexer using slab star coupler-based optical discrete Fourier transform circuit,” Opt. Lett. 36(7), 1140–1142 (2011). [CrossRef] [PubMed]

]. In this design, the number of phase errors for N subcarriers is only N, so that we can expect less impact on inter-subcarrier crosstalk between subcarrier channels.

Figure 2(a)
Fig. 2 Power transfer functions of proposed 4x25GHz all-optical DFT for the ideal case (a), and typical examples with phase errors in coupler-based DFT (b) and in FM-AWG (c), respectively, according to the phase error models in Fig. 1.
shows an example of the ideal power transfer function of a 4x25 GHz all-optical DFT. The transfer function of each subcarrier port is a sinc function and the center of neighbor subcarriers are situated at null point of one another. As shown in the figure, individual subcarriers are partially overlapped with one another due to significant side lobes over the main lobes of other subcarrier channels. However, the interference due to side-lobe crosstalk is canceled by the OFDM property.

In order to investigate the statistical impact of waveguide phase errors indicated with yellow markers in Fig. 1, which is represented by phasor eiφm, on all-optical DFT circuits, we assume that the distribution of phase errors is a truncated Gaussian distribution with mean value of zero [8

8. Y. Chu, X. O. Zheng, H. Zhang, X. Liu, and Y. Guo, “The impact of phase errors on arrayed waveguide gratings,” IEEE J. Sel. Top. Quantum Electron. 8(6), 1122–1129 (2002). [CrossRef]

]:
f(φm)={A   eφm2/2σ2  ,             |φm|2σ           0           ,                 otherwise .
(1)
Here, random variable φm is a phase error of the m-th waveguide and σ is a standard deviation of Gaussian distribution. It is noted that the range of Gaussian random variable is limited to twice of standard deviation, which is assumed as a manufacturing tolerance, for practical modeling. Constant coefficient A normalizes the distribution, such that 2σ+2σf(x)dx=1. Figures 2(b) and 2(c) are the power transfer functions of a coupler-based IDFT and FM-AWG IDFT. Each of these figures is a typical sample case with random phase errors with a standard deviation of λ/20. The transfer function of the FM-AWG shows slight difference from the ideal case but that of the coupler-based IDFT shows significant crosstalks that are responsible for orthogonality degradation. In Fig. 2, the maximum crosstalk value of FM-AWG IDFT is −46 dB at 0 GHz, while that of coupler-based IDFT is −27 dB. This comparison motivates a study for practical implementation with FM-AWG devices. Insertion loss non-uniformity error also introduces penalties on the OFDM process in both types of DFT devices [7

7. S. Lim and J.-K. K. Rhee, “System performance of 2x2 coupler-based all-optical OFDM System,” in Photonics in Switching, OSA Technical Digest (CD) (Optical Society of America, 2010), paper PWF2. http://www.opticsinfobase.org/abstract.cfm?URI=PS-2010-PWF2

,16

16. A. J. Lowery, “Design of Arrayed-Waveguide Grating Routers for use as optical OFDM demultiplexers,” Opt. Express 18(13), 14129–14143 (2010). [CrossRef] [PubMed]

]. However, the phase error penalty is much more critical than insertion loss error in the limit of manufacturing tolerance In our study, we focus only on the phase tolerance and the corresponding manufacturability with reasonable phase error control by post-fabrication compensation techniques [10

10. C. R. Doerr and K. Okamoto, “Advances in silica planar lightwave circuits,” J. Lightwave Technol. 24(12), 4763–4789 (2006). [CrossRef]

,11

11. M. K. Smit and C. Van Dam, “PHASAR-based WDM-devices: principles, design and applications,” IEEE J. Sel. Top. Quantum Electron. 2(2), 236–250 (1996). [CrossRef]

].

3. Comparison of system performance between fundamental mode AWG and coupler-based AOS-OFDM system

The principle idea of all-optical OFDM is that time-varying phase in a symbol period can introduce frequency shift, which is rendered by passing through an inverse DFT circuit [4

4. K. Lee, C. T. D. Thai, and J.-K. K. Rhee, “All optical discrete Fourier transform processor for 100 Gbps OFDM transmission,” Opt. Express 16(6), 4023–4028 (2008). [CrossRef] [PubMed]

]. Ideal phase rotations in the waveforms after an inverse DFT circuit of a 4x25 Gbps OFDM system are shown in Fig. 4
Fig. 4 Waveform examples of 4x25 Gbps OFDM symbols generated with and without random phase errors. Phase rotations are measured with respect to the optical carrier frequency of 0 GHz.
. As for the subcarrier frequency of 0GHz, there is no phase change in time as shown in the ideal case. Subcarriers of ± 25 GHz and 50GHz have maximum phase changes of 1.5π and 3π, respectively, as indicated by red curves. Figure 4 also shows the phase rotation errors of FM-AWG and coupler-based IDFTs with phase errors, which is responsible for waveform distortion. As expected by the transfer function of Fig. 2, a coupler-based IDFT introduces much larger distortion than FM-AWG IDFT. For performance comparison between coupler-based and FM-AWG AOS-OFDM, we have modeled 4-subcarrier 100Gbps and 8-subcarrier 160Gbps AOS-OFDM systems using MATLAB simulations.

In order to investigate phase accuracy requirement based on the system penalty, we introduced random phase errors on every waveguide in both coupler-based and FM-AWG-based IDFTs and FDFTs according to the manufacturing uncertainty model of Eq. (1). The standard deviation of phase errors that are caused by size variations in core dimensions and refractive index variations of cores and claddings is assumed to vary from λ/150 to λ/15. This manufacturing accuracy is typically attainable in the state-of-the-art silica and polymer planar waveguide light circuit manufacturing [9

9. T. Goh, S. Suzuki, and A. Sugita, “Estimation of waveguide phase error in silica-based waveguides,” J. Lightwave Technol. 15(11), 2107–2113 (1997). [CrossRef]

]. In order to evaluate Q factor statistics, we generate 1600 and 4800 different sets of random phases for 4-by-4 and 8-by-8 coupler-based circuit, respectively. In the case of FM-AWG, 800 and 1600 random phases are generated for 4 channel and 8 channel AOS-OFDM system. The reference Q factor of phase-error-free DFTs Q is adjusted to 8.5 dB (in 10 log10 Q) by adjusting the OSNR levels. This reference provides a reasonable OSNR requirement. In Fig. 5(a)
Fig. 5 System performance of the two AOS-OFDM systems (a) and the yield curves of FM-AWG-based 8x20 Gbps AOS-OFDM system (b).
, we have plotted the average Q-factor as a function of the standard deviation of phase error distribution in radian. Initially, Q values of all cases decrease slowly as the phase errors increase, but when phase errors are large, the Q penalty becomes more sensitive to phase error, Besides, in the case of an 8-channel coupler-based OFDM, Q value is about 2.1dB less than an 8-channel FM-AWG-based OFDM system at a phase error standard deviation of λ/20 as compared in the figure. In this result, an FM-AWG-based AOS-OFDM system shows lower sensitivity to phase error compared with a coupler-based system.

There are two reasons that an FM-AWG AOS-OFDM device is less sensitive to phase error. First, the number of waveguides which can cause phase errors in an FM-AWG AOS OFDM is N, which is less than that of a coupler-based DFT circuit, Nlog2 N. The second reason comes from the differences in designs and operation principles of the DFT processes. In the case of coupler-based DFT, it utilizes the lightwave interference between two waveguides and the corresponding interferences are nested [7

7. S. Lim and J.-K. K. Rhee, “System performance of 2x2 coupler-based all-optical OFDM System,” in Photonics in Switching, OSA Technical Digest (CD) (Optical Society of America, 2010), paper PWF2. http://www.opticsinfobase.org/abstract.cfm?URI=PS-2010-PWF2

]. Therefore, optical data couple and interfere with one another at all couplers in a DFT circuit. Hence, the crosstalk components are produced at all couplers by propagating through all of them. On the other hand, an FM-AWG utilizes imaging optics. In this device, after propagating arrayed waveguides, light beams are radiated to the second FPR where all light beams from all outputs of arrayed waveguides interfere constructively and focus at a designated port in the second FPR. Thus phase errors from arrayed waveguides decrease by an average-out effect at each port, even though the output port coupling and the corresponding optical data amplitude change. Therefore, an FM-AWG is relatively less sensitive to phase error. As a result, there is no doubt to conclude that an FM-AWG is more practical and acceptable for applications in an AOS-OFDM system, compared with a coupler-based DFT. Figure 5(b) presents the manufacturing yield curves of an 8-channel FM-AWG example as a function of penalty tolerance. Here, 800 sets of phase error random variables are created and their system performances are evaluated. We count the fraction of qualified system performance. In the figure, the yield of 100% means that the Q-values of all devices with phase errors are higher than 8.5dB, which corresponds to a 1012 BER. For example, if the penalty tolerance is 1dB, the yield indicates the fraction of qualified devices that can achieve system performance with Q-values higher than 7.5dB. This yield curve also shows that FM-AWG is resilient against device manufacturing uncertainty by the measure of AOS-OFDM system performance. Note that this result delivers the manufacturability against waveguide phase error control only.

4. Conclusion

In this paper, we compare system performances of fundamental mode AWG and coupler-based DFT for AOS-OFDM systems. By employing FM-AWG in the AOS-OFDM system, we can minimize the essential problem that an AOS-OFDM system is extremely sensitive to phase errors in DFT circuits. Such errors introduce OFDM transfer function errors in the frequency domain and consequently the time-domain waveform distortion, resulting in degradation of inter-subcarrier orthogonality. The corresponding system performance comparison between FM-AWG and coupler-based AOS-OFDM systems is evaluated using numerical simulations. The simulation result shows that an FM AWG-based OFDM system is less sensitive to phase error, so that it can achieve the more reliable system performance compared with a coupler based AOS-OFDM system. Our statistical modeling results suggest that an AWG-based AOS-OFDM system is a good candidate to deliver a practical system application in all-optical sampling OFDM transmission in the sense of manufacturing feasibility.

Acknowledgment

This work was supported by the IT R&D program of MKE/IITA [2008-F017-02, 100Gbps Ethernet and optical transmission technology development].

References and links

1.

J. Armstrong, “OFDM for optical communication,” J. Lightwave Technol. 27(3), 189–204 (2009). [CrossRef]

2.

Y. Benlachtar, P. M. Watts, R. Bouziane, P. Milder, D. Rangaraj, A. Cartolano, R. Koutsoyannis, J. C. Hoe, M. Püschel, M. Glick, and R. I. Killey, “Generation of optical OFDM signals using 21.4 GS/s real time digital signal processing,” Opt. Express 17(20), 17658–17668 (2009). [CrossRef] [PubMed]

3.

A. Sano, H. Masuda, E. Yoshida, T. Kobayashi, E. Yamada, Y. Miyamoto, F. Inuzuka, Y. Hibino, Y. Takatori, K. Hagimoto, T. Yamada, and Y. Sakamaki, “30x100-Gb/s all-optical OFDM transmission over 1300 km SMF with 10 ROADM nodes,” in Proceedings of IEEE Conference on 33th European Conference on Optical Communication (Institute of Electrical and Electronics Engineers, Berlin, 2007), Paper PDS1.7.

4.

K. Lee, C. T. D. Thai, and J.-K. K. Rhee, “All optical discrete Fourier transform processor for 100 Gbps OFDM transmission,” Opt. Express 16(6), 4023–4028 (2008). [CrossRef] [PubMed]

5.

H. Chen, M. Chen, and S. Xie, “All-optical sampling orthogonal frequency-division multiplexing Scheme for High-Speed Transmission System,” J. Lightwave Technol. 27(21), 4848–4854 (2009). [CrossRef]

6.

W. Li, X. Liang, W. Ma, T. Zhou, B. Huang, and D. Liu, “A planar waveguide optical discrete Fourier transformer design for 160 Gb/s all-optical OFDM systems,” Opt. Fiber Technol. 16(1), 5–11 (2010). [CrossRef]

7.

S. Lim and J.-K. K. Rhee, “System performance of 2x2 coupler-based all-optical OFDM System,” in Photonics in Switching, OSA Technical Digest (CD) (Optical Society of America, 2010), paper PWF2. http://www.opticsinfobase.org/abstract.cfm?URI=PS-2010-PWF2

8.

Y. Chu, X. O. Zheng, H. Zhang, X. Liu, and Y. Guo, “The impact of phase errors on arrayed waveguide gratings,” IEEE J. Sel. Top. Quantum Electron. 8(6), 1122–1129 (2002). [CrossRef]

9.

T. Goh, S. Suzuki, and A. Sugita, “Estimation of waveguide phase error in silica-based waveguides,” J. Lightwave Technol. 15(11), 2107–2113 (1997). [CrossRef]

10.

C. R. Doerr and K. Okamoto, “Advances in silica planar lightwave circuits,” J. Lightwave Technol. 24(12), 4763–4789 (2006). [CrossRef]

11.

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

12.

M. E. Marhic, “Discrete Fourier transforms by single-mode star networks,” Opt. Lett. 12(1), 63–65 (1987). [CrossRef] [PubMed]

13.

A. E. Siegman, “Fiber Fourier optics,” Opt. Lett. 26(16), 1215–1217 (2001). [CrossRef] [PubMed]

14.

Y.-K. Huang, R. Saperstein, and T. Wang, “All-optical OFDM transmission with coupler-based IFFT/FFT and pulse interleaving,” in proceedings of IEEE conference on Lasers and Electro-Optics Society (Institute of Electrical and Electronics Engineers, Acapulco, 2008), pp.408–409.

15.

G. Cincotti, N. Wad, and K. Kitayama, “Characterization of a full encoder/decoder in the AWG configuration for code-based photonic routers,” J. Lightwave Technol. 24(1), 103–112 (2006). [CrossRef]

16.

A. J. Lowery, “Design of Arrayed-Waveguide Grating Routers for use as optical OFDM demultiplexers,” Opt. Express 18(13), 14129–14143 (2010). [CrossRef] [PubMed]

17.

Z. Wang, K. S. Kravtsov, Y.-K. Huang, and P. R. Prucnal, “Optical FFT/IFFT circuit realization using arrayed waveguide gratings and the applications in all-optical OFDM system,” Opt. Express 19(5), 4501–4512 (2011). [CrossRef] [PubMed]

18.

K. Takiguchi, T. Kitoh, A. Mori, M. Oguma, and H. Takahashi, “Optical orthogonal frequency division multiplexing demultiplexer using slab star coupler-based optical discrete Fourier transform circuit,” Opt. Lett. 36(7), 1140–1142 (2011). [CrossRef] [PubMed]

OCIS Codes
(060.0060) Fiber optics and optical communications : Fiber optics and optical communications
(060.4230) Fiber optics and optical communications : Multiplexing
(070.7145) Fourier optics and signal processing : Ultrafast processing

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: May 17, 2011
Revised Manuscript: June 18, 2011
Manuscript Accepted: June 18, 2011
Published: June 29, 2011

Citation
Seong-Jin Lim and June-Koo Kevin Rhee, "System tolerance of all-optical sampling OFDM using AWG discrete Fourier transform," Opt. Express 19, 13590-13597 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-14-13590


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References

  1. J. Armstrong, “OFDM for optical communication,” J. Lightwave Technol. 27(3), 189–204 (2009). [CrossRef]
  2. Y. Benlachtar, P. M. Watts, R. Bouziane, P. Milder, D. Rangaraj, A. Cartolano, R. Koutsoyannis, J. C. Hoe, M. Püschel, M. Glick, and R. I. Killey, “Generation of optical OFDM signals using 21.4 GS/s real time digital signal processing,” Opt. Express 17(20), 17658–17668 (2009). [CrossRef] [PubMed]
  3. A. Sano, H. Masuda, E. Yoshida, T. Kobayashi, E. Yamada, Y. Miyamoto, F. Inuzuka, Y. Hibino, Y. Takatori, K. Hagimoto, T. Yamada, and Y. Sakamaki, “30x100-Gb/s all-optical OFDM transmission over 1300 km SMF with 10 ROADM nodes,” in Proceedings of IEEE Conference on 33th European Conference on Optical Communication (Institute of Electrical and Electronics Engineers, Berlin, 2007), Paper PDS1.7.
  4. K. Lee, C. T. D. Thai, and J.-K. K. Rhee, “All optical discrete Fourier transform processor for 100 Gbps OFDM transmission,” Opt. Express 16(6), 4023–4028 (2008). [CrossRef] [PubMed]
  5. H. Chen, M. Chen, and S. Xie, “All-optical sampling orthogonal frequency-division multiplexing Scheme for High-Speed Transmission System,” J. Lightwave Technol. 27(21), 4848–4854 (2009). [CrossRef]
  6. W. Li, X. Liang, W. Ma, T. Zhou, B. Huang, and D. Liu, “A planar waveguide optical discrete Fourier transformer design for 160 Gb/s all-optical OFDM systems,” Opt. Fiber Technol. 16(1), 5–11 (2010). [CrossRef]
  7. S. Lim and J.-K. K. Rhee, “System performance of 2x2 coupler-based all-optical OFDM System,” in Photonics in Switching, OSA Technical Digest (CD) (Optical Society of America, 2010), paper PWF2. http://www.opticsinfobase.org/abstract.cfm?URI=PS-2010-PWF2
  8. Y. Chu, X. O. Zheng, H. Zhang, X. Liu, and Y. Guo, “The impact of phase errors on arrayed waveguide gratings,” IEEE J. Sel. Top. Quantum Electron. 8(6), 1122–1129 (2002). [CrossRef]
  9. T. Goh, S. Suzuki, and A. Sugita, “Estimation of waveguide phase error in silica-based waveguides,” J. Lightwave Technol. 15(11), 2107–2113 (1997). [CrossRef]
  10. C. R. Doerr and K. Okamoto, “Advances in silica planar lightwave circuits,” J. Lightwave Technol. 24(12), 4763–4789 (2006). [CrossRef]
  11. M. K. Smit and C. Van Dam, “PHASAR-based WDM-devices: principles, design and applications,” IEEE J. Sel. Top. Quantum Electron. 2(2), 236–250 (1996). [CrossRef]
  12. M. E. Marhic, “Discrete Fourier transforms by single-mode star networks,” Opt. Lett. 12(1), 63–65 (1987). [CrossRef] [PubMed]
  13. A. E. Siegman, “Fiber Fourier optics,” Opt. Lett. 26(16), 1215–1217 (2001). [CrossRef] [PubMed]
  14. Y.-K. Huang, R. Saperstein, and T. Wang, “All-optical OFDM transmission with coupler-based IFFT/FFT and pulse interleaving,” in proceedings of IEEE conference on Lasers and Electro-Optics Society (Institute of Electrical and Electronics Engineers, Acapulco, 2008), pp.408–409.
  15. G. Cincotti, N. Wad, and K. Kitayama, “Characterization of a full encoder/decoder in the AWG configuration for code-based photonic routers,” J. Lightwave Technol. 24(1), 103–112 (2006). [CrossRef]
  16. A. J. Lowery, “Design of Arrayed-Waveguide Grating Routers for use as optical OFDM demultiplexers,” Opt. Express 18(13), 14129–14143 (2010). [CrossRef] [PubMed]
  17. Z. Wang, K. S. Kravtsov, Y.-K. Huang, and P. R. Prucnal, “Optical FFT/IFFT circuit realization using arrayed waveguide gratings and the applications in all-optical OFDM system,” Opt. Express 19(5), 4501–4512 (2011). [CrossRef] [PubMed]
  18. K. Takiguchi, T. Kitoh, A. Mori, M. Oguma, and H. Takahashi, “Optical orthogonal frequency division multiplexing demultiplexer using slab star coupler-based optical discrete Fourier transform circuit,” Opt. Lett. 36(7), 1140–1142 (2011). [CrossRef] [PubMed]

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