## What else can an AWG do? |

Optics Express, Vol. 20, Issue 26, pp. B288-B298 (2012)

http://dx.doi.org/10.1364/OE.20.00B288

Acrobat PDF (1183 KB)

### Abstract

The present paper aims to describe other functionalities for an arrayed waveguide grating (AWG)-based device, showing that this widely used configuration can be designed not only to frequency multiplex/demultiplex wavelength division multiplexing (WDM) signals, but also to perform the discrete Fourier transform (DFT) and the discrete fractional Fourier transform (DFrFT) of a signal, in all-optical orthogonal frequency division multiplexing (OFDM) systems. In addition 1 *× N* and *N × N* phased array switches architectures are described, as well as a new configuration to perform polarization diversity demultiplexing. Finally, a general approach, based on an analogy with the finite impulse response (FIR) filter approach, is presented to design optical modulators for any modulation format, using either phase modulators (PM) or electro-absorption modulators (EAM).

© 2012 OSA

## 1. Introduction

*s(t)*is represented along the time axis, and the corresponding Fourier transform (FT)along the frequency axis; therefore, the Fourier transform operator

*F*can be viewed as a change in the representation of a signal corresponding to a π/2 counterclockwise axis rotation.

3. V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. **25**(3), 241–265 (1980). [CrossRef]

*s(t)*on an axis that forms an angle

*p*π/2, with 0<

*p*<2, so that the FrFT operator performs a rotation in the time-frequency plane of an angle

*p*π/2, as shown in Fig. 1 [4

4. L. Almeida, “The fractional Fourier transform and time-frequency representations,” Trans. Sig. Processing **42**(11), 3084–3091 (1994). [CrossRef]

*p*,

*p*= 1, the FrFT coincides with the canonical FT of the signal

*s(t)*of Eq. (1), evaluated at

*u = f*.

5. D. Mendlovic and H. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. **10**(9), 1875–1881 (1993). [CrossRef]

*L*. The medium can be regarded as consisting of infinitesimal layers uniformly distributed along the axial direction, and the FrFT can be physically defined as the functional form of field distribution measured after a distance

*pL*. On the other hand, the FrFT can be also obtained in a bulk two-lens optical system, with a proper choice of the focal length and propagation distance [6

6. H. Ozaktas and D. Mendlovic, “Fractional Fourier transforms and their optical implementation. II,” J. Opt. Soc. Am. **10**(12), 2522–2531 (1993). [CrossRef]

*N*and a

*N*×

*N*phased array switch.

## 2. Discrete Fourier transform

9. M. Smit, “New focusing and dispersive planar component based on an optical phased array,” Electron. Lett. **24**(7), 385–386 (1988). [CrossRef]

*R*equates the slab length

*l*, i.e. the distance between the two surfaces (Fig. 3(a) ). From a functional point of view, this system is equivalent to a bulk-optic configuration composed of two lenses with focal length

*R,*separated of a distance

*l*(Fig. 3(b)); in this case, the field amplitude distribution

*b(x)*at the output plane is proportional to the analog FT of the amplitude

*a(x)*on the input plane, evaluated at the spatial frequency

*x/(λl)*, where

*λ*is the central wavelength [10]

*m*-th output waveguide has the expressionwhere

*d*and

_{i}*d*are the pitches of the input and output waveguide gratings. In the previous expression, I have set

_{o}*N = λl/d*, to obtain the DFT. It is worth to observe that the design guidelines of an AWG to perform the DFT are similar to those corresponding to a conventional frequency demultiplexer [11]; the main difference is that the DFT device has

_{i}d_{o}*N*input/output ports, being

*N*the number of the waveguides in the grating. Although the condition

*N = λl/d*is exactly satisfied only at a single wavelength, the device can be used for broadband application [12]. In addition, the angular dispersion in the slab, due to the frequency dependence of the refractive index, causes a linear phase variation across the output of the grating array that must be carefully considered in the device design [13

_{i}d_{o}13. G. Cincotti, Naoya Wad, and K. Kitayama, “Characterization of a full encoder/decoder in the AWG configuration for code-based photonic routers-Part I: modeling and design,” J. Lightwave Technol. **24**(1), 103–112 (2006). [CrossRef]

14. N. Wada, G. Cincotti, S. Yoshima, N. Kataoka, and K.-i. Kitayama, “Characterization of a full encoder/decoder in the AWG configuration for code-based photonic routers-Part II: experimental results,” J. Lightwave Technol. **24**(1), 113–121 (2006). [CrossRef]

*N = λl/d*, is a 360°/

_{i}d_{o}*N*hybrid, and there is a fixed phase relation between the input and the output fields. For instance, by setting

*N*= 4, the slab coupler is a standard 90° hybrid.

*m*-th output (

*m = 0,1,2..N-1*) of the AWG device are respectively; here

*δ(t)*is the Dirac delta function, and

*T = Nτ*the symbol period. Alternatively, by using the sampling property of the delta function, Eq. (6) can be written aswhere the window function is

*rect*for

_{T}(t) = 1*–T/2<t<T/2*and zero otherwise. According to the convolution theorem, the transfer function can be also written as

15. G. Cincotti, “Design of optical full encoders/decoders for code-based photonic routers,” J. Lightwave Technol. **22**(7), 1642–1650 (2004). [CrossRef]

16. N. Kataoka, G. Cincotti, N. Wada, and K.-i. Kitayama, “Demonstration of asynchronous, 40 Gbps x 4-user DPSK-OCDMA transmission using a multi-port encoder/decoder,” Opt. Express **19**(26), B965–B970 (2011). [CrossRef] [PubMed]

17. A. J. Lowery, “Design of arrayed-waveguide grating routers for use as optical OFDM demultiplexers,” Opt. Express **18**(13), 14129–14143 (2010). [CrossRef] [PubMed]

## 3. Discrete fractional Fourier transform

*m*-th sub-channel waveform isand all the sub-carriers are orthogonal within a symbol period

*T*where * denotes complex conjugation and

*δ*is the Kronecker symbol. To move from an analog to a discrete (digital) representation and use the DFT, the temporal waveform

_{mm’}*t = nτ,*to obtain Eqs. (6) and (8).

19. G. Cincotti, “Generalized fiber Fourier optics,” Opt. Lett. **36**(12), 2321–2323 (2011). [CrossRef] [PubMed]

*m*= 0 and

*p*= 1/8. The FTs (spectra) of the FrFT sub-carriers areand if the parameter

*u*is selected asthen the sub-carriers satisfy the orthogonality condition of Eq. (11). It is evident that the FrFT waveforms coincide with conventional OFDM sub-carriers of Eq. (10) when

_{m}*p*= 1. Finally, sampling at

*t = nτ,*I obtainandthat is plotted in Fig. 4(d).

*m/T*and multiplied by a complex factor

*D*is the dispersion parameter,

*L*the fiber link length.

## 4. QAM modulator

*N*-level electrical waveforms, that drive a conventional IQ modulator. In this case, the high-speed electrical devices required are quite expensive and power hungry, and it is difficult to achieve high linearity and a wide dynamic range for the electrical signals. The second method requires only complementary two-level electrical signals, but a more complex PLC device composed of a nested configuration of Mach Zehnder interferometers (MZI) and a set of phase shifters: in general,

*N*MZIs are required for 2

^{N}–QAM modulation format, along with two trees of asymmetric splitters and a set of phase shifters [21]. For instance, the modulator of Fig. 5(a) generate 16-QAM optical signals, respectively, if the MZIs are driven by

*N*= 4 complementary two-level electrical signals ±

*V*(

_{n}*n*= 0,1,..

*N*-1).

*z*blocks of a FIR filter structure can be optically implemented by a passive 90° hybrid (or slab coupler with suitable parameters), and the filter coefficients a

^{−1}_{0}, a

_{1}= ± 1; a

_{2}, a

_{3}= ± 2, …, a

_{N-2},a

_{N-1}= ± 2

^{N/2-1}by an asymmetric star coupler, as shown in Fig. 5(b). The sign of the coefficients depends on the voltages

*V*(

_{n}*n*= 0,1,2,..

*N-1*) applied to the PM electrodes. The output signal is always taken at port

*m = 1*, so that the phase shifts introduced by the 90° hybrid are

*j*(

^{n}*n = 1,2,3,4*). It is important to observe also that these modulator layouts are completely flexible, so that a 16-QAM configuration can also generate 4-QAM modulation format, if only two ports are connected to the light source. It is also possible to replace PMs with EAMs, making the device layout more compact [22]. Finally, I observe that, with respect to other optical modulator configurations presented in literature, the novel architecture does not require MZIs but only straight PMs or EAMs (therefore the number of electrodes is halved), and the phase shifters are completely eliminated. The proposed architecture does not suffer for chirp effects, and its design is similar to the that one an AWG device to perform the DFT; also in this case, the angular dispersion in the slab, due to the frequency dependence of the refractive index, causes a linear phase variation across the output of the grating array.

## 5. Phased array switch

*N*-level electrical voltages

*V*(

_{n}*n*= 0,1,..

*N-1*), so that they are complex conjugate of the phase factors introduced by the slab couplerall the input power is directed to the

*m*-th port. A phased array switch was firstly designed and fabricated introduced by C. Doerr

*et al*. in 1999 [23

23. C. Doerr and C. Dragone, “Proposed optical cross connect using a planar arrangement of beam steerers,” Photon Technol. Lett. **11**(2), 197–199 (1999). [CrossRef]

*et al*. [24

24. T. Tanemura, M. Takenaka, A. Al Amin, K. Takeda, T. Shioda, M. Sugiyama, and Y. Nakano, “InP–InGaAsP integrated 1×5 optical switch using arrayed phase shifters,” Photon Technol. Lett. **20**(12), 1063–1065 (2008). [CrossRef]

*N*hybrid, and this is a great simplification of the PLC layout. In addition, thanks to the periodic behavior of the hybrid, with respect to the input port number, it is also possible to design a

*N*x

*N*switch configuration, as shown in Fig. 6(b).

## 6. PSK modulator

*z*In this case, I set

_{k}/(1-z_{k-1}).*z*or

_{1}, z_{2},…z_{N-1}= 0*1*; and the device layout is composed of a set of electro-optical (EO) switches and some 360°/

*N*hybrids. The 8-PSK constellation can be generated by the architecture of Fig. 7 , which is composed of three EO switches driven by the electrical voltages

*V*(

_{n}*n*= 0,1,2) and a 180°, 90° and a 45° hybrid. Also in this case, the schematic can be further generalized adding more hybrids and switches to increase the order of the PSK modulation or to reduce the number of switches. It is also possible to consider alternative architectures that integrate PMs [25

25. C. R. Doerr, G. Raybon, L. L. Liming Zhang, A. L. Buhl, J. H. Adamiecki, Sinsky, and N. J. Sauer, “Low-chirp 85-Gb/s duobinary modulator in InP using electroabsorption modulators,” Photon. Technol. Lett. **21**(17), 1199–1201 (2009). [CrossRef]

*z = exp(j2π/M)*, that should be adapted to different advanced modulation formats.

## 7. Polarization multiplexer

26. G. Cincotti, “Polarization gratings: design and applications,” J. Quantum Electron. **39**(12), 1645–1652 (2003). [CrossRef]

*m-*th output can be written as respectively, where

*x*and

*y*axes. It is evident that the polarization state of the signal depends on the output port number

*m*and that the output ports

*m*and

*m*+

*N/2*have orthogonal polarization states. Therefore, this device can be used to perform simultaneously the sub-channel decomposition of an OFDM signal, and dual polarization demultiplexing, as shown in Fig. 8 .

## 4. Conclusions

*N*and

*N*×

*N*phased array switch.

## Acknowledgment

## References and links

1. | I. Tomkos, P. Zakynthinos, E. Palkopoulou, M. Angelou, D. Klonidis, and S. B. Ezra, “Enabling technologies for evolving flexible/elastic optical transmission and expected benefits from their introduction in the networks,” in |

2. | W. Shieh and I. Djordjevic, |

3. | V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl. |

4. | L. Almeida, “The fractional Fourier transform and time-frequency representations,” Trans. Sig. Processing |

5. | D. Mendlovic and H. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. |

6. | H. Ozaktas and D. Mendlovic, “Fractional Fourier transforms and their optical implementation. II,” J. Opt. Soc. Am. |

7. | A. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. |

8. | H. M. Ozaktas, Z. Zalevsky, and M. Kutay, |

9. | M. Smit, “New focusing and dispersive planar component based on an optical phased array,” Electron. Lett. |

10. | J. Goodman, |

11. | C. Madsen and J. Zhao, |

12. | N. Kataoka, N. Wada, G. Cincotti, and K.-I. Kitayama, “2.56 Tbps (40-Gbps x 8-wavelengths 4-OC x 2-POL) asynchronous WDM-OCDMA-PON using a multi-port encoder/decoder,” in |

13. | G. Cincotti, Naoya Wad, and K. Kitayama, “Characterization of a full encoder/decoder in the AWG configuration for code-based photonic routers-Part I: modeling and design,” J. Lightwave Technol. |

14. | N. Wada, G. Cincotti, S. Yoshima, N. Kataoka, and K.-i. Kitayama, “Characterization of a full encoder/decoder in the AWG configuration for code-based photonic routers-Part II: experimental results,” J. Lightwave Technol. |

15. | G. Cincotti, “Design of optical full encoders/decoders for code-based photonic routers,” J. Lightwave Technol. |

16. | N. Kataoka, G. Cincotti, N. Wada, and K.-i. Kitayama, “Demonstration of asynchronous, 40 Gbps x 4-user DPSK-OCDMA transmission using a multi-port encoder/decoder,” Opt. Express |

17. | A. J. Lowery, “Design of arrayed-waveguide grating routers for use as optical OFDM demultiplexers,” Opt. Express |

18. | S. Shimotsu, G. Cincotti, and N. Wada, “Demonstration of a 8x12.5 Gbit/s all-optical OFDM system with an arrayed waveguide grating and waveform reshaper,” in |

19. | G. Cincotti, “Generalized fiber Fourier optics,” Opt. Lett. |

20. | G. Cincotti, “Optical OFDM based on the fractional Fourier transform,” in |

21. | H. Yamazaki, T. Yamada, T. Goh, and S. Mino, “Multilevel optical modulator with PLC and LiNbO3 hybrid integrated circuit,” in |

22. | C. Doerr, P. Winzer, L. Zhang, L. Buhl, and N. Sauer, “Monolithic InP 16-QAM modulator,” in |

23. | C. Doerr and C. Dragone, “Proposed optical cross connect using a planar arrangement of beam steerers,” Photon Technol. Lett. |

24. | T. Tanemura, M. Takenaka, A. Al Amin, K. Takeda, T. Shioda, M. Sugiyama, and Y. Nakano, “InP–InGaAsP integrated 1×5 optical switch using arrayed phase shifters,” Photon Technol. Lett. |

25. | C. R. Doerr, G. Raybon, L. L. Liming Zhang, A. L. Buhl, J. H. Adamiecki, Sinsky, and N. J. Sauer, “Low-chirp 85-Gb/s duobinary modulator in InP using electroabsorption modulators,” Photon. Technol. Lett. |

26. | G. Cincotti, “Polarization gratings: design and applications,” J. Quantum Electron. |

**OCIS Codes**

(230.7390) Optical devices : Waveguides, planar

(230.7400) Optical devices : Waveguides, slab

(250.7360) Optoelectronics : Waveguide modulators

(070.2025) Fourier optics and signal processing : Discrete optical signal processing

(250.6715) Optoelectronics : Switching

**ToC Category:**

Waveguide and Optoelectronic Devices

**History**

Original Manuscript: October 1, 2012

Revised Manuscript: November 5, 2012

Manuscript Accepted: November 5, 2012

Published: November 29, 2012

**Virtual Issues**

European Conference on Optical Communication 2012 (2012) *Optics Express*

**Citation**

Gabriella Cincotti, "What else can an AWG do?," Opt. Express **20**, B288-B298 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-26-B288

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### References

- I. Tomkos, P. Zakynthinos, E. Palkopoulou, M. Angelou, D. Klonidis, and S. B. Ezra, “Enabling technologies for evolving flexible/elastic optical transmission and expected benefits from their introduction in the networks,” in Photonics in Switching (PS) 2012.
- W. Shieh and I. Djordjevic, OFDM for Optical Communications (Elsevier, 2010).
- V. Namias, “The fractional order Fourier transform and its application to quantum mechanics,” J. Inst. Math. Appl.25(3), 241–265 (1980). [CrossRef]
- L. Almeida, “The fractional Fourier transform and time-frequency representations,” Trans. Sig. Processing42(11), 3084–3091 (1994). [CrossRef]
- D. Mendlovic and H. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am.10(9), 1875–1881 (1993). [CrossRef]
- H. Ozaktas and D. Mendlovic, “Fractional Fourier transforms and their optical implementation. II,” J. Opt. Soc. Am.10(12), 2522–2531 (1993). [CrossRef]
- A. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am.10(10), 2181–2186 (1993). [CrossRef]
- H. M. Ozaktas, Z. Zalevsky, and M. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).
- M. Smit, “New focusing and dispersive planar component based on an optical phased array,” Electron. Lett.24(7), 385–386 (1988). [CrossRef]
- J. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1988), Chap 5.
- C. Madsen and J. Zhao, Optical Filter Design and Analysis: A Signal Processing Approach, Par. 4.4.2 (John Wiley & Sons, 1999).
- N. Kataoka, N. Wada, G. Cincotti, and K.-I. Kitayama, “2.56 Tbps (40-Gbps x 8-wavelengths 4-OC x 2-POL) asynchronous WDM-OCDMA-PON using a multi-port encoder/decoder,” in European Conference on Optical Communication (ECOC) postdeadline paper 2011.
- G. Cincotti, Naoya Wad, and K. Kitayama, “Characterization of a full encoder/decoder in the AWG configuration for code-based photonic routers-Part I: modeling and design,” J. Lightwave Technol.24(1), 103–112 (2006). [CrossRef]
- N. Wada, G. Cincotti, S. Yoshima, N. Kataoka, and K.-i. Kitayama, “Characterization of a full encoder/decoder in the AWG configuration for code-based photonic routers-Part II: experimental results,” J. Lightwave Technol.24(1), 113–121 (2006). [CrossRef]
- G. Cincotti, “Design of optical full encoders/decoders for code-based photonic routers,” J. Lightwave Technol.22(7), 1642–1650 (2004). [CrossRef]
- N. Kataoka, G. Cincotti, N. Wada, and K.-i. Kitayama, “Demonstration of asynchronous, 40 Gbps x 4-user DPSK-OCDMA transmission using a multi-port encoder/decoder,” Opt. Express19(26), B965–B970 (2011). [CrossRef] [PubMed]
- A. J. Lowery, “Design of arrayed-waveguide grating routers for use as optical OFDM demultiplexers,” Opt. Express18(13), 14129–14143 (2010). [CrossRef] [PubMed]
- S. Shimotsu, G. Cincotti, and N. Wada, “Demonstration of a 8x12.5 Gbit/s all-optical OFDM system with an arrayed waveguide grating and waveform reshaper,” in European Conference on Optical Communications (ECOC) 2012 Th.1.A.2.
- G. Cincotti, “Generalized fiber Fourier optics,” Opt. Lett.36(12), 2321–2323 (2011). [CrossRef] [PubMed]
- G. Cincotti, “Optical OFDM based on the fractional Fourier transform,” in Proc. SPIE Photonic West, 8284–08, 2012.
- H. Yamazaki, T. Yamada, T. Goh, and S. Mino, “Multilevel optical modulator with PLC and LiNbO3 hybrid integrated circuit,” in Optical Fiber Communication Conference and Exposition (OFC) 2011.
- C. Doerr, P. Winzer, L. Zhang, L. Buhl, and N. Sauer, “Monolithic InP 16-QAM modulator,” in Optical Fiber Communication Conference and Exposition (OFC) 2008 PDP20.
- C. Doerr and C. Dragone, “Proposed optical cross connect using a planar arrangement of beam steerers,” Photon Technol. Lett.11(2), 197–199 (1999). [CrossRef]
- T. Tanemura, M. Takenaka, A. Al Amin, K. Takeda, T. Shioda, M. Sugiyama, and Y. Nakano, “InP–InGaAsP integrated 1×5 optical switch using arrayed phase shifters,” Photon Technol. Lett.20(12), 1063–1065 (2008). [CrossRef]
- C. R. Doerr, G. Raybon, L. L. Liming Zhang, A. L. Buhl, J. H. Adamiecki, Sinsky, and N. J. Sauer, “Low-chirp 85-Gb/s duobinary modulator in InP using electroabsorption modulators,” Photon. Technol. Lett.21(17), 1199–1201 (2009). [CrossRef]
- G. Cincotti, “Polarization gratings: design and applications,” J. Quantum Electron.39(12), 1645–1652 (2003). [CrossRef]

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