## Analysis of optical-signal processing using an arrayed-waveguide grating

Optics Express, Vol. 6, Issue 6, pp. 124-135 (2000)

http://dx.doi.org/10.1364/OE.6.000124

Acrobat PDF (273 KB)

### Abstract

We analyzed optical-signal processing based on time-space conversion in an arrayed-waveguide grating (AWG). General expressions for the electric fields needed to design frequency filters were obtained. We took into account the effects of the waveguides and clearly distinguished the temporal frequency axis from the spatial axis at the focal plane, at which frequency filters were placed. Using the analytical results, we identified the factors limiting the input-pulse width and clarified the windowing effect and the effect ofphase fluctuation in the arrayed waveguide.

© Optical Society of America

## 1. Introduction

1. A. M. Weiner, “Femtosecond optical pulse shaping and processing,” Prog. Quant. Electron. **19**, 161–237 (1995). [CrossRef]

1. A. M. Weiner, “Femtosecond optical pulse shaping and processing,” Prog. Quant. Electron. **19**, 161–237 (1995). [CrossRef]

5. K. Takasago, M. Takekawa, F. Kannari, M. Tani, and K. Sakai, “Accurate pulse shaping of femtosecond lasers using programmable phase-onlymodulator,” Jpn. J. Appl. Phys. **35**, L1430–L1433 (1996). [CrossRef]

6. T. Kurokawa, H. Tsuda, K. Okamoto, K. Naganuma, H. Takenouchi, Y. Inoue, and M. Ishii, “Time-space-conversion optical signal processing using arrayed-waveguide grating,” Electron. Lett. **33**, 1890–1891 (1997). [CrossRef]

7. H. Takenouchi, H. Tsuda, K. Naganuma, T. Kurokawa, Y. Inoue, and K. Okamoto, “Differential processing of ultrashort optical pulses using arrayed-waveguide grating with phase-only filter,” Electron. Lett. **34**, 1245–1246 (1998). [CrossRef]

8. H. Tsuda, K. Okamoto, T. Ishii, K. Naganuma, Y. Inoue, H. Takenouchi, and T. Kurokawa, “Second- and Third-order Dispersion Compensator Using a High-Resolution Arrayed-Waveguide Grating,” IEEE Photon. Technol. Lett. **11**, 569–571 (1999). [CrossRef]

11. A. M. Weiner, J. P. Heritage, and E. M. Kirschner, “High-resolution femtosecond pulse shaping,” J. Opt. Soc. Am. B **5**, 1563–1572 (1988). [CrossRef]

13. J. Paye and A. Migus, “Space-timeWigner functions and their application to the analysis of a pulse shaper,” J. Opt. Soc. Am. B **12**, 1480–1490 (1995). [CrossRef]

## 2. General expressions for optical-signal processing using an AWG

### 2.1 Overview of signal processing based on time-space conversion

*m*, is determined by the difference in path lengths between adjacent waveguides [14]. The analytical results of the DG system cannot be directly applied to the AWG system because the AWG system consists of waveguides. In the following sections we describe general expressions we developed for signal-processing AWG systems.

## 2.2. Spatial field profiles before filtering

*temporal*and

*spatial*frequencies at the focal plane, as discussed below. This will also make it easy to take some temporal-frequency-dependent effects, for example, chromatic dispersion, into account. We do not distinguish between the center optical frequency of the input waveform and the designed center frequency of the AWG because these two frequencies are often the same in practice.

*u*(

*t*) is the complex amplitude of

*f*(

*t*). Its temporal Fourier coefficient is expressed as

*e*(

*x*

_{0}) and its mode field radius as

*w*. At the interface between the input/output (I/O) waveguide and the first slab waveguide, the spatial electric field with temporal frequency

_{IO}*ν*is expressed as

*F*

_{0,ν}(

*x*

_{0}) is diffracted in the first slab waveguide and illuminates the arrayed waveguide. The distribution function of the first slab waveguide can thus be derived as

*n*is the effective index of the slab waveguides and

_{s}*L*is the focal length of the slab waveguides. Parameter

_{f}*α*is important because it relates the space domain to the spatial-frequency domain, as discussed below. Generally,

*α*depends on the temporal frequency; however, we use the approximation

*ν*≅

*ν*

_{0}in the following discussion because

*ν*-

*ν*

_{0}≪

*ν*

_{0}.

*β*(

*x*

_{1}), we can express the electric field at the interface between the first slab waveguide and the arrayed waveguide as

*N*is the number of waveguides in the array,

*d*is the spacing between waveguides along the

*x*

_{1}and

*x*

_{2}axes,

*w*is the mode field radius in the channel waveguides in the array,

_{AW}*δ*(

_{S}*x*) is defined as

*x*) is a rectangular function defined as

*δ*

_{S}(

*x*) represents the discreteness of the arrayed waveguide for repetitions of

*d*. In Eq. (6), the amplitude of the field can be treated approximately as constant within the width of each waveguide in the array. This is because the spatial width of the eigen mode of the waveguides is much smaller than the expansion of the distribution function.

*L*=

*mc*/

*n*

_{c}ν_{0}, where

*n*is the effective index of the channel waveguides in the array and

_{c}*m*is the diffraction order of the arrayed waveguide. This structure produces a temporal-frequency-dependent phase shift written as

*f*

_{2,ν}(

*x*

_{2}):

*ξ*represents the spatial frequency and is defined using Eq. (5),

_{S}(

*ξ*) and B(

*ξ*) are spatial Fourier transforms of

*δ*

_{S}(

*x*) and

*β*(

*x*) expressed as

*x*) is defined as sinc(

*x*)≡sin(

*πx*)/

*πx*. In Eq. (11), B(

*ξ*)*sinc(

*Ndξ*) determines the spot size at the focal plane, and

*δ*(

*ξ*-

*mν*/

*ν*

_{0}

*d*) gives the propagation direction of a beam with a temporal frequency of

*ν*. In other words, each temporal frequency component is centered at a spatial frequency of

*ξ*)*sinc(

*Ndξ*).

## 2.3 Derivation of basic parameters

*γ*, at the focal plane of the second slab waveguide is one of the most important parameters of the AWG because it relates the temporal frequency spectrum of the input pulse spread over the focal plane to the spatial filters (represented in the space domain). From Eqs. (12) and (15), we get

*m*th-order beam but also by the other order beams (i.e., the(

*m*±1)th-order beams, etc.). Therefore, the frequency range in which the AWG can process is limited. The free spectral range (FSR) of the AWG is defined as the frequency range corresponding to the spatial span between the

*m*th-order beam and the (

*m*+1)th-order beam with frequency

*ν*

_{0}. From the coincidence of the direction of the (

*m*+1)th-order diffraction of a beam with frequency

*ν*

_{0}and that of the

*m*th-order beam with frequency

*ν*

_{0}+

*ν*, the FSR is derived by using Eq (15) as follows:

_{FSR}*ν*, is defined by the spatial field profile at the output plane of the arrayed waveguide. From Eq. (11), the shape of the spot at the focal plane is expressed as B(

*ξ*)*sinc(

*Ndξ*) because the envelope of the field at the output of the arrayed waveguide is written as

*β*(

*x*

_{2})·rect(

*x*

_{2}/

*Nd*). The B(

*ξ*)*sinc(

*Ndξ*) means that the frequency resolution depends greatly on the envelope of the field at the output of the arrayed waveguide. In the following, we consider the two extremes. If we assume that the distribution from the I/O waveguide to the array is sufficiently uniform (i.e.,

*β*(

*x*

_{2})≈1/

*N*), Δ

*ν*is mainly determined by the sinc function. In this case, it is reasonable that Δ

*ν*is determined as

*ξ*=1/

*Nd*because the main lobe of sinc(

*Ndξ*) drops to zero at

*ξ*=±1/

*Nd*. From Eq. (15), we get

*β*(

*x*

_{2}) is not uniform, B(

*ξ*), which is approximately expressed as Gaussian, is the major limiting factor of Δ

*ν*. When we denote the spot size of B(

*ξ*) as Δ

*ξ*in the spatial frequency, Δ

*ν*is expressed as

*N*is defined as 1/(

_{eff}*d*Δ

*ξ*) and means the effective number of illuminated waveguides.

*ξ*,

*x*, and

*ν*) are not independent. However, there is not a one-to-one correspondence between

*ν*and

*ξ*or between

*ν*and

*x*because a beam spot with temporal frequency

*ν*has finite size. Therefore, frequency and space are coupled in signal processing using an AWG.

## 2.4 Spatial field profiles after filtering

*H*(

*ξ*) is placed at the focal plane, the input signal is modulated by the filter. The spatial field of the processed signal can be derived by reversing the previous discussion. The electric fields at the planes after filtering are expressed as follows:

*g*

_{2,ν}(

*x*

_{2}) :

*ξ*=0, the temporal frequency component of the output signal,

*V*, can be derived by considering the coupling between field

_{ν}*G*

_{0,ν}(

*ξ*) and the eigen-mode function of the I/O waveguide. Because

*x*

_{0}=

*αξ*at the interface, the eigen-mode function,

*e*(

*x*

_{0}), is expressed as

*e*(

*αξ*). Therefore,

*V*is expressed as

_{ν}*g*(

*t*) can be obtained as the temporal inverse Fourier transform of

*V*=

_{ν}*V*(

*ν*-

*ν*

_{0}).

## 3. Performance of signal-processing AWG

*N*, diffraction order

*m*, and distribution function

*β*(

*x*

_{1}). The phase fluctuation in the array is another important factor because the shape and loss of the output signal are greatly affected by the phase characteristics.

8. H. Tsuda, K. Okamoto, T. Ishii, K. Naganuma, Y. Inoue, H. Takenouchi, and T. Kurokawa, “Second- and Third-order Dispersion Compensator Using a High-Resolution Arrayed-Waveguide Grating,” IEEE Photon. Technol. Lett. **11**, 569–571 (1999). [CrossRef]

15. M. Kawachi, “Silica waveguides on silicon and their application to integrated-optic components,” Optic. Quant. Electron. **22**, 391–416 (1990). [CrossRef]

*ν*and Δ

_{FSR}*ν*. Therefore, there are some limitations on the temporal width of the input pulses to be processed. The minimum width is determined by

*ν*, and the temporal resolution is given by the inverse of

_{FSR}*ν*:

_{FSR}*ν*. The inverse of Δ

*ν*gives the maximum time span,

*T*

_{0}, in which signals can be processed:

*T*

_{0}have frequency components smaller than Δ

*ν*. Therefore, they cannot be spatially decomposed into their frequency components.

*N*and

*m*, limit the width of the input pulse, as shown by Eqs. (25) and (26). In designing signal-processing AWGs, the smaller the

*m*, the higher the temporal resolution. Moreover, a large

*N*is needed to obtain a large

*T*

_{0}. In designing AWG devices, the dimension is determined by parameter,

*Nm*, because the maximum path-length difference of the arrayed waveguide is expressed by

*Nm*·

*λ*/

*n*.

_{c}8. H. Tsuda, K. Okamoto, T. Ishii, K. Naganuma, Y. Inoue, H. Takenouchi, and T. Kurokawa, “Second- and Third-order Dispersion Compensator Using a High-Resolution Arrayed-Waveguide Grating,” IEEE Photon. Technol. Lett. **11**, 569–571 (1999). [CrossRef]

*Nm*was estimated to be around 3×10

^{4}. If AWGs whose relative refractive index difference is 1.5% or higher are fabricated on a 6-inch wafer, higher-performance (

*Nm*⋍1×10

^{5}) AWGs can be made. This means that, for example, we can design AWGs with

*N*=2000 and

*m*=50 (i.e.,

*T*

_{0}=500 ps and Δ

*t*=0.25 ps at a wavelength of 1.55 µm). Such performance should be sufficient for future high-speed communications.

*T*

_{0}. In practice, however, the stripe mirror is illuminated by a few (or more than a few) temporal frequency components because each component has a finite spot size, as shown in Fig. 3(b). Therefore, the temporal frequency spectrum reflected by the filter is like that shown in Fig. 3(c), and the temporal shape of the output is restricted, as shown in Fig. 3(d). Therefore, it is reasonable to define the figure of merit,

*F*, as the ratio of the maximum point of the output waveform to the minimum point. In the following discussion, we use parameter

*a*/

*Nd*, where

*a*is the spot size of the Gaussian beam,

*A*exp(-

*x*

^{2}/

*a*

^{2}), illuminating the arrayed waveguide.

*a*/

*Nd*. The larger the

*a*/

*Nd*, the more the beam profile of the focal plane resembles a sinc function and the more it spreads spatially. However, the effect of crosstalk becomes weaker because the crosstalk components in the electric fields barely couple to the single-mode I/O waveguide and become zero at the center of the stripe. The smaller the

*a*/

*Nd*, the larger the loss. Because there is a trade-off between

*F*and loss, the previously reported AWG was designed with an

*a*/

*Nd*of 0.57, as shown in Fig. 4 [6

6. T. Kurokawa, H. Tsuda, K. Okamoto, K. Naganuma, H. Takenouchi, Y. Inoue, and M. Ishii, “Time-space-conversion optical signal processing using arrayed-waveguide grating,” Electron. Lett. **33**, 1890–1891 (1997). [CrossRef]

16. H. Takahashi, K. Oda, and H. Toba, “Impact of crosstalk in an arrayed-waveguide multiplexer on N×N optical interconnection,” J. Lightwave Technol. **14**, 1097–1105 (1996). [CrossRef]

17. K. Takada, H. Yamada, and Y. Inoue, “Origin of channel crosstalk in 100-GHz-spaced silica-based arrayed-waveguide grating multiplexer,” Electron. Lett. **31**, 1176–1177 (1995). [CrossRef]

*m*and

*N*, the larger the total phase error in the waveguide because longer waveguides are needed to obtain a larger maximum path-length difference. The standard deviation of the phase error, σ, in a silica-based waveguide with a relative refractive index difference of 0.75% is typically 0.8×10

^{-2}rad/mm [18

18. T. Goh, S. Suzuki, and A. Sugita, “Estimation of Waveguide Phase Error in Silica-BasedWaveguides,” J. Lightwave Technol. **15**, 2107–2113 (1997). [CrossRef]

7. H. Takenouchi, H. Tsuda, K. Naganuma, T. Kurokawa, Y. Inoue, and K. Okamoto, “Differential processing of ultrashort optical pulses using arrayed-waveguide grating with phase-only filter,” Electron. Lett. **34**, 1245–1246 (1998). [CrossRef]

15. M. Kawachi, “Silica waveguides on silicon and their application to integrated-optic components,” Optic. Quant. Electron. **22**, 391–416 (1990). [CrossRef]

*R*

^{2}) calculated from the output pulse shape for σ=0.8 and 0.2×10

^{-2}rad/mm, where the diffraction order is fixed at

*m*=72 and the input pulse width was 1 ps. For σ=0.8×10

^{-2}(typical value), the output waveform was distorted and the excess loss increased when

*N*exceeded 500 (

*mN*>3.6×10

^{4}). A Smaller phase fluctuation is needed for higher-performance signal-processing AWGs. If

*N*⋍2000 (

*mN*⋍1.5×10

^{5}), for example, optical waveguides with a standard deviation of the phase error of 0.2×10

^{-2}is needed to obtain non-distorted output waveforms.

## 4. Conclusion

## Acknowledgments

## References and links

1. | A. M. Weiner, “Femtosecond optical pulse shaping and processing,” Prog. Quant. Electron. |

2. | A. M. Weiner, D. E. Leaird, D. H. Reitze, and E. G. Paek, “Femtosecond spectral holography,” IEEE J. of Quant. Electron. |

3. | M. C. Nuss, M. Li, T.H. Chiu, A.M. Weiner, and A. Partovi, “Time-to-spacemappingof femtosecond pulses,” Opt. Lett. |

4. | P. C. Sun, Y. T. Mazurenko, W. S. C. Chang, P.K.L. Yu, and Y. Fainman, “All-optical parallel-to-serial conversion by holographic spatial-to-temporal frequency encoding,” Opt. Lett. |

5. | K. Takasago, M. Takekawa, F. Kannari, M. Tani, and K. Sakai, “Accurate pulse shaping of femtosecond lasers using programmable phase-onlymodulator,” Jpn. J. Appl. Phys. |

6. | T. Kurokawa, H. Tsuda, K. Okamoto, K. Naganuma, H. Takenouchi, Y. Inoue, and M. Ishii, “Time-space-conversion optical signal processing using arrayed-waveguide grating,” Electron. Lett. |

7. | H. Takenouchi, H. Tsuda, K. Naganuma, T. Kurokawa, Y. Inoue, and K. Okamoto, “Differential processing of ultrashort optical pulses using arrayed-waveguide grating with phase-only filter,” Electron. Lett. |

8. | H. Tsuda, K. Okamoto, T. Ishii, K. Naganuma, Y. Inoue, H. Takenouchi, and T. Kurokawa, “Second- and Third-order Dispersion Compensator Using a High-Resolution Arrayed-Waveguide Grating,” IEEE Photon. Technol. Lett. |

9. | H. Takenouchi, H. Tsuda, C. Amano, T. Goh, K. Okamoto, and T. Kurokawa, “An optical phase-shift keying direct detection receiver using a high-resolution arrayed-waveguide grating,” in Technical Digest of Optical Fiber Conference (OFC) ‘99, paper TuO4. |

10. | H. Tsuda, H. Takenouchi, T. Ishii, K. Okamoto, T. Goh, K. Sato, A. Hirano, T. Kurokawa, and C. Amano, “Photonic spectral encoder/decoder using an arrayed-waveguide grating for coherent optical code division multiplexing,” in Technical Digest of Optical Fiber Conference (OFC) ‘99, paper PD32. |

11. | A. M. Weiner, J. P. Heritage, and E. M. Kirschner, “High-resolution femtosecond pulse shaping,” J. Opt. Soc. Am. B |

12. | A.M. Weiner, D. E. Leaird, J. S. Patel, and J. R. Wullert, “Programmable shaping of femtosecond pulsesby use of a 128-element liquid-crystal phase modulator,” IEEE J. Quant. Electron. |

13. | J. Paye and A. Migus, “Space-timeWigner functions and their application to the analysis of a pulse shaper,” J. Opt. Soc. Am. B |

14. | H. Takenouchi, H. Tsuda, C. Amano, T. Goh, K. Okamoto, and T. Kurokawa, “Differential processing using an arrayed-waveguide grating,” IEICE Trans. Commun. |

15. | M. Kawachi, “Silica waveguides on silicon and their application to integrated-optic components,” Optic. Quant. Electron. |

16. | H. Takahashi, K. Oda, and H. Toba, “Impact of crosstalk in an arrayed-waveguide multiplexer on N×N optical interconnection,” J. Lightwave Technol. |

17. | K. Takada, H. Yamada, and Y. Inoue, “Origin of channel crosstalk in 100-GHz-spaced silica-based arrayed-waveguide grating multiplexer,” Electron. Lett. |

18. | T. Goh, S. Suzuki, and A. Sugita, “Estimation of Waveguide Phase Error in Silica-BasedWaveguides,” J. Lightwave Technol. |

**OCIS Codes**

(060.4510) Fiber optics and optical communications : Optical communications

(070.6020) Fourier optics and signal processing : Continuous optical signal processing

(230.1150) Optical devices : All-optical devices

(230.7370) Optical devices : Waveguides

**ToC Category:**

Research Papers

**History**

Original Manuscript: January 18, 2000

Published: March 13, 2000

**Citation**

Hirokazu Takenouchi, Hiroyuki Tsuda, and Takashi Kurokawa, "Analysis of optical-signal processing using an arrayed-waveguide grating," Opt. Express **6**, 124-135 (2000)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-6-6-124

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### References

- A. M. Weiner, "Femtosecond optical pulse shaping and processing," Prog. Quant. Electron. 19,161-237 (1995). [CrossRef]
- A. M. Weiner, D. E. Leaird, D. H. Reitze and E. G. Paek, "Femtosecond spectral holography," IEEE J. of Quant. Electron. 28, 2251-2261 (1992). [CrossRef]
- M. C. Nuss,M.Li, T. H. Chiu, A. M.Weiner, and A.Partovi,"Time-to-space mapping of femtosecond pulses," Opt. Lett. 19, 664-666 (1994). [CrossRef] [PubMed]
- P. C. Sun, Y. T. Mazurenko, W. S. C. Chang, P.K.L. Yu and Y. Fainman, "All-optical parallel-to-serial conversion by holographic spatial-to-temporal frequency encoding," Opt. Lett. 20, 1728-1730 (1995). [CrossRef] [PubMed]
- K. Takasago, M. Takekawa, F. Kannari, M. Tani and K. Sakai, "Accurate pulse shaping of femtosecond lasers using programmable phase-only modulator," Jpn. J. Appl. Phys. 35, L1430-L1433 (1996). [CrossRef]
- T. Kurokawa, H. Tsuda, K. Okamoto, K. Naganuma, H. Takenouchi, Y. Inoue and M. Ishii, "Time-space-conversion optical signal processing using arrayed-waveguide grating," Electron. Lett. 33, 1890-1891 (1997). [CrossRef]
- H. Takenouchi, H. Tsuda, K. Naganuma, T. Kurokawa, Y. Inoue and K. Okamoto, "Differential processing of ultrashort optical pulses using arrayed-waveguide grating with phase-only filter," Electron. Lett. 34, 1245-1246 (1998). [CrossRef]
- H. Tsuda, K. Okamoto, T. Ishii, K. Naganuma, Y. Inoue, H. Takenouchi and T. Kurokawa, "Second- and Third-order Dispersion Compensator Using a High-Resolution Arrayed-Waveguide Grating," IEEE Photon. Technol. Lett. 11, 569-571 (1999). [CrossRef]
- H. Takenouchi, H. Tsuda, C. Amano, T. Goh, K. Okamoto and T. Kurokawa, "An optical phase-shift keying direct detection receiver using a high-resolution arrayed-waveguide grating," in Technical Digest of Optical Fiber Conference (OFC) '99, paper TuO4.
- H. Tsuda, H. Takenouchi, T. Ishii, K. Okamoto, T. Goh, K. Sato, A. Hirano, T. Kurokawa and C. Amano, "Photonic spectral encoder/decoder using an arrayed-waveguide grating for coherent optical code division multiplexing," in Technical Digest of Optical Fiber Conference (OFC) '99, paper PD32.
- A. M. Weiner, J. P. Heritage and E. M. Kirschner, "High-resolution femtosecond pulse shaping," J. Opt. Soc. Am. B 5, 1563-1572 (1988). [CrossRef]
- A. M. Weiner, D. E. Leaird,J.S.Patel and J.R.Wullert,"Programmable shaping of femtosecond pulses by use of a 128-element liquid-crystal phase modulator," IEEE J. Quant. Electron. 28, 908-920 (1992). [CrossRef]
- J. Paye and A. Migus, "Space-time Wigner functions and their application to the analysis of a pulse shaper," J. Opt. Soc. Am. B 12, 1480-1490 (1995). [CrossRef]
- H. Takenouchi, H. Tsuda, C. Amano, T. Goh, K. Okamoto and T. Kurokawa, "Differential processing using an arrayed-waveguide grating," IEICE Trans. Commun. E82-B, 1252-1258 (1999).
- M. Kawachi, "Silica waveguides on silicon and their application to integrated-optic components," Optic. Quant. Electron. 22, 391-416 (1990). [CrossRef]
- H. Takahashi, K. Oda and H. Toba, "Impact of crosstalk in an arrayed-waveguide multiplexer on N�N optical interconnection," J. Lightwave Technol. 14, 1097-1105 (1996). [CrossRef]
- K. Takada, H. Yamada and Y. Inoue, "Origin of channel crosstalk in 100-GHz-spaced silica-based arrayed-waveguide grating multiplexer," Electron. Lett. 31, 1176-1177 (1995). [CrossRef]
- T. Goh, S. Suzuki and A. Sugita, "Estimation of Waveguide Phase Error in Silica-Based Waveguides," J. Lightwave Technol. 15, 2107-2113 (1997). [CrossRef]

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