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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 6 — Mar. 25, 2013
  • pp: 6928–6942
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Full field model for interleave-chirped arrayed waveguide gratings

Bernardo Gargallo and Pascual Muñoz  »View Author Affiliations


Optics Express, Vol. 21, Issue 6, pp. 6928-6942 (2013)
http://dx.doi.org/10.1364/OE.21.006928


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Abstract

In this paper, a theoretical model for an Interleave-Chirped Arrayed Waveguide Grating (IC-AWG) is presented. The model describes the operation of the device by means of a field (amplitude and phase) transfer response. The validation of the model is accomplished by means of simulations, using parameters from previously fabricated devices. A novel design procedure is derived from the model, and it is later on employed to demonstrate the design of colorless universal IC-AWGs. The model can be readily applied to the analysis and design of future multi-wavelength optical coherent communications receivers and optical waveform analyzers.

© 2013 OSA

1. Introduction

Coherent optical communications have revived in the recent years, due to the availability of high-speed digital electronic processors [1

1. K. Kikuchi, “Coherent optical communications: historical perspectives and future directions,” in High Spectral Density Optical Communication Technology, M. Nakazawa, K. Kikuchi, and T. Miyazaki, eds (Springer, 2010), Chap. 2 [CrossRef] .

]. Together with high performance photonic integrated cirtuis (PICs), real-time coherent communications operating at aggregate speeds of up to 500 Gb/s have been demonstrated [2

2. R. Nagarajan, e.a., “Terabit/s class InP photonic integrated circuits,” Semicond. Sci. Tech. 27, 094003 (2012) [CrossRef] .

, 3

3. J. T. Rahn, e.a., “250 Gb/s real-time PIC-based super-channel transmission over a gridless 6000 km terrestrial link,” in Proc. Opt. Fiber Comm. conference paper PDP5D.5 (2012).

]. Within the toolset for the production of integrated coherent receivers, a very relevant part is the analysis and design models, together with numerical tools. Currently, the core of integrated optics based coherent receivers are Multi-Mode Interference (MMI) couplers, designed to operate as 90° hybrids. The MMI theory is very well known, and extensive models exist to calculate the field transfer functions of such devices [4

4. L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 4, 615–627 (1995) [CrossRef] .

6

6. J. M. Heaton and R. M. Jenkins, “General matrix theory of self-imaging in multimode interference (MMI) couplers,” IEEE Photon. Technol. Lett. 11, 212–214 (1999) [CrossRef] .

]. Moreover, numerical tools can be readily employed to optimize the performance of such devices, as Beam Propagation Method (BPM) in combination of Effective Index Method (EIM) [7

7. J. Van Roey, J. van der Donk, and P. Lagasse, “Beam-propagation method: analysis and assessment,” J. Opt. Soc. Am. 71, 803–810 (1981) [CrossRef] .

]. The rectangular structure and paraxial nature of the MMIs result in modest computation times for those algorithms/tools, using commodity personal computers.

Similar to conventional Intensity Modulated Direct Detection (IM-DD) systems, Wavelength Division Multiplexing (WDM) has been recently brought into PICs for coherent optical communications. The natural combination of on chip demultiplexers, as the Arrayed Waveguide Grating, and optical MMI 90° hybrids have been demonstrated in [2

2. R. Nagarajan, e.a., “Terabit/s class InP photonic integrated circuits,” Semicond. Sci. Tech. 27, 094003 (2012) [CrossRef] .

], whereas off-chip optical polarization splitters are used to feed the PIC. Nonetheless, the solution proposed by Doerr e.a. [8

8. C. R. Doerr, L. Zhang, and P. J. Winzer, “Monolithic InP multiwavelength coherent receiver using a chirped arrayed waveguide grating,” J. Lightwave Technol. 29, 536–541 (2011) [CrossRef] .

] in which the AWG arms are modified to enable three fold operation (demultiplexer, polarization splitter, 90° hybrid), is envisaged as the most compact solution for these future on-chip architectures. This device is named Interleave-Chirped AWG (IC-AWG). Although in [8

8. C. R. Doerr, L. Zhang, and P. J. Winzer, “Monolithic InP multiwavelength coherent receiver using a chirped arrayed waveguide grating,” J. Lightwave Technol. 29, 536–541 (2011) [CrossRef] .

] an experimental demonstration of the device is given, a model for the analysis and design of the IC-AWG does not exist in the literature. The purpose of this paper is to report the development of the optical field model for the analysis and design of IC-AWGs. The paper is structured as follows. In Section 2, the theoretical equations describing the full field (amplitude and phase) transfer function for an IC-AWG are developed, taking as base our work in [9

9. P. Muñoz, D. Pastor, and J. Capmany, “Modeling and design of arrayed waveguide gratings,” J. Lightwave Technol. 20, 661–674 (2002) [CrossRef] .

]. The model is validated in Section 3 using as IC-AWG construction parameters those of the fabricated device reported in [8

8. C. R. Doerr, L. Zhang, and P. J. Winzer, “Monolithic InP multiwavelength coherent receiver using a chirped arrayed waveguide grating,” J. Lightwave Technol. 29, 536–541 (2011) [CrossRef] .

]. In Section 4, a design procedure for the IC-AWG is proposed, in which the aim is to maximize the spectral usage for the wavelength channels and polarizations. The design procedure is used in Section 5 to demonstrate a colorless universal IC-AWG. Finally the outlook and conclusions are given in Section 6.

2. Theoretical model

2.1. AWG layout

A schematic design of a 2 × N Arrayed Waveguide Grating (AWG) is shown in Fig. 1(a). It consists of two input waveguides, followed by a free propagation region (FPR). At the end of this FPR, the field is coupled in the arrayed waveguides. In a conventional AWG design, the length of a consecutive waveguides in the array differ by a constant value, Δl, which is an integer (m) multiple of the design wavelength (λ0) within the waveguide, as shown in Fig. 1(b). The accumulated phase shift depends on wavelength/frequency, which combined with the second FPR, leads to light focusing at different output positions for each wavelength. Moreover, the field discretization imposed in the arms, causes diffraction in different orders (...,m−1,m,m+1,...), resulting in different Brillouin Zones (BZs) [10

10. H. Takahashi, K. Oda, H. Toba, and Y. Inoue, “Transmission characteristics of arrayed waveguide N × N wavelength multiplexer,” J. Lightwave Technol. 13, 447–455 (1995) [CrossRef] .

]. In an Interleave-Chirped AWG (IC-AWG), the array is divided in M subsets. Within each subset, whose waveguides are spaced Mdω at x1 and x2, an additional floor length (related to λ0) is introduced [11

11. Y. Wan and R. Hui, “Design of WDM cross connect based on interleaved AWG (IAWG) and a phase shifter array,” J. Lightwave Technol. 25, 1390–1400 (2007) [CrossRef] .

]. The division of the array in M subsets reduces the original BZ size by a factor M, as will be formulated in detail in the following sections. Besides, introducing different floor length in each subset, results in out of phase interference at x3, for the fields diffracted by each subset. This can be seen in Fig. 1(c), where the width of the focusing zones has been reduced by a factor M compared to Fig. 1(b). Each group of this arrayed waveguides with the same incremental length is called subarray. Hence, if for example a M = 4 chirp pattern is used, the first subarray will be composed of the arrayed waveguides number {1,5,9,...,N−3}, the second by {2,6,10,...,N−2}... In this section, the model of the IC-AWG extending the Fourier optics AWG model presented in [9

9. P. Muñoz, D. Pastor, and J. Capmany, “Modeling and design of arrayed waveguide gratings,” J. Lightwave Technol. 20, 661–674 (2002) [CrossRef] .

] is formulated. Table 1 shows a summary of the variables used in the formulation.

Fig. 1 (a) IC-AWG layout. (b) Field focusing points without and (c) with chirp. Abbreviations: subscripts i, w and o stand for input, arrayed and output waveguides, respectively; ω: waveguide width; d: waveguide spacing; Lf : focal length; FPR: free propagation region; l0: shortest AW length; Δl: incremental length; λ: wavelength; nc: AW effective index; BZ: Brillouin Zone.

Table 1. Summary of the variables used in the formulation. Subscripts i, g and o stand for input, arrayed and output waveguides, respectively.

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2.2. Field at the input waveguide

The field in an input waveguide placed at the middle of the first FPR, can be approximated by a power normalized Gaussian function
bi(x0)=2πωi24e(x0ωi)2
(1)
where ωi is the mode field radius and x0 is the spatial coordinate at the input plane, as is shown in Fig. 1(a). Although the Gaussian approximation is suitable to clarify the understanding of the device and enables to obtain a final analytical expression, it fails in predicting the response outside the band, thus is necessary the use of real modes to obtain simulation responses closer to the real physical response, as detailed in [9

9. P. Muñoz, D. Pastor, and J. Capmany, “Modeling and design of arrayed waveguide gratings,” J. Lightwave Technol. 20, 661–674 (2002) [CrossRef] .

].

2.3. First Free Propagation Region

This field is radiated to the first FPR, and the light spatial distribution can be obtained by the spatial Fourier transform of the input distribution, using the paraxial approximation [9

9. P. Muñoz, D. Pastor, and J. Capmany, “Modeling and design of arrayed waveguide gratings,” J. Lightwave Technol. 20, 661–674 (2002) [CrossRef] .

, 12

12. J. W. Goodman, “Introduction to Fourier optics,” in Classic Textbook Reissue Series, W. Stephen, ed. (New York: McGraw-Hill, 1988), Chap. 5, pp. 83–90.

]:
Bi(x1)={bi(x0)}|u=x1α=2πωi2α24e(πωi(x1α))2
(2)
where u is the spatial frequency domain variable of the Fourier transform and α is the equivalent to the wavelength focal length product in Fourier optics propagation [12

12. J. W. Goodman, “Introduction to Fourier optics,” in Classic Textbook Reissue Series, W. Stephen, ed. (New York: McGraw-Hill, 1988), Chap. 5, pp. 83–90.

], expressed as:
α=cLfnsν
(3)
being Lf the slab length, ns the effective index of the FPR, ν the frequency and c the speed of light in vacuum. The paraxial approximation holds in the Fraunhofer diffraction region, which corresponds to the AWG slab lengths accomplishing Lf(πWx2)/(4λ)[12

12. J. W. Goodman, “Introduction to Fourier optics,” in Classic Textbook Reissue Series, W. Stephen, ed. (New York: McGraw-Hill, 1988), Chap. 5, pp. 83–90.

]. For typical waveguide widths Wx = 1.5 μm and wavelength value λ = 1550 nm, this approximation is valid for Lf ≫ 1.14 μm. The total field distribution for an arbitrary number of illuminated arrayed waveguides (AWs), placed at the x1 plane, is:
f1(x1)=2πωg24k=0M1[rBi(Mrdω+kdω)bg(x1Mrdωkdω)]
(4)
where dω is the spacing between arrayed waveguides, Fig. 1(a), r is the AW number and ωg is the mode field radius of the arrayed waveguide, M is the length of the chirp pattern (and the number of waveguide array subsets) and bg(x) is the Gaussian approximation as in Eq. (1) for the field in the AW. Particularizing for a set of an arbitrary number N of illuminated waveguides:
f1(x1)=2πωg24k=0M1[Π(x1Ndω)Bi(x1)r=+δ(x1Mrdωkdω)]bg(x1)
(5)
with ⊗ being the convolution, and Π (x1/Ndω) being a truncation function defined as:
Π(x1Ndω)={1,if|x|Ndω20,otherwise
(6)

2.4. Field at the arrayed waveguides

2.5. Second Free Propagation Region

2.6. Field at the output waveguides

At the output waveguides, the field can be obtained through the following overlap integral:
t0,q(ν)=+f3(x3,ν)b0(x3qd0)x3
(21)
where b0 is the fundamental mode profile at the output waveguide (OW), q is the OW number and d0 is the OW spacing.

2.7. Arbitrary IW position

All this formulation can be easily extended to the case with more than one input waveguide (IW), not only for the case with one input placed at the central position of the first FPR. The field at the IW can be expressed then as
bi,p(x0)=2πωi24e(x0pdiωi)2=bi,p(x0pdi)
(22)
where p is the IW number and di is the IW spacing. At the output plane the field will be
f3(x3,ν)=2πωg2α24Bg(x3)1Mk=0M1[ψk(ν)ejβΔlkMr=+ej2πrkMfM(x3rαMdω+ncνΔlαcMdω+pdi)]
(23)
and the field in the output waveguides is
tp,q(ν)=+f3,p(x3,ν)b0(x3qd0)x3
(24)

2.8. Modeling polarization dependence

This model can also be extended to the case where two polarizations are supported by the waveguiding structures. Although polarization dependence is minimized in conventional AWG [13

13. L. H. Spiekman, M. R. Amersfoort, A. H. de Vreede, F. P. G. M. van Ham, A. Kuntze, J. W. Pedersen, P. Demeester, and M. K. Smit, “Design and realization of polarization independent phased array wavelength demultiplexers using different array orders for TE and TM,” J. Lightwave Technol. 14, 991–995 (1996) [CrossRef] .

], it can be exploited in emerging applications, as discussed in the introduction. Through Eq. (23) the shift between polarizations is given by:
Δxpol=mdω(αTM(ν)nc,TM(ν)αTM(ν)νnc,TE(ν0)ν0αTE(ν)+nc,TE(ν)αTE(ν)νnc,TE(ν0)ν0)
(25)
The equation indicates that at the output plane each polarization is going to be focused at different position. The distance between TE and TM polarization focusing points, at a given ν, depends on the spacing, length and effective index of the AWs and length of the FPR. The frequency shift corresponding to the aforementioned distances, which in the literature is known as Polarization Dependent Wavelength Shift (PDWS) [13

13. L. H. Spiekman, M. R. Amersfoort, A. H. de Vreede, F. P. G. M. van Ham, A. Kuntze, J. W. Pedersen, P. Demeester, and M. K. Smit, “Design and realization of polarization independent phased array wavelength demultiplexers using different array orders for TE and TM,” J. Lightwave Technol. 14, 991–995 (1996) [CrossRef] .

], is given by:
Δνpol=nc,TE(νo)νonc,TM(ν+Δνpol)ναTE(ν)nc,TE(νo)νonc,TM(ν+Δνpol)αTM(ν+Δνpol)+αTE(ν)nc,TE(ν)νnc,TM(ν+Δνpol)αTM(ν+Δνpol)
(26)
Equation (26) shows the wavelength separation between both polarizations as a function of the effective index of the AWs and the FPRs length. For ν = ν0 it reduces to:
Δνpol=nc,TE(ν0)ν0nc,TM(ν0+Δνpol)ν0
(27)
Equation (27) is extremely important in the design of the IC-AWG, since it shows that the PDWS only depends on the effective index of each polarization, and ν0. For the design of IC-AWGs, described in Section 4, it will restrict the number of channels and their frequency spacing.

3. Simulation of a fabricated and reported device

In this section, the model is particularized to the case of the device reported in [8

8. C. R. Doerr, L. Zhang, and P. J. Winzer, “Monolithic InP multiwavelength coherent receiver using a chirped arrayed waveguide grating,” J. Lightwave Technol. 29, 536–541 (2011) [CrossRef] .

]. In [8

8. C. R. Doerr, L. Zhang, and P. J. Winzer, “Monolithic InP multiwavelength coherent receiver using a chirped arrayed waveguide grating,” J. Lightwave Technol. 29, 536–541 (2011) [CrossRef] .

], it is mentioned that it is possible to find a M = 4 chirp pattern that produces four equal-power outputs with the relations of a 90° optical hybrid, being this chirp pattern [0, 3λ0/ (8nc), 0, −λ0/ (8nc)], where λ0 is the design central wavelength and nc the effective index of the AWs. This chirp pattern can be easily introduced in the AWs through the lc,k parameter in Eq. (7), i.e. lc,0 = 0, lc,1 = 3λ0/ (8nc), lc,2 = 0 and lc,3 = −λ0/ (8nc). The 90° hybrid operation consists in taking two input ligthwaves with complex amplitudes a and b, and obtaining at the output four signals of the form (a + b)/2, (a + jb)/2, (ab)/2 and (ajb)/2. The AWs are InP ridge waveguides, with a InGaAsP core 160 nm thick and a bandgap of 1.4 μm. The waveguide width is 2.45 μm, with an etching of 0.2 μm and a cladding height of 1 μm [14

14. C. R. Doerr, L. Zhang, L. Buhl, V. I. Kopp, D. Neugroschl, and G. Weiner, “Tapered dual-core fiber for efficient and robust coupling to InP photonic integrated circuits,” Proc. OFC paper OThN5 (2009).

]. The effective indexes for the waveguide and FPR cross-section were calculated, both for TE and TM, using a film-mode matching (FMM) [15

15. M. Lohmeyer, “Wave-matching-method for mode analysis of dielectric waveguides,” Opt. Quantum Electron. 29, 907–922 (1997) [CrossRef] .

] commercial software [16

16. FieldDesigner™, PhoeniX Software, http://www.phoenixbv.com.

]. Using equation (27), is possible to calculate the PDWS for these waveguides, obtaining 5.9 nm. In [8

8. C. R. Doerr, L. Zhang, and P. J. Winzer, “Monolithic InP multiwavelength coherent receiver using a chirped arrayed waveguide grating,” J. Lightwave Technol. 29, 536–541 (2011) [CrossRef] .

], four channels with a 185 GHz spacing are used. The IC-AWG has 100 arms, and FPR lengths of 2 mm. The AW and OW spacing is estimated to be 4.74 μm. The OW number is 32 and the grating order is m = 21. Consequently, the incremental length between arrayed waveguides Δl in each subarray is 40.69 μm. To produce four equal-power outputs with the relations of a 90° optical hybrid, the chirp pattern used is [0, 3λ0/ (8nc), 0, −λ0/ (8nc)] [11

11. Y. Wan and R. Hui, “Design of WDM cross connect based on interleaved AWG (IAWG) and a phase shifter array,” J. Lightwave Technol. 25, 1390–1400 (2007) [CrossRef] .

]. The width of the BZs is 1/4 of the BZs of the AWG without chirp. Placing two input waveguides, spaced by 1/4 of the BZ, i.e. with π/2 phase shift between them (the full BZ FSR corresponds to 2π), enables the operation as 90° hybrid. Hence, the two input waveguides are placed in the positions di,1 = 25.4 μm and di,2 = −25.4 μm.

Equations (23) and (24) are used to calculate the normalized field at the output plane x3 for each channel and both polarizations for the input waveguide placed at the position di,1 (Fig. 2). The other input, placed at the position di,2, has identical response. Figure 2 shows how TE and TM, for a given λ, are focused at different points. Between them, a given number of additional channels can be placed, limited by the PDWS.

Fig. 2 f3 (x3, λ) using input waveguide 1 for each wavelength channel and polarization.

In Figs. 3(a) and 3(b) the transfer function at output waveguides 1 to 8 for each polarization is shown. As can be seen, the first 4 output waveguides have a maximum in the response for TE polarization at the wavelengths of the chosen channels, and the second group of 4 output waveguides have a maximum for TM polarization at the same channels. This response is repeated for the four groups of 8 output waveguides. In other words, the output waveguides 1, 9, 17 and 25 have a maximum in the response at 1550 nm and TE polarization. The same holds for TM polarization and waveguides 5, 13, 21 and 29. Then, the output waveguides 2, 10, 18 and 26 have a maximum at 1551.5 nm for TE polarization. This is repeated for each group of 4 output waveguides, so at the end there are four output waveguides for each channel and polarization. These four waveguides have the relations of a 90° optical hybrid as is shown in Figs. 4 and 5. In Fig. 4 the power and phase from the two AWG inputs (p=0,1) are plotted for each of the four output waveguides 1, 9, 17 and 25 from Fig. 4(a)–4(d) respectively. In the figures, the in-band phase response differences have been labeled. The corresponding vectorial plot at the channel center wavelength (1550 nm) and TE polarization is given in Fig. 5. As can be seen, in the 1st output waveguide the phase shift between the field in the two inputs is 135° at 1550 nm and TE polarization. At the outputs 9, 17 and 25, these phase shifts at the same wavelength and polarization are 45°, −45° and −135°, respectively.

Fig. 3 Transfer function tp,q (λ) for input waveguide p = 1 at (a) output waveguides 1 to 4 at TE polarization and (b) output waveguides 5 to 8 at TM polarization.
Fig. 4 Power and phase transfer functions tp,q (λ) for input waveguides p = 0, 1 and output waveguides (a) 1, (b) 9, (c) 17 and (d) 25 for TE polarization.
Fig. 5 Vectorial representation of the field at the output waveguides

In Table 2 a summary of the phase relations between the two inputs for each output waveguide is given. As can be seen, each group of 4 output waveguides have the necessary hybrid relations, and it is accomplished for each channel and polarization.

Table 2. Number of the output waveguide for each phase shift, channel and polarization

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4. Design procedure

5. Simulation of a device using the design procedure

Fig. 6 Field at the output plane x3 using input waveguide 1 for each channel and polarization

Moreover, the loss nonuniformity fixed to 1 dB is accomplished at the 32 output positions where output waveguides will be, as can be seen in Fig. 7 where there are represented the transfer function at the corresponding output waveguides of channel 1 for both polarizations. Table 3 shows a summary of the design parameters and the deviation about the target value.

Fig. 7 Transfer function at (a) output waveguides 1 to 4 at TE polarization and (b) output waveguides 5 to 8 at TM polarization

Table 3. Summary of the IAWG design example parameters. Subscripts i, g and o stand for input, arrayed and output waveguides, respectively.

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6. Outlook and conclusion

Fig. 8 Schematic of IC-AWG with (a) two input waveguides, obtaining at the output waveguides response for each polarization in two channels per OW, and with (b) four input waveguides, obtaining at the output waveguides response in two channels but at the same polarization state.

Fig. 9 (a) Schematic of a coherent detector using a four input waveguides IC-AWG with optical switches. (b) Schematic of a coherent detector using a two input waveguides IC-AWG with polarization splitter. Abbreviations: LO, local oscillator; PD array, photodetector array; Pol splitter, polarization splitter.

Possible drawbacks of this device are the larger number of output channels required compared to a regular AWG. For each input wavelength, the IC-AWG requires 8 output waveguides (two polarizations and four different phases). This can be limiting depending on the integration technology. AWGs have been reported with 32 outputs in InP [8

8. C. R. Doerr, L. Zhang, and P. J. Winzer, “Monolithic InP multiwavelength coherent receiver using a chirped arrayed waveguide grating,” J. Lightwave Technol. 29, 536–541 (2011) [CrossRef] .

] and with 40 outputs in SOI [19

19. S. T. S. Cheung, B. Guan, S. S. Djordjevic, K. Okamoto, and S. J. B. Yoo, “Low-loss and high contrast Silicon-on-Insulator (SOI) arrayed waveguide grating,” in CLEO: Science and Innovations, OSA Technical Digest (online) (Optical Society of America, 2012), paper CM4A.5.

]. This number of output ports has to be divided by 8 to obtain the possible output channels in an IC-AWG. Also, there may be walk-off problems due to the large slab couplers when using it in a colorless application, but they might be overcome with design techniques similar to those reported in [20

20. P. Bernasconi, C. Doerr, C. Dragone, M. Cappuzzo, E. Laskowski, and A. Paunescu, “Large N×N waveguide grating routers,” J. Lightwave Technol. 18, 985–991 (2000) [CrossRef] .

] for cyclic AWGs.

In conclusion, this paper provides a theoretical design and analysis model for an IC-AWG. The model is able to describe the operation as 90° hybrid, channel demultiplexing and polarization splitting. The model has been validated by means of parameters reported in the literature for fabricated coherent optical receivers. The model has been also used to develop a design procedure, which is a novel tool to obtain the most important physical parameters of the IC-AWG from targetted high level specifications. Finally, the design procedure and the accompanying discussions show how universal (colorless) devices can be designed and analyzed by means of the proposed model.

Acknowledgments

The authors acknowledge financial support by the Spanish MICINN Project TEC2010-21337, acronym ATOMIC; project FEDER UPVOV10-3E-492 and project FEDER UPVOV08-3E-008. B. Gargallo acknowledges financial support through FPI grant BES-2011-046100.

References and links

1.

K. Kikuchi, “Coherent optical communications: historical perspectives and future directions,” in High Spectral Density Optical Communication Technology, M. Nakazawa, K. Kikuchi, and T. Miyazaki, eds (Springer, 2010), Chap. 2 [CrossRef] .

2.

R. Nagarajan, e.a., “Terabit/s class InP photonic integrated circuits,” Semicond. Sci. Tech. 27, 094003 (2012) [CrossRef] .

3.

J. T. Rahn, e.a., “250 Gb/s real-time PIC-based super-channel transmission over a gridless 6000 km terrestrial link,” in Proc. Opt. Fiber Comm. conference paper PDP5D.5 (2012).

4.

L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 4, 615–627 (1995) [CrossRef] .

5.

M. Bachmann, P. Besse, and H. Melchior, “General self-imaging properties in N × N multimode interference couplers including phase relations,” Appl. Opt. 33, 3905–3911 (1994) [CrossRef] [PubMed] .

6.

J. M. Heaton and R. M. Jenkins, “General matrix theory of self-imaging in multimode interference (MMI) couplers,” IEEE Photon. Technol. Lett. 11, 212–214 (1999) [CrossRef] .

7.

J. Van Roey, J. van der Donk, and P. Lagasse, “Beam-propagation method: analysis and assessment,” J. Opt. Soc. Am. 71, 803–810 (1981) [CrossRef] .

8.

C. R. Doerr, L. Zhang, and P. J. Winzer, “Monolithic InP multiwavelength coherent receiver using a chirped arrayed waveguide grating,” J. Lightwave Technol. 29, 536–541 (2011) [CrossRef] .

9.

P. Muñoz, D. Pastor, and J. Capmany, “Modeling and design of arrayed waveguide gratings,” J. Lightwave Technol. 20, 661–674 (2002) [CrossRef] .

10.

H. Takahashi, K. Oda, H. Toba, and Y. Inoue, “Transmission characteristics of arrayed waveguide N × N wavelength multiplexer,” J. Lightwave Technol. 13, 447–455 (1995) [CrossRef] .

11.

Y. Wan and R. Hui, “Design of WDM cross connect based on interleaved AWG (IAWG) and a phase shifter array,” J. Lightwave Technol. 25, 1390–1400 (2007) [CrossRef] .

12.

J. W. Goodman, “Introduction to Fourier optics,” in Classic Textbook Reissue Series, W. Stephen, ed. (New York: McGraw-Hill, 1988), Chap. 5, pp. 83–90.

13.

L. H. Spiekman, M. R. Amersfoort, A. H. de Vreede, F. P. G. M. van Ham, A. Kuntze, J. W. Pedersen, P. Demeester, and M. K. Smit, “Design and realization of polarization independent phased array wavelength demultiplexers using different array orders for TE and TM,” J. Lightwave Technol. 14, 991–995 (1996) [CrossRef] .

14.

C. R. Doerr, L. Zhang, L. Buhl, V. I. Kopp, D. Neugroschl, and G. Weiner, “Tapered dual-core fiber for efficient and robust coupling to InP photonic integrated circuits,” Proc. OFC paper OThN5 (2009).

15.

M. Lohmeyer, “Wave-matching-method for mode analysis of dielectric waveguides,” Opt. Quantum Electron. 29, 907–922 (1997) [CrossRef] .

16.

FieldDesigner™, PhoeniX Software, http://www.phoenixbv.com.

17.

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

18.

N. K. Fontaine, R. P. Scott, L. Zhou, F. M. Soares, J. P. Heritage, and S. J. B. Yoo, “Real-time full-field arbitrary optical waveform measurement,” Nat. Photonics 4, 248–254 (2010) [CrossRef] .

19.

S. T. S. Cheung, B. Guan, S. S. Djordjevic, K. Okamoto, and S. J. B. Yoo, “Low-loss and high contrast Silicon-on-Insulator (SOI) arrayed waveguide grating,” in CLEO: Science and Innovations, OSA Technical Digest (online) (Optical Society of America, 2012), paper CM4A.5.

20.

P. Bernasconi, C. Doerr, C. Dragone, M. Cappuzzo, E. Laskowski, and A. Paunescu, “Large N×N waveguide grating routers,” J. Lightwave Technol. 18, 985–991 (2000) [CrossRef] .

OCIS Codes
(060.1660) Fiber optics and optical communications : Coherent communications
(070.2580) Fourier optics and signal processing : Paraxial wave optics
(130.3120) Integrated optics : Integrated optics devices
(130.7408) Integrated optics : Wavelength filtering devices
(130.5440) Integrated optics : Polarization-selective devices

ToC Category:
Integrated Optics

History
Original Manuscript: January 30, 2013
Revised Manuscript: March 5, 2013
Manuscript Accepted: March 6, 2013
Published: March 13, 2013

Citation
Bernardo Gargallo and Pascual Muñoz, "Full field model for interleave-chirped arrayed waveguide gratings," Opt. Express 21, 6928-6942 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-6-6928


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References

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  13. L. H. Spiekman, M. R. Amersfoort, A. H. de Vreede, F. P. G. M. van Ham, A. Kuntze, J. W. Pedersen, P. Demeester, and M. K. Smit, “Design and realization of polarization independent phased array wavelength demultiplexers using different array orders for TE and TM,” J. Lightwave Technol.14, 991–995 (1996). [CrossRef]
  14. C. R. Doerr, L. Zhang, L. Buhl, V. I. Kopp, D. Neugroschl, and G. Weiner, “Tapered dual-core fiber for efficient and robust coupling to InP photonic integrated circuits,” Proc. OFC paper OThN5 (2009).
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  16. FieldDesigner™, PhoeniX Software, http://www.phoenixbv.com .
  17. M. K. Smit and C. van Dam, “PHASAR-based WDM-devices: Principles, design and applications,” IEEE J. Sel. Top. Quant.2, 236–250 (1996). [CrossRef]
  18. N. K. Fontaine, R. P. Scott, L. Zhou, F. M. Soares, J. P. Heritage, and S. J. B. Yoo, “Real-time full-field arbitrary optical waveform measurement,” Nat. Photonics4, 248–254 (2010). [CrossRef]
  19. S. T. S. Cheung, B. Guan, S. S. Djordjevic, K. Okamoto, and S. J. B. Yoo, “Low-loss and high contrast Silicon-on-Insulator (SOI) arrayed waveguide grating,” in CLEO: Science and Innovations, OSA Technical Digest (online) (Optical Society of America, 2012), paper CM4A.5.
  20. P. Bernasconi, C. Doerr, C. Dragone, M. Cappuzzo, E. Laskowski, and A. Paunescu, “Large N×N waveguide grating routers,” J. Lightwave Technol.18, 985–991 (2000). [CrossRef]

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