OSA's Digital Library

Optics Express

Optics Express

  • Editor: J. H. Eberly
  • Vol. 9, Iss. 7 — Sep. 24, 2001
  • pp: 328–338
« Show journal navigation

Analysis and design of arrayed waveguide gratings with MMI couplers

P. Muñoz, D. Pastor, and J. Capmany  »View Author Affiliations


Optics Express, Vol. 9, Issue 7, pp. 328-338 (2001)
http://dx.doi.org/10.1364/OE.9.000328


View Full Text Article

Acrobat PDF (511 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We present an extension of the AWG model and design procedure described in [1] to incorporate multimode interference, MMI, couplers. For the first time to our knowledge, a closed formula for the passing bands bandwidth and crosstalk estimation plots are derived.

© Optical Society of America

1 Introduction

Arrayed waveguide gratings are called to be key devices in the implementation of optical layer functions, due to their spatial and spectral particular characterisitics [1

1. P. Muñoz, D. Pastor, and J. Capmany, “Modeling and designing arrayed waveguide gratings,” J. Light. Tech. (Submitted)

, 2

2. H. Takenouchi, H. Tsuda, and T. Kurokawa, “Analysis of optical-signal processing using an arrayed-waveguide grating,” Opt. Express 6124–135 (2000), http://www.opticsexpress.org/oearchive/source/19103.htm [CrossRef] [PubMed]

]. The potential can be increased if the AWG response is optimized in order to perform filtering as close as possible to the ideal filter. More ideal filtering can be attained through different ways proposed in the literature, such as parabolic waveguide horns [3

3. K. Okamoto and A. Sugita, “Flat spectral response arrayed-waveguide grating multiplexer with parabolic waveguide horns,” Electron. Lett. 32, 1661–1662 (1996). [CrossRef]

], arrayed waveguide amplitude and phase apodisation [4

4. C. Dragone, “Efficient techniques for widening the passband of a wavelength router,” J. Light. Tech. 16, 1895–1906 (1998). [CrossRef]

], the use of Y-branches [5

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

] and multimode interference, MMI, couplers [6

6. J. Soole e.a., “Use of multimode interference couplers to broaden the passband of wavelength-dispersive integrated WDM filters,” IEEE Photon. Technol. Lett. 8, 1340–1342 (1996). [CrossRef]

]. Several AWG models and design procedures have been developed in the literature [1

1. P. Muñoz, D. Pastor, and J. Capmany, “Modeling and designing arrayed waveguide gratings,” J. Light. Tech. (Submitted)

8

8. F. Pizzato, G. Perrone, and I. Montroset, “Arrayed waveguide grating demultiplexers: a new efficient numerical analysis approach,” in Silicon-based Optoelectronics, D.C. Houghton and E. A. Fitzgerald, eds., Proc. SPIE3630, 198–206, 1999.

]. In particular we extend the one developed in [1

1. P. Muñoz, D. Pastor, and J. Capmany, “Modeling and designing arrayed waveguide gratings,” J. Light. Tech. (Submitted)

] to include MMI couplers, still holding the analytical flavour of the model, and expressions for the bandwidth, crosstalk plots and design hints are presented.

Fig. 1. AWG physical layout. Insets, waveguide parameters (left) and FPR coupler layout (right)

2 AWG Theoretical Model Review

This section reviews the AWG model previously developed in [1

1. P. Muñoz, D. Pastor, and J. Capmany, “Modeling and designing arrayed waveguide gratings,” J. Light. Tech. (Submitted)

] for standard AWG’s. The AWG physical layout is shown in Fig. 1, where the abbreviations used along this paper are defined. It consists of a set of input and output waveguides, IW’s and OW’s respectively. The two sets of waveguides are joined by two free space couplers, or free propagation regions, FPR’s, [9

9. C. Dragone e.a., “Efficient N×N star couplers using Fourier optics,” J. Light. Technol. 7, 479–489 (1989). [CrossRef]

, 10

10. C. Dragone e.a., “Efficient multichannel integrated optics star coupler on silicon,” IEEE Photon. Technol. Lett. 1, 241–243 (1989). [CrossRef]

] connected by a set of waveguides named arrayed waveguides, AW’s. The AW’s act as a grating between the two FPR’s [11

11. C. Dragone, C. Edwards, and R. Kistler, “Integrated optics N×N multiplexer on silicon,” IEEE Photon. Technol. Lett. 3, 896–898 (1991). [CrossRef]

]. The lenght of each waveguide in the array is increased by a fixed amount, Δl, with respect to that preceeding it. The inset on the upper left corner of Fig. 1 shows the waveguide layout with its corresponding parameters, the waveguide width, Wx, the gap between waveguides, Gx, and the waveguide spacing, dx, where x=i,w, o, corresponding to IW’s, AW’s and OW’s respectively. The inset on the upper right side of the figure, shows the FPR’s layout. It consists of two sets of waveguides, the IW’s (OW’s) and the AW’s. The AW’s are positioned over a circumference of radius Lf, which is called the focal length, whose center is located in the central IW (OW), CIW (COW). The rest of the IW’s are located over a circumference of diameter Lf, called the Rowland circle [5

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

], as shown in the figure.

It is costumary on the literature to describe the field inside the waveguides by a Gaussian function [2

2. H. Takenouchi, H. Tsuda, and T. Kurokawa, “Analysis of optical-signal processing using an arrayed-waveguide grating,” Opt. Express 6124–135 (2000), http://www.opticsexpress.org/oearchive/source/19103.htm [CrossRef] [PubMed]

, 5

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

, 7

7. H. Takahashi e.a., “Transmission characterisitics of arrayed waveguide N×N wavelength multiplexer,” J. Light. Tech. 13, 447–455 (1995). [CrossRef]

]. Consider the spatial field distribution in the central IW, CIW, described by the following power normalized Gaussian function [2

2. H. Takenouchi, H. Tsuda, and T. Kurokawa, “Analysis of optical-signal processing using an arrayed-waveguide grating,” Opt. Express 6124–135 (2000), http://www.opticsexpress.org/oearchive/source/19103.htm [CrossRef] [PubMed]

]:

βi(x0)=2πωi24e(x0ωi)2
(1)

Bi(x1)=2πωi2α24e(πωix1α)2
(2)

where α is:

α=cLfnsν0
(3)

with c the light speed, Lf the focal length of the FPR’s, ns its refractive index and ν 0 the central design frequency of the AWG, i.e., the frequency corresponding to the center of the passing band from the CIW to the central output waveguide, COW. The total field distribution for the arrayed waveguides can be derived from the summation of the fundamental modes in the waveguides, each one weighted by a factor corresponding to the overlap integral between Eq. (2) and the following expression[13

13. A.W. Snyder and J.D. Love,Optical Waveguide Theory, (Chapman& Hall, New York, 1983).

]:

βg(x1)=2πωg24e(x1ωg)2
(4)

f1(x1)=[Π(x1Ndw)Bi(x1)δω(x1)]2πωg24βg(x1)
(5)

with Π (x1Ndw), being the pi function, whose expression is:

Π(x1Ndω)={1x1Ndω20otherwise
(6)

and δω (x 1) is a summation of delta function

δw(x1)=r=+δ(x1rdw)
(7)

This description of the slab-array interface is an approximation in order to obtain suitable closed expressions for the device response. In order to obtain the proper transition loss in the interface under computer simulation, the corresponding overlap integrals have to be performed.

The length difference between two consecutive AW’s, Δl, is set to an integer multiple, m, of the design wavelength in the waveguides:

Δl=mλ0nc=mcncν0
(8)

m is known as the grating order, and n c is the refraction index in the waveguides. The value of Δl ensures that the light wave from the CIW (p=0), focuses on to the central output waveguide, COW (q=0), at the design frequency ν 0. The constant length increment between consecutive waveguides, is incorporated into Eq. (5) to yield the field distribution over x 2:

f2(x2,ν)=[Bi(x2)Π(x2Ndw)δw(x2)ϕ(x2,ν)]2πωg24βg(x2)
(9)

where ϕ(x 2, ν) is defined as:

ϕ(x2,ν)=ψ(ν)ej2πmνν0x2dw
(10)
ψ(ν)=ei2πν(ncl0c+mNν02)
(11)

To obtain the field distribution over x 3 in front of the OW’s, the Fourier transform of Eq. (9) is used to yield:

f3(x3,ν)=2πωg2α24Bg(x3)ψ(ν)r=fM(x3rαdw+νγ)
(12)

where γ is the frequency-spatial dispersion parameter, FSDP, relating the temporal frequencies of the input waveform to the spatial position at the output plane:

γ=dων0αm
(13)

Bg(x3)=F{βg(x2)}u=x3α=2πωg24e(πωgx3α)2
(14)

The function fM (x 3) is the Fourier transform of the truncated Gaussian function, corresponding to the first two terms at the left hand side of Eq. (9):

fM(x3)=(α28πωi2)14e(x3ωi)2[erf(πωiNdw2α+ix3α)+erf(πωiNdw2αix3α)]
(15)

The result of Eq. (12) encloses all the information of the AWG response. The loss non uniformity is introduced by the term Bg (x 3). The baseline temporal delay of the waveforms travelling through the AWG is incorporated by ψ (ν). The summation term on the right hand side of the equation, is responsible of the shape of the passing bands, and the spatial repetition of the response over the focal plane of the second FPR, depending both on the space coordinate x 3 and the frequency. The spatial repetition period of the response is called Spatial Free Spectral Range, SFSR:

Δx3,FSR=αdw
(16)

which is a measure of the distance between the different diffraction orders of the AWG over the focal plane of the second FPR, for a given frequency. The Frequency Free Spectral Range is the frequency difference between two adjacent spatial diffraction orders that makes them focus to the same point in the x 3 plane [1

1. P. Muñoz, D. Pastor, and J. Capmany, “Modeling and designing arrayed waveguide gratings,” J. Light. Tech. (Submitted)

]:

ΔνFSR=νm
(17)

It is possible to rewrite Eq. (12) depending on these latter parameters (using Eqs. (13), (16) and (17)):

f3(x3,ν)=2πωg2α24Bg(x3)ψ(ν)r=fM(x3Δx3,FSR[rνΔνFSR,0])
(18)

where ΔνFSR ,0 is the FSR for the AWG design frequency, ν 0. This expression illustrates how for a given diffraction order, r, different frequencies focus to different points on the x 3 plane. For ν=ν 0, the order r=m is focused to the COW, as pointed previously.

Fig. 2. MMI coupler layout

Finally, it is possible to calculate the energy from Eq. (18) coupled to the fundamental mode in each OW, evaluating its overlap integral with the field distribution over x 3, described by Eq. (18):

t0,q(ν)=+f3(x3,ν)βo(x3qdo)x3
(19)

where q is the OW number, d o is the OW spacing and β o (x 3) is the OW mode profile similar to Eq. (1). This expression corresponds to the field transmission coefficiente from the CIW, number 0, to an arbitrary OW, q. It is possible to derive an expression for an arbitrary pair of IW-OW, tp,q (ν) [1

1. P. Muñoz, D. Pastor, and J. Capmany, “Modeling and designing arrayed waveguide gratings,” J. Light. Tech. (Submitted)

].

3 IW’s with MMI couplers

For the MMI to give a flat bandpass response, we consider a two-fold imaging configuration, where, as stated in [6

6. J. Soole e.a., “Use of multimode interference couplers to broaden the passband of wavelength-dispersive integrated WDM filters,” IEEE Photon. Technol. Lett. 8, 1340–1342 (1996). [CrossRef]

] the output field from the MMI device is composed by the superposition of two spatially shifted versions of the input field to the MMI. Since this field is given by a Gaussian, the output field distribution can be expressed as the sumation of two Gaussian functions, as ilustrated in Fig. 2. The proposed MMI has a length, Lm:

Lm=3π8(ζ0ζ1)
(20)

where ζ0 and ζ1 are the effective indices of the fundamental and first order MMI coupler modes respectively. With this configuration, a center fed Gaussian field like the one in Eq. (1), is converted into a double Gaussian one, whose normalized power expression is [6

6. J. Soole e.a., “Use of multimode interference couplers to broaden the passband of wavelength-dispersive integrated WDM filters,” IEEE Photon. Technol. Lett. 8, 1340–1342 (1996). [CrossRef]

]:

βi(x0)=[2ωiπ2(1+eΔxm22ωi2)]12[e(x012Δxmωi)2+e(x0+12Δxmωi)2]
(21)

Bi(x1)=[2α2πωi(1+eΔxm22ωi2)]14[eiπΔxmx1α+eiπΔxmx1α]e(πωix1α)2
(22)

The expression for the field distribution over x 3, Eq. (12), can be easily modified using the shifting properties of the Fourier transform:

f3(x3,ν)=πωg22α2(1+eΔxm22ωi2)4Bg(x3)ψ(ν)r=[fM(x3rαdw+νγ+Δxm2)]
+fM(x3rαdw+νγΔxm2)]
(23)

The field for any OW can be calculated inserting this expression into Eq. (19).

4 Modified Design Equations

4.1 Channel Bandpass 3 dB Bandwidth

A new expression to evaluate the channel bandpass 3 dB bandwidth of an AWG with MMI couplers, can be derived under the simplifications detailed in [1

1. P. Muñoz, D. Pastor, and J. Capmany, “Modeling and designing arrayed waveguide gratings,” J. Light. Tech. (Submitted)

], which reduce the field distribution over x 3, Eq. (23), to the Gaussian functions inside fM (x 3). Eq. (19) with q=0, for the transmission between the CIW-COW pair, can be used to derive the expression for the channel bandpass 3 dB bandwidth:

t0,0(Δν)+βi(x3Δνγ)βo(x3)x3
(24)

The fall from the maximum to 3 dB can be calculated using a normalized to the maximum version of the later result:

t0,0,n(Δν)=t0,0(Δν)t0,0(0)=e12(Δνωoγ)2cosh(ΔxmΔν2ωo2γ)
(25)

For 1:1 imaging between the input plane, x 0, and the ouput plane, x 3, with identical input and output waveguides [6

6. J. Soole e.a., “Use of multimode interference couplers to broaden the passband of wavelength-dispersive integrated WDM filters,” IEEE Photon. Technol. Lett. 8, 1340–1342 (1996). [CrossRef]

]:

Δxm=2ωi
(26)

This condition reduces Eq. (25) to:

t0,0,n(x)=e12x2cosh(x)=10320
(27)

with x=Δνωoγ. The zero of this last equation is located at x=1.6173, and hence, with Δνν′bw/2, the expression for the bandwidth is:

Δνbω=2γωo1.6173
(28)

which compared to the one obtained in [1

1. P. Muñoz, D. Pastor, and J. Capmany, “Modeling and designing arrayed waveguide gratings,” J. Light. Tech. (Submitted)

] without MMI’s:

Δνbω=2γωo0.8311
(29)

proves that an increase of nearly two times in the bandwidth is attained for the same AWG configuration.

4.2 Cross Talk

The cross talk can be defined as the ratio between the amount of energy coupled from one IW to an OW, i.e. the “desired” OW, and the amount of energy coupled from the same IW to and adjacent OW to the desired OW, as described in [1

1. P. Muñoz, D. Pastor, and J. Capmany, “Modeling and designing arrayed waveguide gratings,” J. Light. Tech. (Submitted)

]. Hence, it is possible to calculate the cross talk using the following expression:

t0,1(σ,σo)=+βi(un)βo(un)un
(30)

σ=απNdwωo
(31)
σo=doωo
(32)

In Eq. (30), βi (un)=F {Bi (x 1n)}, where Bi (x 1n) and βo (un) are the illumination and the OW mode profile respectively, for the normalized coordinates. For the case of MMI couplers, the modified illumination from Eq. (22) must be used instead of Bi (x 1n). It is possible to write a similar normalized expression in this case, assuming that ωi=ωo and Δxm=2ωi:

Bi(x1n)=e(x1nσ)2(ei2x1nσ+ei2x1nσ)
(33)

βo(un)=e(πσunσo)2
(34)

In both expressions constants have been neglected since the cross talk is defined as the ratio (t0,1(σ,σo)t0,1(σ,0)). The integral in Eq. (30) is solved numericaly yielding the following engineering plots. Fig. 3 shows a surface plot for the cross talk and Fig. 4 shows a parametrized plot of the cross talk versus σ for some values of the ratio σo. The interpretation of this result is analog to the one corresponding to the ordinary AWG. Low uniformity of the illumination, Eq. (33), using small values for σ, yields low cross talk values. In general, reducing σ lowers the side lobes of β′i (un), but also widens its main lobe. For small values of σo, the cross talk value shows no dependence on σ, since despite the side lobes are lower, significant energy is coupled the main lobe of β′ i (un).

For example, if σo=5 in Fig. 4, the lower σ, the lower the cross talk. A reduction of σ can be attained for instance increasing the number of AW’s, N, in Eq. (31), but it is not the only way due to the complex dependencies among the AWG parameters, as described in the design procedure elaborated in [1

1. P. Muñoz, D. Pastor, and J. Capmany, “Modeling and designing arrayed waveguide gratings,” J. Light. Tech. (Submitted)

].

Fig. 3. Cross talk level @ Δνc with MMI at the IW’s

The results in the latter figures are a lowerbound of the cross talk, since the field in the waveguides is assumed to be a Gaussian function, but in fact, “real” fields show exponential decaying tails outside the waveguides [13

13. A.W. Snyder and J.D. Love,Optical Waveguide Theory, (Chapman& Hall, New York, 1983).

], so a higher cross talk penalty has to be expected.

5 MMI-AWG Simulation Results

In order to compare two AWG’s, one with ordinary waveguides, and one with MMI at the IW’s, the design procedure described in [1

1. P. Muñoz, D. Pastor, and J. Capmany, “Modeling and designing arrayed waveguide gratings,” J. Light. Tech. (Submitted)

] is followed starting in both cases with the same High Level Requirements, HLR, summarized in Table 1. For the MMI-based AWG, the new bandwidth equation and cross-talk plot are used. This yields different Physical Parameters, PhP, than for the ordinary AWG. The common PhP used for both cases are the refractive indices of the FPR’s and AW’s, ns=1.4529 and n c=1.453 respectively, the normalized waveguide frequency, V=3, the waveguide width, Wx=14µm, the number of AW’s, N=128 and the shortest waveguide length, l 0=26mm. The module and delay CIW-COW responses obtained for the MMI-based AWG are ploted in Fig. 6. The desired HLR are well matched after simulation as Table 1 states.

Table 1. High Level Requirements for the designed AWG’s

table-icon
View This Table

Both module responses from the ordinary and MMI-based AWG’s, are compared in Fig. 7. For the same HLR, a cross-talk reduction and optimized filtering, flateness and bandpass shape, are achieved, though some extra insertion loss is reported.

Fig. 4. Cross talk level @ Δνc with MMI at the IW’s
Fig. 5. MMI-based 1×16 frequency cyclic AWG module response versus detunning from the design frequency
Fig. 6. MMI-based AWG module (blue) and delay (green) response versus detunning from the design frequency
Fig. 7. MMI-based (red) and ordinary (blue) AWG module responses

6 Conclusions

Acknowledgements

The authors wish to acknowledge financial support from the Spanish CICYT via projects TEL-99-0437 and TIC98-0346. P. Muñoz wishes to acknowledge an FPI grant funding from UPV.

References and links

1.

P. Muñoz, D. Pastor, and J. Capmany, “Modeling and designing arrayed waveguide gratings,” J. Light. Tech. (Submitted)

2.

H. Takenouchi, H. Tsuda, and T. Kurokawa, “Analysis of optical-signal processing using an arrayed-waveguide grating,” Opt. Express 6124–135 (2000), http://www.opticsexpress.org/oearchive/source/19103.htm [CrossRef] [PubMed]

3.

K. Okamoto and A. Sugita, “Flat spectral response arrayed-waveguide grating multiplexer with parabolic waveguide horns,” Electron. Lett. 32, 1661–1662 (1996). [CrossRef]

4.

C. Dragone, “Efficient techniques for widening the passband of a wavelength router,” J. Light. Tech. 16, 1895–1906 (1998). [CrossRef]

5.

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

6.

J. Soole e.a., “Use of multimode interference couplers to broaden the passband of wavelength-dispersive integrated WDM filters,” IEEE Photon. Technol. Lett. 8, 1340–1342 (1996). [CrossRef]

7.

H. Takahashi e.a., “Transmission characterisitics of arrayed waveguide N×N wavelength multiplexer,” J. Light. Tech. 13, 447–455 (1995). [CrossRef]

8.

F. Pizzato, G. Perrone, and I. Montroset, “Arrayed waveguide grating demultiplexers: a new efficient numerical analysis approach,” in Silicon-based Optoelectronics, D.C. Houghton and E. A. Fitzgerald, eds., Proc. SPIE3630, 198–206, 1999.

9.

C. Dragone e.a., “Efficient N×N star couplers using Fourier optics,” J. Light. Technol. 7, 479–489 (1989). [CrossRef]

10.

C. Dragone e.a., “Efficient multichannel integrated optics star coupler on silicon,” IEEE Photon. Technol. Lett. 1, 241–243 (1989). [CrossRef]

11.

C. Dragone, C. Edwards, and R. Kistler, “Integrated optics N×N multiplexer on silicon,” IEEE Photon. Technol. Lett. 3, 896–898 (1991). [CrossRef]

12.

G.P. Agrawal,Fiber-Optic Communications Systems, (John Wiley and Sons, New York, 1997).

13.

A.W. Snyder and J.D. Love,Optical Waveguide Theory, (Chapman& Hall, New York, 1983).

14.

L.B. Soldano and E.C. Pennings, “Optical multi-mode interference devices based on self-imaging: Principles and applications,” J. Light. Technol. 13, 615–627 (1995). [CrossRef]

OCIS Codes
(060.4230) Fiber optics and optical communications : Multiplexing
(070.2580) Fourier optics and signal processing : Paraxial wave optics
(070.6110) Fourier optics and signal processing : Spatial filtering
(230.1950) Optical devices : Diffraction gratings
(230.7390) Optical devices : Waveguides, planar
(350.2460) Other areas of optics : Filters, interference

ToC Category:
Research Papers

History
Original Manuscript: July 20, 2001
Published: September 24, 2001

Citation
Pascual Munoz, Daniel Pastor, and Jose Capmany, "Analysis and design of arrayed waveguide gratings with MMI couplers," Opt. Express 9, 328-338 (2001)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-7-328


Sort:  Journal  |  Reset  

References

  1. P. Munoz, D. Pastor and J. Capmany, "Modeling and designing arrayed waveguide gratings," J. Light. Tech. (Submitted).
  2. H. Takenouchi, H. Tsuda and T. Kurokawa, "Analysis of optical-signal processing using an arrayed-waveguide grating," Opt. Express 6, 124-135 (2000), http://www.opticsexpress.org/oearchive/source/19103.htm [CrossRef] [PubMed]
  3. K. Okamoto and A. Sugita, "Flat spectral response arrayed-waveguide grating multiplexer with parabolic waveguide horns," Electron. Lett. 32, 1661-1662 (1996). [CrossRef]
  4. C. Dragone, "Efficient techniques for widening the passband of a wavelength router," J. Light. Tech. 16, 1895-1906 (1998). [CrossRef]
  5. M. K. Smit and C. van Dam, "PHASAR-based WDM-devices: Principles, design and applications," J. Sel. Top. Quant. Electron. 2, 236-250 (1996). [CrossRef]
  6. J. Soole e.a., "Use of multimode interference couplers to broaden the passband of wavelength-dispersive integrated WDM filters," IEEE Photon. Technol. Lett. 8, 1340-1342 (1996). [CrossRef]
  7. H. Takahashi e.a., "Transmission characterisitics of arrayed waveguide N x N wavelength multiplexer," J. Light. Tech. 13, 447-455 (1995). [CrossRef]
  8. F. Pizzato, G. Perrone and I. Montroset, "Arrayed waveguide grating demultiplexers: a new efficient numerical analysis approach," in Silicon-based Optoelectronics, Houghton, D.C., Fitzgerald, E. A., eds., Proc. SPIE 3630, 198-206, 1999.
  9. C. Dragone e.a., "Efficient N x N star couplers using Fourier optics," J. Light. Technol. 7, 479-489 (1989). [CrossRef]
  10. C. Dragone e.a., "Efficient multichannel integrated optics star coupler on silicon," IEEE Photon. Technol. Lett. 1, 241-243 (1989). [CrossRef]
  11. C. Dragone, C. Edwards and R. Kistler, "Integrated optics N x N multiplexer on silicon," IEEE Photon. Technol. Lett. 3, 896-898 (1991). [CrossRef]
  12. G. P. Agrawal, Fiber-Optic Communications Systems, (John Wiley and Sons, New York, 1997).
  13. A. W. Snyder and J. D. Love, Optical Waveguide Theory, (Chapman & Hall, New York, 1983).
  14. L. B. Soldano and E. C. Pennings, "Optical multi-mode interference devices based on self-imaging: Principles and applications," J. Light. Technol. 13, 615-627 (1995). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited