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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 16 — Aug. 6, 2007
  • pp: 9883–9892
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All-optical switch based on the local nonlinear Mach-Zehnder interferometer

Yaw-Dong Wu, Min-Lun Huang, Mao-Hsiung Chen, and Rong-Zhan Tasy  »View Author Affiliations


Optics Express, Vol. 15, Issue 16, pp. 9883-9892 (2007)
http://dx.doi.org/10.1364/OE.15.009883


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Abstract

We proposed a new all-optical switch by using the phase modulation of spatial solitons. The proposed structure is composed of the nonlinear Mach-Zehnder interferometer (MZI) with the straight control waveguide, the uniform nonlinear medium and the nonlinear output waveguides. The local nonlinear MZI functions like a phase shifter. The light-induced index changes in the local nonlinear MZI make the output signal beam routing in the uniform nonlinear medium. The all-optical switching scheme employs angular deflection of spatial solitons controlled by phase modulation created in the local nonlinear MZI. By properly launching the control power and increasing the length of the uniform nonlinear medium, this device can be generalized to a 1×N all-optical switch. It would be a potential key component in the applications of ultra-high-speed optical communications and optical data processing system.

© 2007 Optical Society of America

1. Introduction

Demands for ultra-high-speed optical signal processing and computing are increasing dramatically. All-optical ultrafast switching devices and logic gates based on the optical Kerr effect in a nonlinear waveguide have been of particular interest for high-bit rate optical communication and optical computing systems. The interest in nonlinear waveguide device, which has been growing steadily in recent years, stems from their potential use for ultrafast all optical signal processing and optical computing systems. The confinement of optical beam in the small core area and its diffractionless propagation over long distance increase the efficiency of the nonlinear interaction and permit the use of relatively weak nonlinearities. It has been shown that these nonlinear waveguides possess a variety of novel and exciting features such as power-dependent propagation constants and field profiles leading to novel feasibilities for all-optical signal processing and optical computing.

Several all-optical switching and logic devices using optical nonlinearity have ever been proposed and implemented [1

1. Y. D. Wu, M. H. Chen, and C. H. Chu, “All-optical logic device using bent nonlinear taperred Y-junction waveguide structure,” Fiber and Integrated Optics 20, 517 (2001).

12

12. Y. D. Wu, “New all-optical switch based on the spatial soliton repulsion,” Opt. Express 14, 4005 (2006), http://www.opticsexpress.org/abstract.cfm?id=89605. [CrossRef] [PubMed]

]. Most of the conventional all-optical devices are based on uniformly nonlinear structure. Therefore, the whole of the waveguide has optical nonlinearity uniformly. In optical waveguide structure of uniform nonlinearity, several interesting optical properties have been shown, however, there are still more attractive propagation characteristics in waveguide structures combined a nonlinear material with a linear one. Some theoretical studies about the optical waveguide structures made from linear and nonlinear materials have been proposed, for example, a waveguide structure composed of linear films bounded by nonlinear media [13

13. Y. H. Pramono and Endarko, “Nonlinear waveguides for optical logic and computation,” Journal of Nonlinear Optical Physics & Materials 10, 209 (2001). [CrossRef]

19

19. C. W. Kuo, S. Y. Chen, M. H. Chen, C. F. Chang, and Y. D. Wu, “Analyzing multilayer optical waveguide with all nonlinear layers,” Opt. Express 15, 2499 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-5-2499. [CrossRef] [PubMed]

] or a nonlinear film sandwitched between linear media [20

20. A. D. Boardman and P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. 22, 319 (1986). [CrossRef]

24

24. Y. D. Wu and M. H. Chen, “Method for analyzing multilayer nonlinear optical waveguide,” Opt. Express 13, 7982 (2005), http://www.opticsexpress.org/abstract.cfm?id=85750. [CrossRef] [PubMed]

].

Recently, there has been great interesting in the Mach-Zehnder waveguide interferometer device [25

25. X. F. Liu, M. L. Ke, B. C. Qiu, A. C. Bryce, and J. H. Marsh, “Fabrication of monolithically integrated Mach-Zehnder asymmetric interferometer switch,” Indium Phosphide and Related Materials,2000. Conference Proceedings. 2000 International Conference 412 (2000).

28

28. T. Yabu, M. Geshiro, T. Kitamura, K. Nishida, and S. Sawa, “All-optical logic gates containing a two-mode nonlinear waveguide,” IEEE J. Quantum Electron. 38, 37 (2002). [CrossRef]

]. The Mach-Zehnder waveguide interferometer device has been developed for use of modulating, switching, and logic gates, etc. Most of them are operated by the principles of electric-optic effect where the change of refractive index in the arms of Mach-Zehnder interferometer is produced by applying the external electric field. The all-optical Mach-Zehnder waveguide interferometer device has been presented previously [29

29. A. M. Kan’an and P. Likam wa, “Ultrafast all-optical switching not limited by the carrier lifetime in an integrated multiple-quantum-well Mach-Zehnder interferometer,” J. Opt. Soc. Am. B 14, 3217 (1997). [CrossRef]

30

30. Y. H. Pramono and Endarko, “Nonlinear waveguides for optical logic and computation,” Journal of Nonlinear Optical Physics & Materials 10, 209 (2001). [CrossRef]

]. In this paper, we propose a new 1×N all-optical switching device by using the local nonlinear MZI with a straight control waveguide based on the phase modulation of spatial solitons. The light-induced index changes in the MZI break the symmetry of the structure and make the output signal beam routing in the uniform nonlinear medium. By properly launching the control power and increasing the length of the uniform nonlinear medium, the numerical results show that this device can be generalized to a 1×N all-optical switch.

2. Analysis process

The proposed structure is composed of the local nonlinear MZI with the straight control waveguide, and the uniform nonlinear medium, as shown in Fig. 1. The scheme employs angular deflection of spatial solitons controlled by the phase modulation created in the local nonlinear MZI which functions like a phase shifter. The presence of the evanescent tail of the optical beam in the straight control waveguide will introduce the phase difference between the two arms of the local nonlinear MZI, and will lead to a final position shift of the signal beam at the end of the proposed structure. The nonlinear MZI with two local nonlinear waveguides is used to propagate the signal beam. Since the control beam is useless in the following process, a lossy medium at the end of a straight control waveguide is used to attenuate it. For the local nonlinear MZI, the branching angle is θ, w1 the width of the control waveguide, w2 the width of the Mach-Zehnder waveguide, L 1 the length of local nonlinear waveguides, L 2 the length of Mach-Zehnder waveguide, L 3 the length of the uniform nonlinear medium, and l the distance between the local control waveguide and the nearest signal waveguide.

Fig. 1. The proposed all-optical Mach-Zehnder waveguide interferometer structure.
Fig. 2. The structure of multilayer optical waveguides with nonlinear guiding films.

For simplicity, we consider the transverse electric polarized waves propagating along the z direction. The wave equation can be reduced to

2Eyi=ni2c22Eyit2,i=1,2,,m
(1)

with solutions of the form

Eyi(x,z,t)=εi(x)exp[j(ωtβk0z)]
(2)

where ω is the angular frequency, k 0 is the wave number in the free space, and β is the effective refractive index. For a Kerr-type nonlinear medium [54–56], the square of the refractive index of the guiding film can be expressed as[31

31. C. T. Seaton, X. Mai, G. I. Stegeman, and N. G. Winful, “Nonlinear guided wave applications,” Opt. Eng. 24, 593 (1985).

33

33. T. T. Shi and S. Chi, “Beam propagation method analysis of transverse-electric waves propagating in a nonlinear tapered planar waveguide,” J. Opt. Soc. Am. B 8, 2318 (1991). [CrossRef]

]:

ni2=n0i2+αiεi(x)2,i=2,4,,m1
(3)

where n 0i and αi are the linear refractive index and the nonlinear coefficient of the i-th layer nonlinear guiding film, respectively. The transverse electric field in each layer can be expressed as:

ε1(x)=Esexp(p1x)inthesubstrate
(4)
εi(x)=EI(i2)exp{pi[x(i12)d(i32)w]}+EI(i1){exppi[x(i12)d(i12)w]}
i=3,5,,m2intheinteractionlayers
(5)
εi(x)=bicn{Ai[x(i21)(d+w)+x0i]li}
i=2,4,,m1intheguidingfilm,forβ<ni
(6)
εi(x)=bicn{Ai[x(i21)(d+w)+x0i]li}
i=2,4,,m1intheguidingfilm,forβ<ni
(7)
εm(x)=Ecexp{pm[x(m12)d(m32)w]}inthecladding
(8)

where cn is a Jacobian elliptic function, and the constants d and w are the widths of the guiding film and the interaction layer, respectively.

The constants pi, bi, Ai, li, b′i, A′i, and l′i can be expressed as

pi=k0β2ni2, bi2=qi4+2αik02Kiqi2αik02, Ai=[(ai2+bi2)(αik022)]12, li=bi2(ai2+bi2), bi2=Qi4+2αik02Ki+Qi2αik02, Ai=[(ai2+bi2)(αik022)]12, li=bi2(ai2+bi2),

where the constants ai, a′i, qi, Qi, Ki, x 0 i and x′ 0 i are all constants which can be determined by a numerical method on a computer.

3. Numerical results and discussions

Fig. 3. The position shift Δd as a function of the normalized control power Pc/P0.
Fig. 4. The evolutions of the signal beam propagating along the structure (a) with no control beam, (b) with the control power at Pc=0.0573P0, Δd=-12µm, (c) with the control power at Pc=0.0528P0, Δd=-6µm, (d) with the control power at Pc=0.0246P0, Δd=6µm, (e) with the control power at Pc=0.021P0, Δd=12µm.

The proposed 1×N all-optical switching device is composed of the local nonlinear MZI with a straight control waveguide, a uniform nonlinear medium, and N nonlinear output waveguides, as shown in Fig. 5. L 4 is denoted the length of the nonlinear output waveguides. The beam propagation method is used to simulate the evolution of optical waves propagating along the proposed structure. For the simulation, we propose a 1×7 all-optical switching device with the local nonlinear MZI, as shown in Fig. 6. The numerical data have been calculated with the values: θ=0.53°, w1=1µm, w2=2µm, L 1=4000µm, L 2=9000µm, L 3=3000µm, L 4=2000µm, l=3.5µm, λ=1.55µm, nf 0=nu 0=1.55, nc=nc 0=1.545, α=6.3786×10-12 m 2/V 2, P0=23W/m, and the weak control power is varied from Pc=0.01P0 to Pc=0.063P0. When there is no control beam, the output signal beam will propagate straight through the central output waveguide A, as shown in Fig. 7 (a). When the control beam is on, the output signal beam will be switched from one output waveguide to another, as shown in Figs. 7 (b)~(g). When the control power reaches Pc=0.061P0, the output signal beam will be switched to the left output waveguide B with the position shift Δd=-30µm, as shown in Fig. 7 (b), when the control power reaches Pc=0.0578P0, the output signal beam will be switched to the left output waveguide C with the position shift Δd=-20µm, as shown in Fig. 7 (c), when the control power reaches Pc=0.0529P0, the output signal beam will be switched to the left output waveguide D with the position shift Δd=-10µm, as shown in Fig. 7 (d), when the control power reaches Pc=0.025P0, the output signal beam will be switched to the right output waveguide E with the position shift Δd=10µm, as shown in Fig. 7 (e), when the control power reaches Pc=0.021P0, the output signal beam will be switched to the right output waveguide F with the position shift Δd=20µm, as shown in Fig. 7 (f), and when the control power reaches Pc=0.0118P0, the output signal beam will be switched to the right output waveguide G with the position shift Δd=30µm, as shown in Fig. 7 (g). For the numerical results shown above, we confirm that the proposed device could function as a new 1×N all-optical switching device by properly launching the control power and increasing the length of the uniform nonlinear medium.

Fig. 5. The proposed structure of a 1×N all-optical switching device.
Fig. 6. The proposed structure of a 1×7 all-optical switching device.
Fig. 7. The evolutions of the signal beam propagating along the structure (a) with no control beam, (b) with the control power at Pc=0.061P0, Δd=-30µm, (c) with the control power at Pc=0.0578P0, Δd=-20µm, (d) with the control power at Pc=0.0529P0, Δd=-10µm, (e) with the control power at Pc=0.025P0, Δd=10µm, (f) with the control power at Pc=0.021P0, Δd=20µm, (g) with the control power at Pc=0.0118P0, Δd=30µm.

4. Conclusions

We proposed a new all-optical switch by using the position shift of the spatial solitons controlled by phase modulation created in the local nonlinear MZI with a straight control waveguide. The local nonlinear MZI functions like a phase shifter. The presence of the evanescent tail of the optical beam in the straight control waveguide will introduce the phase difference between the two arms of the local nonlinear MZI, and will make the output signal beam switching from one output waveguide to another. By properly launching the control power and increasing the length of the uniform nonlinear medium, this device can be generalized to a 1×N all-optical switch. It would be a potential key component in the applications of ultra-high-speed optical communications and optical data processing systems.

Acknowledgment

This work was partly supported by National Science Council R. O. C. and Ministry of Education R. O. C. under Grant No. 95C9031 and 95TSFC9031.

References and links

1.

Y. D. Wu, M. H. Chen, and C. H. Chu, “All-optical logic device using bent nonlinear taperred Y-junction waveguide structure,” Fiber and Integrated Optics 20, 517 (2001).

2.

M. O. Twati and T.J.F Pavlasek, “A three-wavelength Mach-Zehnder optical demultiplexer by on step ion-exchange in glass,” Opt. Commun. 206, 327 (2002). [CrossRef]

3.

Y. D. Wu, H. J. Chen, and Tasi, “Novel all-optical switching device with localized nonlinearity,” Optical Society of America, Optics in Computing Devices 297 (2002).

4.

Y. D. Wu, M. H. Chen, and R. Z. Tasy, “A new all-optical switching device by using the nonlinear Mach-Zehnder interferometer with a control waveguides,” Proceedings CLEO/Pacific Rim Conference on Laser and Electro-OpticsI292 (2003).

5.

S. Cao, J. Chen, J. N. Damask, C. R. Doerr, L. Guiziou, G. Harvey, Y. Hibino, H. Li, s. Suzuki, K. Y. Wu, and P. Xie, “Interleaver technology: comparisons and applications requirements,” J. Lightwave Technol. 22, 281 (2004). [CrossRef]

6.

Y. L. Lee et al., “Channel-selective wavelength conversion and tuning in periodically poled Ti:LiNbO3 waveguides,” Opt. Expr. 12, 2649 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe1212-2649. [CrossRef]

7.

Y. D. Wu, “Nonlinear all-optical switching device by using the spatial soliton collision,” Fiber and Integrated Optics 23, 387 (2004). [CrossRef]

8.

Y. D. Wu, “Coupled-soliton all-optical logic device with two parallel tapered waveguide,” Fiber and Integrated Optics 23, 405 (2004). [CrossRef]

9.

Y. D. Wu, “New all-optical wavelength auto-router based on spatial solitons,” Opt. Express 12, 4172 (2004), http://www.opticsexpress.org/abstract.cfm?id=89605. [CrossRef] [PubMed]

10.

Y. D. Wu, “All-Optical Logic Gates by Using Multibranch Waveguide Structure With Localized Optical Nonlinearity,” IEEE J. Sel. Top. Quantum. Electron. 11, 307 (2005). [CrossRef]

11.

Y. D. Wu, “All-optical 1×N All-Optical Switching Device by Using the Phase Modulation of Spatial Solitons,” Appl. Optics 44, 4144 (2005) [CrossRef]

12.

Y. D. Wu, “New all-optical switch based on the spatial soliton repulsion,” Opt. Express 14, 4005 (2006), http://www.opticsexpress.org/abstract.cfm?id=89605. [CrossRef] [PubMed]

13.

Y. H. Pramono and Endarko, “Nonlinear waveguides for optical logic and computation,” Journal of Nonlinear Optical Physics & Materials 10, 209 (2001). [CrossRef]

14.

C. T. Steaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and D. Smith, “Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media,” IEEE J. Quantum Electron. 21, 774 (1985). [CrossRef]

15.

S. She and S. Zhang, “Analysis of nonlinear TE waves in a periodic refractive index waveguide with nonlinear cladding,” Opt. Commun. 161, 141 (1999). [CrossRef]

16.

Y. D. Wu, M. H. Chen, and H. J. Tasi, “A general method for analyzing the multilayer optical waveguide with nonlinear cladding and substrate”, SPIE Design, Fabrication, and Characterization of Photonic Device ▫ 4594, 323 (2001).

17.

Y. D. Wu and M. H. Chen, “Analyzing multiplayer optical waveguides with nonlinear cladding and substrates,” J. Opt. Soc. Am. B 19, 1737 (2002). [CrossRef]

18.

Y. D. Wu and M. H. Chen, “The fundamental theory of the symmetric three layer nonlinear optical waveguide structures and the numerical simulation,” J. Nat. Kao. Uni. of App. Sci. 32, 133 (2002).

19.

C. W. Kuo, S. Y. Chen, M. H. Chen, C. F. Chang, and Y. D. Wu, “Analyzing multilayer optical waveguide with all nonlinear layers,” Opt. Express 15, 2499 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-5-2499. [CrossRef] [PubMed]

20.

A. D. Boardman and P. Egan, “Optically nonlinear waves in thin films,” IEEE J. Quantum Electron. 22, 319 (1986). [CrossRef]

21.

H. Murata, M. Izutsu, and T. Sueta, “Optical bistability and all-optical switching in novel waveguide functions with localized optical nonlinearity,” J. Light. Technol. 16, 833 (1998). [CrossRef]

22.

Y. D. Wu and Y. C. Jang, “Analyzing and numerical study of seven-layer optical saveguide with localized nonlinear central guiding film,” Proceedings Electrical and Information Engineering Symposium24 (2003).

23.

Y. D. Wu, “Analyzing Multilayer Optical Waveguides with a Localized Arbitrary Nonlinear Guiding Film,” IEEE J. Quantum. Electron. 40, 529 (2004). [CrossRef]

24.

Y. D. Wu and M. H. Chen, “Method for analyzing multilayer nonlinear optical waveguide,” Opt. Express 13, 7982 (2005), http://www.opticsexpress.org/abstract.cfm?id=85750. [CrossRef] [PubMed]

25.

X. F. Liu, M. L. Ke, B. C. Qiu, A. C. Bryce, and J. H. Marsh, “Fabrication of monolithically integrated Mach-Zehnder asymmetric interferometer switch,” Indium Phosphide and Related Materials,2000. Conference Proceedings. 2000 International Conference 412 (2000).

26.

H. Ehlers, M. Schlak, and U. H. P. Fischer, “Multi-fiber-chip-coupling modules for monolithically integrated Mach-Zehnder interferometers for TDM/WDM communication systems,” Optical Fiber Communication Conference and Exhibit 3, WDD66-1 (2001).

27.

L. Pavelescu, “Simplified design relationships for silicon integrated optical pressure sensors based on Mach-Zehnder interferometry with antiresonant reflecting optical waveguides,” Semiconductor Conference, 2001, CAS 2001 Proceedings, International 1, 201 (2001). [CrossRef]

28.

T. Yabu, M. Geshiro, T. Kitamura, K. Nishida, and S. Sawa, “All-optical logic gates containing a two-mode nonlinear waveguide,” IEEE J. Quantum Electron. 38, 37 (2002). [CrossRef]

29.

A. M. Kan’an and P. Likam wa, “Ultrafast all-optical switching not limited by the carrier lifetime in an integrated multiple-quantum-well Mach-Zehnder interferometer,” J. Opt. Soc. Am. B 14, 3217 (1997). [CrossRef]

30.

Y. H. Pramono and Endarko, “Nonlinear waveguides for optical logic and computation,” Journal of Nonlinear Optical Physics & Materials 10, 209 (2001). [CrossRef]

31.

C. T. Seaton, X. Mai, G. I. Stegeman, and N. G. Winful, “Nonlinear guided wave applications,” Opt. Eng. 24, 593 (1985).

32.

R. A. Sammut, Q. Y. Li, and C. Pask, “Variational approximations and mode stability in planar nonlinear waveguides,” J. Opt. Soc, Am. B 9, 884 (1992). [CrossRef]

33.

T. T. Shi and S. Chi, “Beam propagation method analysis of transverse-electric waves propagating in a nonlinear tapered planar waveguide,” J. Opt. Soc. Am. B 8, 2318 (1991). [CrossRef]

34.

Y. Chung and N. Dagli, “As assessment of finite difference beam propagation method,” IEEE J. Quantum Electron. 26, 529 (1994).

OCIS Codes
(190.0190) Nonlinear optics : Nonlinear optics
(190.3270) Nonlinear optics : Kerr effect
(230.1150) Optical devices : All-optical devices

ToC Category:
Optical Devices

History
Original Manuscript: May 24, 2007
Revised Manuscript: June 28, 2007
Manuscript Accepted: June 28, 2007
Published: July 23, 2007

Citation
Yaw-Dong Wu, Min-Lun Huang, Mao-Hsiung Chen, and Rong-Zhan Tasy, "All-optical switch based on the local nonlinear Mach-Zehnder interferometer," Opt. Express 15, 9883-9892 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-16-9883


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References

  1. Y. D. Wu, M. H. Chen, and C. H. Chu, "All-optical logic device using bent nonlinear taperred Y-junction waveguide structure," Fiber and Integrated Optics 20, 517 (2001).
  2. M. O. Twati, T.J.F Pavlasek, "A three-wavelength Mach-Zehnder optical demultiplexer by on step ion-exchange in glass," Opt. Commun. 206, 327 (2002). [CrossRef]
  3. Y. D. Wu, M. H. Chen, and H. J. Tasi, "Novel all-optical switching device with localized nonlinearity," Optical Society of America, Optics in Computing Devices 297 (2002).
  4. Y. D. Wu, M. H. Chen, and R. Z. Tasy, "A new all-optical switching device by using the nonlinear Mach-Zehnder interferometer with a control waveguides," Proceedings CLEO/Pacific Rim Conference on Laser and Electro-Optics I 292 (2003).
  5. S. Cao, J. Chen, J. N. Damask, C. R. Doerr, L. Guiziou, G. Harvey, Y. Hibino, H. Li, s. Suzuki, K. Y. Wu, and P. Xie, "Interleaver technology: comparisons and applications requirements," J. Lightwave Technol. 22, 281 (2004). [CrossRef]
  6. Y. L. Lee et al., "Channel-selective wavelength conversion and tuning in periodically poled Ti:LiNbO3 waveguides," Opt. Express 12, 2649 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe1212-2649. [CrossRef]
  7. Y. D. Wu, "Nonlinear all-optical switching device by using the spatial soliton collision," Fiber and Integrated Optics 23, 387 (2004). [CrossRef]
  8. Y. D. Wu, "Coupled-soliton all-optical logic device with two parallel tapered waveguide," Fiber and Integrated Optics 23, 405 (2004). [CrossRef]
  9. Y. D. Wu, "New all-optical wavelength auto-router based on spatial solitons," Opt. Express 12, 4172 (2004), http://www.opticsexpress.org/abstract.cfm?id=89605. [CrossRef] [PubMed]
  10. Y. D. Wu, "All-Optical Logic Gates by Using Multibranch Waveguide Structure With Localized Optical Nonlinearity," IEEE J. Sel. Top. Quantum. Electron. 11, 307 (2005). [CrossRef]
  11. Y. D. Wu, "All-optical 1×N All-Optical Switching Device by Using the Phase Modulation of Spatial Solitons," Appl. Optics 44, 4144 (2005) [CrossRef]
  12. Y. D. Wu, " New all-optical switch based on the spatial soliton repulsion," Opt. Express 14, 4005 (2006), http://www.opticsexpress.org/abstract.cfm?id=89605. [CrossRef] [PubMed]
  13. Y. H. Pramono, and Endarko, "Nonlinear waveguides for optical logic and computation," Journal of Nonlinear Optical Physics & Materials 10, 209 (2001). [CrossRef]
  14. C. T. Steaton, J. D. Valera, R. L. Shoemaker, G. I. Stegeman, J. T. Chilwell, and D. Smith, "Calculations of nonlinear TE waves guided by thin dielectric films bounded by nonlinear media," IEEE J. Quantum Electron. 21, 774 (1985). [CrossRef]
  15. S. She and S. Zhang, "Analysis of nonlinear TE waves in a periodic refractive index waveguide with nonlinear cladding," Opt. Commun. 161, 141 (1999). [CrossRef]
  16. Y. D. Wu, M. H. Chen, and H. J. Tasi, "A general method for analyzing the multilayer optical waveguide with nonlinear cladding and substrate", SPIE Design, Fabrication, and Characterization of Photonic Device II 4594, 323 (2001).
  17. Y. D. Wu and M. H. Chen, "Analyzing multiplayer optical waveguides with nonlinear cladding and substrates," J. Opt. Soc. Am. B 19, 1737 (2002). [CrossRef]
  18. Y. D. Wu and M. H. Chen, "The fundamental theory of the symmetric three layer nonlinear optical waveguide structures and the numerical simulation," J. Nat. Kao. Uni. of App. Sci. 32, 133 (2002).
  19. C. W. Kuo, S. Y. Chen, M. H. Chen, C. F. Chang, and Y. D. Wu, "Analyzing multilayer optical waveguide with all nonlinear layers," Opt. Express 15, 2499 (2007), http://www.opticsinfobase.org/abstract.cfm?URI=oe-15-5-2499. [CrossRef] [PubMed]
  20. A. D. Boardman and P. Egan, "Optically nonlinear waves in thin films," IEEE J. Quantum Electron. 22, 319 (1986). [CrossRef]
  21. H. Murata, M. Izutsu, and T. Sueta, "Optical bistability and all-optical switching in novel waveguide functions with localized optical nonlinearity," J. Light. Technol. 16, 833 (1998). [CrossRef]
  22. Y. D. Wu and Y. C. Jang, "Analyzing and numerical study of seven-layer optical saveguide with localized nonlinear central guiding film," Proceedings Electrical and Information Engineering Symposium 24 (2003).
  23. Y. D. Wu, "Analyzing Multilayer Optical Waveguides with a Localized Arbitrary Nonlinear Guiding Film," IEEE J. Quantum. Electron. 40, 529 (2004). [CrossRef]
  24. Y. D. Wu, and M. H. Chen, "Method for analyzing multilayer nonlinear optical waveguide," Opt. Express 13, 7982 (2005), http://www.opticsexpress.org/abstract.cfm?id=85750. [CrossRef] [PubMed]
  25. X. F. Liu, M. L. Ke, B. C. Qiu, A. C. Bryce, and J. H. Marsh, "Fabrication of monolithically integrated Mach-Zehnder asymmetric interferometer switch," Indium Phosphide and Related Materials,2000. Conference Proceedings. 2000 International Conference 412 (2000).
  26. H. Ehlers, M. Schlak, and U. H. P. Fischer, "Multi-fiber-chip-coupling modules for monolithically integrated Mach-Zehnder interferometers for TDM/WDM communication systems," Optical Fiber Communication Conference and Exhibit 3, WDD66-1 (2001).
  27. L. Pavelescu, "Simplified design relationships for silicon integrated optical pressure sensors based on Mach-Zehnder interferometry with antiresonant reflecting optical waveguides,"Semiconductor Conference, 2001, CAS 2001 Proceedings, International 1, 201 (2001). [CrossRef]
  28. T. Yabu, M. Geshiro, T. Kitamura, K. Nishida, and S. Sawa, "All-optical logic gates containing a two-mode nonlinear waveguide," IEEE J. Quantum Electron. 38, 37 (2002). [CrossRef]
  29. A. M. Kan’an and P. Likam wa, "Ultrafast all-optical switching not limited by the carrier lifetime in an integrated multiple-quantum-well Mach-Zehnder interferometer," J. Opt. Soc. Am. B 14, 3217 (1997). [CrossRef]
  30. Y. H. Pramono and Endarko, "Nonlinear waveguides for optical logic and computation," Journal of Nonlinear Optical Physics & Materials 10, 209 (2001). [CrossRef]
  31. C. T. Seaton, X. Mai, G. I. Stegeman, N. G. Winful, "Nonlinear guided wave applications," Opt. Eng. 24, 593 (1985).
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