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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 14 — Jul. 9, 2007
  • pp: 9040–9047
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Numerical study of nonlinear interactions in a multimode waveguide

T. Chaipiboonwong, P. Horak, J. D. Mills, and W. S. Brocklesby  »View Author Affiliations


Optics Express, Vol. 15, Issue 14, pp. 9040-9047 (2007)
http://dx.doi.org/10.1364/OE.15.009040


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Abstract

Multimode nonlinear pulse propagation within a Ta2O5 rectangular rib waveguide has been numerically simulated. The study provides information relating to both the localized spectral evolution along the waveguide and the transverse spectral variation across the guide. The results explain measurements from our previous near-field scanning microscopy experiments that were designed to map continuum generation along and across such waveguides, and that deviated significantly from simple theory. The simulations predict an increased nonlinear phase modulation compared to that occurring in nonlinear single-mode waveguides, due to intermodal nonlinear effects such as cross-phase modulation, leading to an enhanced spectral broadening.

© 2007 Optical Society of America

1. Introduction

2. Waveguide characteristics

The waveguide in the simulation is a ridge of Ta2O5, 0.5 μm high, 4.2 μm wide, with length 6 mm, on a layer of SiO2 on a Si wafer, as displayed in Fig. 1(a). The dispersion constants of the propagating modes are determined by the effective index method [10

10. K. S. Chiang, K. M. Lo, and K. S. Kwok, “Effective-index method with built-in perturbation correction for integrated optical waveguides,” J. Lightwave Technol. 14, 223–228 (1996). [CrossRef]

] in which the wavelength-dependent refractive indices of Ta2O5 and SiO2 are given by the Sellmeier equation whose coefficients are provided by the literature [11

11. D. Smith and P. Baumeister, “Refractive index of some oxide and fluoride coating materials,” Appl. Opt. 18, 111–115 (1979). [CrossRef] [PubMed]

, 12

12. M. Jerman, Z. Qiao, and D. Mergel, “Refractive index of thin films of SiO2, ZrO2, and HfO2 as a function of the films’ mass density,” Appl. Opt. 44, 3006–3012 (2005). [CrossRef] [PubMed]

]. The laser wavelength for the study is 800 nm, and the polarization is such that the electric field aligns along the y–axis. Each mode, denoted by TMmn, has indices m and n which identify the mode field distribution along the x– and y–axes respectively. In total, the guide is able to support around 20 modes for these waveguide parameters. Some examples of modeled mode intensity profiles are shown in Fig. 1(b). Due to the fact that the guide’s width is ~ 8 times greater than its height, there are many more variations of the mode field distribution along the x-axis than along the y-axis, which consists only the first symmetric (n = 0) and antisymmetric modes (n = 1).

Fig. 1. (a) Waveguide structure used in both simulation and previous experiments, (b) examples of modeled mode intensity distribution, (c) and (d) GVD parameter β2 for some symmetric and antisymmetric modes respectively.

Figures 1(c) and 1(d) show the calculated group velocity dispersion β2 for some symmetric and antisymmetric modes respectively. As can be seen, at the laser wavelength 800 nm, all the symmetric modes are in the anomalous regime (β2 < 0) except for the fundamental mode TM00, whereas the antisymmetric modes are in the normal regime (β2 > 0).

3. Simulation results

The multimode nonlinear pulse propagation in this study can be described by the adapted NLS equation for an N-mode field which can be written as

A(p)ztz=(β1(1)β1(p))A(p)zttiβ2(p)22A(p)ztt2+i[q=1Nγ(p)(q)A(q)zt2]A(p)zt
(1)

The coupling nonlinear parameter γ(p)(q) in Eq. (1) depends on the nonlinear refractive index n 2 of the guide material, which is 7.23×10-19 m2/W [7

7. C-Y. Tai, J. S. Wilkinson, N. M. B Perney, M. C. Netti, F. Cattaneo, C. E. Finlayson, and J. J. Baumberg, “Determination of nonlinear refractive index in a Ta2O 5 rib waveguide using self-phase modulation,” Opt. Express 12, 5110–5116 (2004). [CrossRef] [PubMed]

], and on overlapping integrals of the transverse mode intensities [13

13. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2001).

].

γ(p)(q)=n2k0h(p)(q)+F(p)xy2F(q)xy2dxdy(+F(p)xy2dxdy)(+F(q)xy2dxdy)
(2)

Table 1. b1 and b2 at 800 nm wavelength.

table-icon
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Fig. 2. (a) Spectral evolution of multimode pulses along the length of the waveguide. Included also is the SPM spectrum of a single-mode pulse. (b) Details of contributing modes and their intensity ratio.

Fig. 3. (a) Simulated time profile at various propagation distances for the spectrum of 7-mode pulse in Fig. 2 (b) Relative integrated intensity collected by the NSOM probe in the simulation.

Fig. 4. (a) Root-mean-square (RMS) spectral width along the length of the waveguide of the multimode spectra shown in Fig. 2 except for an additional 3-mode curve (violet) which is the mixing of TM00, TM10 and TM20 with intensity ratio 0.5:0.1:0.4. (b) RMS spectral width of individual modes contributing to the three-mode pulse (green) from (a).

So far, propagation losses have been neglected in our analysis. However, based on the above discussion, we can easily predict the spectral modifications if higher-order modes exhibit significant losses. If losses are small on the length scales where temporal walk-off occurs, no significant spectral changes are expected since all broadening due to XPM occurs at an early stage of pulse propagation. On the other hand, if higher-order mode losses are large over much shorter distances, no significant XPM takes place and the observed spectra will resemble the single-mode spectrum of Fig. 2(a).

Fig. 5. (a) Spectrum across the center of the waveguide with three-mode mixing TM00:TM10:TM20 with relative intensity ratio 0.7:0.15:0.15. (b), (c) and (d) Individual modal contributions.

4. Conclusion

References and links

1.

I. Hartl, X. D. Li, C. Chudoba, R. K. Hganta, T. H. Ko, J. G. Fujimoto, J. K. Ranka, and R. S. Windeler, “Ultrahigh-resolution optical coherence tomography using continuum generation in an air-silica microstructure optical fiber,” Opt. Lett. 26, 608–610 (2001). [CrossRef]

2.

S. A. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, and J. L. Hall, “Direct link between microwave and optical frequencies with a 300 THz femtosecond laser comb,” Phys. Rev. Lett. 84, 5102–5105 (2000). [CrossRef] [PubMed]

3.

R. Holzwarth, T. Udem, T. W. Haensch, J. C. Knight, W. J. Wadsworth, and P. S. J. Russell, “Optical frequency synthesizer for precision spectroscopy,” Phys. Rev. Lett. 85, 2264–2267 (2000). [CrossRef] [PubMed]

4.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis,” Science 288, 635–639 (2000). [CrossRef] [PubMed]

5.

S. Spalter, H.Y. Hwang, J. Zimmermann, G. Lenz, T. Katsufuji, S.-W. Cheong, and R. E. Slusher, “Strong self-phase modulation in planar chalcogenide glass waveguides,” Opt. Lett. 27, 363–265 (2002). [CrossRef]

6.

Y. Ruan, W. Li, R. Jarvis, N. Madsen, A. Rode, and B. Luther-Davis, “Fabrication and characterization of low loss rib chalcogenide waveguides made by dry etching,” Opt. Express 12, 5140–5145 (2004). [CrossRef] [PubMed]

7.

C-Y. Tai, J. S. Wilkinson, N. M. B Perney, M. C. Netti, F. Cattaneo, C. E. Finlayson, and J. J. Baumberg, “Determination of nonlinear refractive index in a Ta2O 5 rib waveguide using self-phase modulation,” Opt. Express 12, 5110–5116 (2004). [CrossRef] [PubMed]

8.

J. D. Mills, T. Chaipiboonwong, W. S. Brocklesby, M. D. B. Charlton, M. E. Zoorob, C. Netti, and J. J. Baumberg, “Observation of the developing optical continuum along a nonlinear waveguide,” Opt. Lett. 31, 2459–2461 (2006). [CrossRef] [PubMed]

9.

J. D. Mills, T. Chaipiboonwong, W. S. Brocklesby, M. D. B. Charlton, C. Netti, M. E. Zoorob, and J. J. Baumberg, “Group velocity measurement using spectral interference in near-field scanning optical microscopy,” Appl. Phys. Lett. 89, 051101–1–051101–3 (2006). [CrossRef]

10.

K. S. Chiang, K. M. Lo, and K. S. Kwok, “Effective-index method with built-in perturbation correction for integrated optical waveguides,” J. Lightwave Technol. 14, 223–228 (1996). [CrossRef]

11.

D. Smith and P. Baumeister, “Refractive index of some oxide and fluoride coating materials,” Appl. Opt. 18, 111–115 (1979). [CrossRef] [PubMed]

12.

M. Jerman, Z. Qiao, and D. Mergel, “Refractive index of thin films of SiO2, ZrO2, and HfO2 as a function of the films’ mass density,” Appl. Opt. 44, 3006–3012 (2005). [CrossRef] [PubMed]

13.

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2001).

OCIS Codes
(190.4360) Nonlinear optics : Nonlinear optics, devices
(190.4390) Nonlinear optics : Nonlinear optics, integrated optics
(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

ToC Category:
Nonlinear Optics

History
Original Manuscript: March 30, 2007
Revised Manuscript: May 16, 2007
Manuscript Accepted: May 19, 2007
Published: July 6, 2007

Citation
T. Chaipiboonwong, P. Horak, J. D. Mills, and W. S. Brocklesby, "Numerical study of nonlinear interactions in a multimode waveguide," Opt. Express 15, 9040-9047 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-14-9040


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References

  1. I. Hartl, X. D. Li, C. Chudoba, R. K. Hganta, T. H. Ko, J. G. Fujimoto, J. K. Ranka, and R. S. Windeler, "Ultrahigh-resolution optical coherence tomography using continuum generation in an air-silica microstructure optical fiber," Opt. Lett. 26, 608-610 (2001). [CrossRef]
  2. S. A. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, and J. L. Hall, "Direct link between microwave and optical frequencies with a 300 THz femtosecond laser comb," Phys. Rev. Lett. 84, 5102-5105 (2000). [CrossRef] [PubMed]
  3. R. Holzwarth, T. Udem, T. W. Haensch, J. C. Knight, W. J. Wadsworth, and P. S. J. Russell, "Optical frequency synthesizer for precision spectroscopy," Phys. Rev. Lett. 85, 2264-2267 (2000). [CrossRef] [PubMed]
  4. D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, "Carrier-Envelope Phase Control of FemtosecondMode-Locked Lasers and Direct Optical Frequency Synthesis," Science 288, 635- 639 (2000). [CrossRef] [PubMed]
  5. S. Spalter, H. Y. Hwang, J. Zimmermann, G. Lenz, T. Katsufuji. S.-W. Cheong, and R. E. Slusher, "Strong selfphase modulation in planar chalcogenide glass waveguides," Opt. Lett. 27, 363-265 (2002). [CrossRef]
  6. Y. Ruan, W, Li, R. Jarvis, N. Madsen, A. Rode, and B. Luther-Davis, "Fabrication and characterization of low loss rib chalcogenide waveguides made by dry etching," Opt. Express 12, 5140-5145 (2004). [CrossRef] [PubMed]
  7. C-Y. Tai, J. S. Wilkinson, N. M. B Perney, M. C. Netti, F. Cattaneo, C. E. Finlayson, and J. J. Baumberg, "Determination of nonlinear refractive index in a Ta2O 5 rib waveguide using self-phase modulation," Opt. Express 12, 5110-5116 (2004). [CrossRef] [PubMed]
  8. J. D. Mills, T. Chaipiboonwong, W. S. Brocklesby,M. D. B. Charlton, M. E. Zoorob, C. Netti, and J. J. Baumberg, "Observation of the developing optical continuum along a nonlinear waveguide," Opt. Lett. 31, 2459-2461 (2006). [CrossRef] [PubMed]
  9. J. D. Mills, T. Chaipiboonwong, W. S. Brocklesby, M. D. B. Charlton, C. Netti, M. E. Zoorob, J. J. Baumberg, "Group velocity measurement using spectral interference in near-field scanning optical microscopy," Appl. Phys. Lett. 89, 051101-1-051101-3 (2006). [CrossRef]
  10. K. S. Chiang and K. M. Lo and K. S. Kwok, "Effective-index method with built-in perturbation correction for integrated optical waveguides," J. Lightwave Technol. 14, 223-228 (1996). [CrossRef]
  11. D. Smith and P. Baumeister, "Refractive index of some oxide and fluoride coating materials," Appl. Opt. 18, 111-115 (1979). [CrossRef] [PubMed]
  12. M. Jerman, Z. Qiao, and D. Mergel, "Refractive index of thin films of SiO2, ZrO2, and HfO2 as a function of the films’ mass density," Appl. Opt. 44, 3006-3012 (2005). [CrossRef] [PubMed]
  13. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, 2001).

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