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

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

2006

2005

2004

2002

2001

2000

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]

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]

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]

1996

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]

1979

Baumberg, J. J.

Baumeister, P.

Brocklesby, W. S.

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]

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]

Cattaneo, F.

Chaipiboonwong, T.

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]

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]

Charlton, M. D. B.

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]

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]

Chiang, K. S.

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]

Chudoba, C.

Cundiff, S. T.

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]

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]

Diddams, S. A.

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]

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]

Finlayson, C. E.

Fujimoto, J. G.

Haensch, T. W.

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]

Hall, J. L.

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]

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]

Hartl, I.

Hganta, R. K.

Holzwarth, R.

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]

Hwang, H.Y.

Jerman, M.

Jones, D. J.

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]

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]

Katsufuji, T.

Knight, J. C.

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]

Ko, T. H.

Kwok, K. S.

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]

Lenz, G.

Li, X. D.

Lo, K. M.

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]

Mergel, D.

Mills, J. D.

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]

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]

Netti, C.

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]

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]

Netti, M. C.

Perney, N. M. B

Qiao, Z.

Ranka, J. K.

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]

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]

Ruan, Y.

Russell, P. S. J.

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]

Smith, D.

Spalter, S.

Stentz, A.

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]

Tai, C-Y.

Udem, T.

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]

Wadsworth, W. J.

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]

Wilkinson, J. S.

Windeler, R. S.

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]

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]

Ye, J.

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]

Zimmermann, J.

Zoorob, M. E.

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]

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]

Appl. Opt.

J. Lightwave Technol.

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]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

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]

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]

Science

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]

Other

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]

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

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Figures (5)

Fig. 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.

Fig. 2.
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.
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.
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).

Fig. 5.
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.

Tables (1)

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Table 1. b1 and b2 at 800 nm wavelength.

Equations (2)

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A ( p ) z t z = ( β 1 ( 1 ) β 1 ( p ) ) A ( p ) z t t i β 2 ( p ) 2 2 A ( p ) z t t 2 + i [ q = 1 N γ ( p ) ( q ) A ( q ) z t 2 ] A ( p ) z t
γ ( p ) ( q ) = n 2 k 0 h ( p ) ( q ) + F ( p ) x y 2 F ( q ) x y 2 dxdy ( + F ( p ) x y 2 dxdy ) ( + F ( q ) x y 2 dxdy )

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