Abstract

We investigated the dispersion characteristics of submicron sized AlGaAs waveguides. Numerical simulations shows that the tight confinement of the optical waves in such nanowires leads to strong variations of the dispersion characteristics compared to classic, weakly guided waveguides of the same material system. We found numerically that the investigated structure has negative GVD for the TE mode provided the waveguide width is between 670 nm and 280 nm. Experimental data obtained from 300 μm - 1 mm long wires confirms the numerical results.

© 2007 Optical Society of America

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    [CrossRef]

2006

2005

S. F. Preble, Q. Xu, B. S. Schmidt, and M. Lipson, "Ultrafast all-optical modulation on a silicon chip," Opt. Lett. 30, 2891-2893 (2005).
[CrossRef] [PubMed]

M. A. Foster, J. M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. L. Gaeta, "Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation," Appl. Phys. B 81, 363-367 (2005).
[CrossRef]

2004

2000

S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigg, "The refractive index of AlxGa1-xAs below the band gap: Accurate determination and empirical modeling," J. Appl. Phys. 87, 7825-7837 (2000).
[CrossRef]

1998

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

1996

1994

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Alhemyari, C. C. Yang, C. H. Lin, H. H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, "Algaas Below Half Bandgap - the Silicon of Nonlinear-Optical Materials," Int. J. Nonlinear Opt. Phys. 3, 347-371 (1994).
[CrossRef]

1992

J. S. Aitchison, K. Alhemyari, C. N. Ironside, R. S. Grant, and W. Sibbett, "Observation of Spatial Solitons in Algaas Wave-Guides," Electron Lett. 28, 1879-1880 (1992).
[CrossRef]

1991

J. S. Aitchison, A. H. Kean, C. N. Ironside, A. Villeneuve, and G. I. Stegeman, "Ultrafast All-Optical Switching in Al0.18ga0.82as Directional Coupler in 1.55 Mu-M Spectral Region," Electron Lett. 27, 1709-1710 (1991).
[CrossRef]

Aitchison, J. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

J. U. Kang, G. I. Stegeman, and J. S. Aitchison, "One-dimensional spatial soliton dragging, trapping, and all-optical switching in AlGaAs waveguides," Opt. Lett. 21, 189-191 (1996).
[CrossRef] [PubMed]

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Alhemyari, C. C. Yang, C. H. Lin, H. H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, "Algaas Below Half Bandgap - the Silicon of Nonlinear-Optical Materials," Int. J. Nonlinear Opt. Phys. 3, 347-371 (1994).
[CrossRef]

J. S. Aitchison, K. Alhemyari, C. N. Ironside, R. S. Grant, and W. Sibbett, "Observation of Spatial Solitons in Algaas Wave-Guides," Electron Lett. 28, 1879-1880 (1992).
[CrossRef]

J. S. Aitchison, A. H. Kean, C. N. Ironside, A. Villeneuve, and G. I. Stegeman, "Ultrafast All-Optical Switching in Al0.18ga0.82as Directional Coupler in 1.55 Mu-M Spectral Region," Electron Lett. 27, 1709-1710 (1991).
[CrossRef]

Alhemyari, K.

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Alhemyari, C. C. Yang, C. H. Lin, H. H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, "Algaas Below Half Bandgap - the Silicon of Nonlinear-Optical Materials," Int. J. Nonlinear Opt. Phys. 3, 347-371 (1994).
[CrossRef]

J. S. Aitchison, K. Alhemyari, C. N. Ironside, R. S. Grant, and W. Sibbett, "Observation of Spatial Solitons in Algaas Wave-Guides," Electron Lett. 28, 1879-1880 (1992).
[CrossRef]

Almeida, V. R.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, "All-optical control of light on a silicon chip," Nature 431,1081-1084 (2004).
[CrossRef] [PubMed]

Barrios, C. A.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, "All-optical control of light on a silicon chip," Nature 431,1081-1084 (2004).
[CrossRef] [PubMed]

Bergman, K.

Birks, T. A.

Bolger, J. A.

Boyd, A. R.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Cao, Q.

M. A. Foster, J. M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. L. Gaeta, "Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation," Appl. Phys. B 81, 363-367 (2005).
[CrossRef]

Christodoulides, D. N.

De Angelis, C.

Dudley, J. M.

M. A. Foster, J. M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. L. Gaeta, "Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation," Appl. Phys. B 81, 363-367 (2005).
[CrossRef]

Eggleton, B. J.

Eisenberg, H. S.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

El-Ganainy, R.

Foster, M. A.

A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, "Tailored anomalous group-velocity dispersion in silicon channel waveguides," Opt. Express 14, 4357-4362 (2006).
[CrossRef] [PubMed]

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, "Broad-band optical parametric gain on a silicon photonic chip," Nature 441, 960-963 (2006).
[CrossRef] [PubMed]

M. A. Foster, J. M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. L. Gaeta, "Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation," Appl. Phys. B 81, 363-367 (2005).
[CrossRef]

Gaeta, A. L.

A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, "Tailored anomalous group-velocity dispersion in silicon channel waveguides," Opt. Express 14, 4357-4362 (2006).
[CrossRef] [PubMed]

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, "Broad-band optical parametric gain on a silicon photonic chip," Nature 441, 960-963 (2006).
[CrossRef] [PubMed]

M. A. Foster, J. M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. L. Gaeta, "Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation," Appl. Phys. B 81, 363-367 (2005).
[CrossRef]

Gehrsitz, S.

S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigg, "The refractive index of AlxGa1-xAs below the band gap: Accurate determination and empirical modeling," J. Appl. Phys. 87, 7825-7837 (2000).
[CrossRef]

Gourgon, C.

S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigg, "The refractive index of AlxGa1-xAs below the band gap: Accurate determination and empirical modeling," J. Appl. Phys. 87, 7825-7837 (2000).
[CrossRef]

Grant, R. S.

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Alhemyari, C. C. Yang, C. H. Lin, H. H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, "Algaas Below Half Bandgap - the Silicon of Nonlinear-Optical Materials," Int. J. Nonlinear Opt. Phys. 3, 347-371 (1994).
[CrossRef]

J. S. Aitchison, K. Alhemyari, C. N. Ironside, R. S. Grant, and W. Sibbett, "Observation of Spatial Solitons in Algaas Wave-Guides," Electron Lett. 28, 1879-1880 (1992).
[CrossRef]

Herres, N.

S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigg, "The refractive index of AlxGa1-xAs below the band gap: Accurate determination and empirical modeling," J. Appl. Phys. 87, 7825-7837 (2000).
[CrossRef]

Ironside, C. N.

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Alhemyari, C. C. Yang, C. H. Lin, H. H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, "Algaas Below Half Bandgap - the Silicon of Nonlinear-Optical Materials," Int. J. Nonlinear Opt. Phys. 3, 347-371 (1994).
[CrossRef]

J. S. Aitchison, K. Alhemyari, C. N. Ironside, R. S. Grant, and W. Sibbett, "Observation of Spatial Solitons in Algaas Wave-Guides," Electron Lett. 28, 1879-1880 (1992).
[CrossRef]

J. S. Aitchison, A. H. Kean, C. N. Ironside, A. Villeneuve, and G. I. Stegeman, "Ultrafast All-Optical Switching in Al0.18ga0.82as Directional Coupler in 1.55 Mu-M Spectral Region," Electron Lett. 27, 1709-1710 (1991).
[CrossRef]

Iwanow, R.

Kang, J.

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Alhemyari, C. C. Yang, C. H. Lin, H. H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, "Algaas Below Half Bandgap - the Silicon of Nonlinear-Optical Materials," Int. J. Nonlinear Opt. Phys. 3, 347-371 (1994).
[CrossRef]

Kang, J. U.

Kean, A. H.

J. S. Aitchison, A. H. Kean, C. N. Ironside, A. Villeneuve, and G. I. Stegeman, "Ultrafast All-Optical Switching in Al0.18ga0.82as Directional Coupler in 1.55 Mu-M Spectral Region," Electron Lett. 27, 1709-1710 (1991).
[CrossRef]

Kennedy, G. T.

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Alhemyari, C. C. Yang, C. H. Lin, H. H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, "Algaas Below Half Bandgap - the Silicon of Nonlinear-Optical Materials," Int. J. Nonlinear Opt. Phys. 3, 347-371 (1994).
[CrossRef]

Kibler, B.

M. A. Foster, J. M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. L. Gaeta, "Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation," Appl. Phys. B 81, 363-367 (2005).
[CrossRef]

Lee, B. G.

Lee, D.

M. A. Foster, J. M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. L. Gaeta, "Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation," Appl. Phys. B 81, 363-367 (2005).
[CrossRef]

Leon-Saval, S. G.

Lin, C. H.

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Alhemyari, C. C. Yang, C. H. Lin, H. H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, "Algaas Below Half Bandgap - the Silicon of Nonlinear-Optical Materials," Int. J. Nonlinear Opt. Phys. 3, 347-371 (1994).
[CrossRef]

Lin, H. H.

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Alhemyari, C. C. Yang, C. H. Lin, H. H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, "Algaas Below Half Bandgap - the Silicon of Nonlinear-Optical Materials," Int. J. Nonlinear Opt. Phys. 3, 347-371 (1994).
[CrossRef]

Lipson, M.

Lize, Y. K.

Locatelli, A.

Magi, E. C.

Makris, K. G.

Manolatou, C.

Mason, M. W.

Modotto, D.

Mokhov, S.

Morandotti, R.

Panepucci, R. R.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, "All-optical control of light on a silicon chip," Nature 431,1081-1084 (2004).
[CrossRef] [PubMed]

Pozzi, F.

Preble, S. F.

Reinhart, F. K.

S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigg, "The refractive index of AlxGa1-xAs below the band gap: Accurate determination and empirical modeling," J. Appl. Phys. 87, 7825-7837 (2000).
[CrossRef]

Russell, P. S. J.

Schmidt, B. S.

Sharping, J. E.

A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, "Tailored anomalous group-velocity dispersion in silicon channel waveguides," Opt. Express 14, 4357-4362 (2006).
[CrossRef] [PubMed]

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, "Broad-band optical parametric gain on a silicon photonic chip," Nature 441, 960-963 (2006).
[CrossRef] [PubMed]

Sibbett, W.

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Alhemyari, C. C. Yang, C. H. Lin, H. H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, "Algaas Below Half Bandgap - the Silicon of Nonlinear-Optical Materials," Int. J. Nonlinear Opt. Phys. 3, 347-371 (1994).
[CrossRef]

J. S. Aitchison, K. Alhemyari, C. N. Ironside, R. S. Grant, and W. Sibbett, "Observation of Spatial Solitons in Algaas Wave-Guides," Electron Lett. 28, 1879-1880 (1992).
[CrossRef]

Sigg, H.

S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigg, "The refractive index of AlxGa1-xAs below the band gap: Accurate determination and empirical modeling," J. Appl. Phys. 87, 7825-7837 (2000).
[CrossRef]

Silberberg, Y.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Siviloglou, G. A.

Small, B. A.

Sorel, M.

Stanley, C. R.

Stegeman, G. I.

G. A. Siviloglou, S. Suntsov, R. El-Ganainy, R. Iwanow, G. I. Stegeman, D. N. Christodoulides, R. Morandotti, D. Modotto, A. Locatelli, C. De Angelis, F. Pozzi, C. R. Stanley, and M. Sorel, "Enhanced third-order nonlinear effects in optical AlGaAs nanowires," Opt. Express 14, 9377-9384 (2006).
[CrossRef] [PubMed]

J. U. Kang, G. I. Stegeman, and J. S. Aitchison, "One-dimensional spatial soliton dragging, trapping, and all-optical switching in AlGaAs waveguides," Opt. Lett. 21, 189-191 (1996).
[CrossRef] [PubMed]

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Alhemyari, C. C. Yang, C. H. Lin, H. H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, "Algaas Below Half Bandgap - the Silicon of Nonlinear-Optical Materials," Int. J. Nonlinear Opt. Phys. 3, 347-371 (1994).
[CrossRef]

J. S. Aitchison, A. H. Kean, C. N. Ironside, A. Villeneuve, and G. I. Stegeman, "Ultrafast All-Optical Switching in Al0.18ga0.82as Directional Coupler in 1.55 Mu-M Spectral Region," Electron Lett. 27, 1709-1710 (1991).
[CrossRef]

Steinvurzel, P.

Suntsov, S.

Ta'eed, V. G.

Trebino, R.

M. A. Foster, J. M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. L. Gaeta, "Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation," Appl. Phys. B 81, 363-367 (2005).
[CrossRef]

Turner, A. C.

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, "Broad-band optical parametric gain on a silicon photonic chip," Nature 441, 960-963 (2006).
[CrossRef] [PubMed]

A. C. Turner, C. Manolatou, B. S. Schmidt, M. Lipson, M. A. Foster, J. E. Sharping, and A. L. Gaeta, "Tailored anomalous group-velocity dispersion in silicon channel waveguides," Opt. Express 14, 4357-4362 (2006).
[CrossRef] [PubMed]

Villeneuve, A.

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Alhemyari, C. C. Yang, C. H. Lin, H. H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, "Algaas Below Half Bandgap - the Silicon of Nonlinear-Optical Materials," Int. J. Nonlinear Opt. Phys. 3, 347-371 (1994).
[CrossRef]

J. S. Aitchison, A. H. Kean, C. N. Ironside, A. Villeneuve, and G. I. Stegeman, "Ultrafast All-Optical Switching in Al0.18ga0.82as Directional Coupler in 1.55 Mu-M Spectral Region," Electron Lett. 27, 1709-1710 (1991).
[CrossRef]

Vonlanthen, A.

S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigg, "The refractive index of AlxGa1-xAs below the band gap: Accurate determination and empirical modeling," J. Appl. Phys. 87, 7825-7837 (2000).
[CrossRef]

Wadsworth, W. J.

Xu, Q.

Yang, C. C.

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Alhemyari, C. C. Yang, C. H. Lin, H. H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, "Algaas Below Half Bandgap - the Silicon of Nonlinear-Optical Materials," Int. J. Nonlinear Opt. Phys. 3, 347-371 (1994).
[CrossRef]

Appl. Phys. B

M. A. Foster, J. M. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. L. Gaeta, "Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation," Appl. Phys. B 81, 363-367 (2005).
[CrossRef]

Electron Lett.

J. S. Aitchison, A. H. Kean, C. N. Ironside, A. Villeneuve, and G. I. Stegeman, "Ultrafast All-Optical Switching in Al0.18ga0.82as Directional Coupler in 1.55 Mu-M Spectral Region," Electron Lett. 27, 1709-1710 (1991).
[CrossRef]

J. S. Aitchison, K. Alhemyari, C. N. Ironside, R. S. Grant, and W. Sibbett, "Observation of Spatial Solitons in Algaas Wave-Guides," Electron Lett. 28, 1879-1880 (1992).
[CrossRef]

Int. J. Nonlinear Opt. Phys.

G. I. Stegeman, A. Villeneuve, J. Kang, J. S. Aitchison, C. N. Ironside, K. Alhemyari, C. C. Yang, C. H. Lin, H. H. Lin, G. T. Kennedy, R. S. Grant, and W. Sibbett, "Algaas Below Half Bandgap - the Silicon of Nonlinear-Optical Materials," Int. J. Nonlinear Opt. Phys. 3, 347-371 (1994).
[CrossRef]

J. Appl. Phys.

S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen, and H. Sigg, "The refractive index of AlxGa1-xAs below the band gap: Accurate determination and empirical modeling," J. Appl. Phys. 87, 7825-7837 (2000).
[CrossRef]

Nature

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, "Broad-band optical parametric gain on a silicon photonic chip," Nature 441, 960-963 (2006).
[CrossRef] [PubMed]

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Phys. Rev. Lett.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

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[CrossRef]

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Figures (11)

Fig. 1.
Fig. 1.

Structure

Fig. 2.
Fig. 2.

Example of a TE mode for 250 nm wide waveguide. (Arrow: electric field)

Fig. 3.
Fig. 3.

GVD dependence on waveguide width and core height at a wavelength of 1.55 μm.

Fig. 4.
Fig. 4.

GVB vs. wavelength and width. (h = 0.5 μm)

Fig. 5.
Fig. 5.

Effective index of TE00 and TM00 vs. waveguide width for a wavelength of 1.55 μm.

Fig. 6.
Fig. 6.

(a) Nano-wire general structure with tapering and feed. (b) The length parameters for the four different designs.

Fig. 7.
Fig. 7.

(a) SEM image of fabricated nanowires. (b) Fringe period determined from measured spectra

Fig. 8.
Fig. 8.

Group index ng vs. waveguide width for a wavelength of 1.55 μm.

Fig. 9.
Fig. 9.

GVD vs. waveguide width for wavelength of 1.55 μm.

Fig. 10.
Fig. 10.

GVD vs. wavelength for width of 668 nm, 323 nm and 280 nm. (core height = 0.5 μm)

Fig. 11.
Fig. 11.

Effective area vs. waveguide width for a core height of 500 nm and a wavelength of 1.55 μm

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

Λ = Λ 0 + Λ 1 ( λ λ 0 ) + 1 2 Λ 2 ( λ λ 0 ) 2
d 2 β d λ 2 = π Λ 0 2 d Λ 1 .
n g = λ 2 2 π
β″ = λ 3 ( 2 πc ) 2 [ 2 + d 2 β d λ 2 ]
β total d = β feed d feed + β taper d taper + β wire d wire
A eff = [ I ( x , y ) d x d y ] 2 C o r e I 2 ( x , y ) d x d y

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