Abstract

Starting from the condition that optical signals propagate causally, we derive Kramers-Kronig relations for waveguides. For hollow waveguides with perfectly conductive walls, the modes propagate causally and Kramers-Kronig relations between the real and imaginary part of the mode indices exist. For dielectric waveguides, there exists a Kramers-Kronig type relation between the real mode index of a guided mode and the imaginary mode indices associated with the evanescent modes. For weakly guiding waveguides, the Kramers-Kronig relations are particularly simple, as the modal dispersion is determined solely from the profile of the corresponding mode field.

© 2005 Optical Society of America

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References

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  1. R. de L. Kronig, “On the theory of dispersion of X-rays,” J. Opt. Soc. Am. Rev. Sci. Instrum. 12, 547–557 (1926).
    [CrossRef]
  2. H. A. Kramers, “La diffusion de la lumi´ere par les atomes,” Atti Congr. Int. Fis. Como 2, 545–557 (1927).
  3. H. M. Nussenzveig, Causality and Dispersion Relations (Academic Press, New York, 1972).
  4. P. W. Milonni, “Controlling the speed of light pulses,” J. Phys. B: At. Mol. Opt. Phys. 35, R31–R56 (2002).
    [CrossRef]
  5. V. Lucarini, F. Bassani, K.-E. Peiponen, and J. J. Saarinen, “Dispersion theory and sum rules in linear and nonlinear optics,” Riv. Nuovo Cim. 26, 1–120 (2003).
  6. V. V. Shevchenko, “Spectral expansions in eigenfunctions and associated functions of a non-self-adjoint Sturm-Liouville problem on the whole real line,” Differ. Equations. 15, 1431–1443 (1979).
  7. V. V. Shevchenko, “On the completeness of spectral expansion of the electromagnetic field in the set of dielectric circular rod waveguide eigen waves,” Radio Science 17, 229–231 (1982).
    [CrossRef]
  8. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, New York, 1983).
  9. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1999).
  10. D. K. Cheng, Field and Wave Electromagnetics (Addison-Wesley, Reading, Massachusetts, 1989).
  11. Z. H. Wang, “Free Space Mode Approximation of Radiation Modes for Weakly Guiding Planar Optical Waveguides,” IEEE J. Quantum Electron. 34, 680–685 (1998).
    [CrossRef]
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  13. F. Brechet, P. Leproux, P. Roy, J. Marcou, and D. Pagnoux, “Analysis of bandpass filtering behaviour of singlemode depressed-core-index photonic-bandgap fibre,” Electron. Lett. 36, 870–872 (2000).
    [CrossRef]
  14. J. Riishede, J. Lægsgaard, J. Broeng, and A. Bjarklev, “All-silica photonic bandgap fibre with zero dispersion and a large mode area at 730 nm,” J. Opt. A: Pure Appl. Opt. 6, 667–670 (2004).
    [CrossRef]
  15. A. Argyros, T. A. Birks, S. G. Leon-Saval, C. M. B. Cordeiro, F. Luan, and P. St. J. Russell, “Photonic bandgap with an index step of one percent,” Opt. Express 13, 309–314 (2005), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-309">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-309</a>.
    [CrossRef] [PubMed]

Atti Congr. Int. Fis. Como (1)

H. A. Kramers, “La diffusion de la lumi´ere par les atomes,” Atti Congr. Int. Fis. Como 2, 545–557 (1927).

Differ. Equations (1)

V. V. Shevchenko, “Spectral expansions in eigenfunctions and associated functions of a non-self-adjoint Sturm-Liouville problem on the whole real line,” Differ. Equations. 15, 1431–1443 (1979).

Electron. Lett. (1)

F. Brechet, P. Leproux, P. Roy, J. Marcou, and D. Pagnoux, “Analysis of bandpass filtering behaviour of singlemode depressed-core-index photonic-bandgap fibre,” Electron. Lett. 36, 870–872 (2000).
[CrossRef]

IEEE J. Quantum Electron. (1)

Z. H. Wang, “Free Space Mode Approximation of Radiation Modes for Weakly Guiding Planar Optical Waveguides,” IEEE J. Quantum Electron. 34, 680–685 (1998).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

J. Riishede, J. Lægsgaard, J. Broeng, and A. Bjarklev, “All-silica photonic bandgap fibre with zero dispersion and a large mode area at 730 nm,” J. Opt. A: Pure Appl. Opt. 6, 667–670 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. Rev. Sci. Instrum. (1)

R. de L. Kronig, “On the theory of dispersion of X-rays,” J. Opt. Soc. Am. Rev. Sci. Instrum. 12, 547–557 (1926).
[CrossRef]

J. Phys. B (1)

P. W. Milonni, “Controlling the speed of light pulses,” J. Phys. B: At. Mol. Opt. Phys. 35, R31–R56 (2002).
[CrossRef]

Opt. Express (1)

Radio Science (1)

V. V. Shevchenko, “On the completeness of spectral expansion of the electromagnetic field in the set of dielectric circular rod waveguide eigen waves,” Radio Science 17, 229–231 (1982).
[CrossRef]

Riv. Nuovo Cim. (1)

V. Lucarini, F. Bassani, K.-E. Peiponen, and J. J. Saarinen, “Dispersion theory and sum rules in linear and nonlinear optics,” Riv. Nuovo Cim. 26, 1–120 (2003).

Other (4)

H. M. Nussenzveig, Causality and Dispersion Relations (Academic Press, New York, 1972).

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

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1999).

D. K. Cheng, Field and Wave Electromagnetics (Addison-Wesley, Reading, Massachusetts, 1989).

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Figures (7)

Fig. 1.
Fig. 1.

The real and imaginary part of the mode index for a hollow waveguide with perfectly conducting walls is shown in (a). The derivative of the real part of the mode index with respect to frequency is shown in (b). Also shown are results obtained using the Kramers-Kronig relations, where the real part of the mode index is determined from the imaginary part and vice versa. In all cases the results from the Kramers-Kronig relations are in excellent agreement with the exact results. We find that any discrepancy is only dependent on the numerical resolution in the calculations.

Fig. 2.
Fig. 2.

Real and imaginary part of the mode index for a dielectric-filled waveguide with perfectly conducting walls. Exact results and results obtained using the Kramers-Kronig relations for waveguides are shown.

Fig. 3.
Fig. 3.

The fundamental mode in a planar index-guiding waveguide for ωd/c = 40 (a), and the resulting integral Eq. (38) over evanescent modes (b).

Fig. 4.
Fig. 4.

Mode index for the fundamental TE mode in a dielectric waveguide (a). In (b) dn0r/dω is shown, based on a solution of the wave equation, together with results based on the Kramers-Kronig approach.

Fig. 5.
Fig. 5.

Mode index for the second order symmetric TE mode in a dielectric waveguide (a). In (b) dn2r/dω is shown, based on a solution of the wave equation, together with results based on the Kramers-Kronig approach.

Fig. 6.
Fig. 6.

The refractive index profile of a symmetric Bragg reflection waveguide is shown in (a). The fundamental guided mode for ωΛ/c = 35 is shown in (b).

Fig. 7.
Fig. 7.

The mode index n0r of the fundamental guided mode as a function of frequency is shown in (a) together with the band edges of the first bandgap. dnr 0/dω is shown in (b), based on a solution of the wave equation, together with results based on the Kramers-Kronig approach.

Equations (44)

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E ( z , t ) = Ω 1 ( Ω 1 ) j A j ( ω ) ψ j ( ω ) exp [ i β j ( ω ) z ] exp ( iωt ) d ω
+ Ω 2 ( Ω 2 ) B ( ω ) χ ( z , ω ) exp [ iωz c ] exp ( iωt ) d ω .
ψ i * ( ω ) ψ j ( ω ) d y ψ i ( ω ) | ψ j ( ω ) = δ ij ,
j ψ | ψ j ( ω ) 2 = 1
E ( 0 , t ) = ψ src A in ( ω ) exp ( iωt ) d ω .
E ( L , t ) = ψ det A out ( ω ) exp ( iωt ) d ω .
A out ( ω ) = G ( ω ) A in ( ω ) ,
G ( ω ) = { j ψ det ψ j ( ω ) ψ j ( ω ) ψ src exp [ i β j ( ω ) L ] if ω Ω 1 , ψ det χ ( L , ω ) exp [ iωL c ] if ω Ω 2 .
a out ( t ) = 1 2 π g ( t t ) a in ( t ) d t ,
G ( ω ) = 1 2 π L c g ( t ) exp ( iωt ) d t ,
G ˜ ( ω ) G ( ω ) exp ( iωL c ) = 1 2 π 0 g ( t + L c ) exp ( iωt ) d t .
ψ src = ψ det = ψ 0 ( ω 0 ) ,
G ˜ ( ω ) = { j ψ 0 ( ω 0 ) ψ j ( ω ) 2 exp { i [ n j ( ω ) 1 ] ωL c } if ω Ω 1 ψ 0 ( ω 0 ) χ ( L , ω ) if ω Ω 2 ,
n j ( ω ) = c β j ( ω ) ω .
F ( ω ) c iL [ G ˜ ( ω ) 1 ] = { j ψ 0 ( ω 0 ) ψ j ( ω ) 2 [ n j ( ω ) 1 ] ω if ω Ω 1 c iL [ ψ 0 ( ω 0 ) χ ( L , ω ) 1 ] if ω Ω 2 ,
Re F ( ω ) = 2 ω π P 0 Im F ( ω ) ω 2 ω 2 d ω
Im F ( ω ) = 2 π P 0 ω Re F ( ω ) ω 2 ω 2 d ω ,
c j ( ω 0 , ω ) ψ 0 ( ω 0 ) | ψ j ( ω ) 2 .
2 ω π P 0 Im F ( ω ) ω 2 ω 2 d ω = 2 ω π P Ω 1 j c j ( ω 0 , ω ) n j i ( ω ) ω ω 2 ω 2 d ω
2 ωc πL Ω 2 Re ψ 0 ( ω 0 ) χ ( L , ω ) 1 ω 2 ω 2 d ω ,
δ ( ω 0 , ω ) = 2 c πL Ω 2 Re ψ 0 ( ω 0 ) χ ( L , ω ) 1 ω 2 ω 2 d ω ,
j c j ( ω 0 , ω ) [ n j r ( ω ) 1 ] = 2 π P Ω 1 j c j ( ω 0 , ω ) n j i ( ω ) ω ω 2 ω 2 d ω + δ ( ω 0 , ω ) ,
d n 0 r ( ω ) d ω ω = ω 0 = 4 ω 0 π P Ω 1 j c j ( ω 0 , ω ) n j i ( ω ) ω ( ω 2 ω 0 2 ) 2 d ω + δ ( ω 0 , ω ) ω ω = ω 0 .
δ ω = 4 πL Ω 2 Re ψ 0 ( ω 0 ) χ ( L , ω ) 1 ( ω 2 ω 2 ) 2 d ω 8 πL Ω 2 d ω ( ω 2 ω 2 ) 2 ,
d n 0 r ( ω ) d ω = 4 ω π P Ω 1 j c j ( ω , ω ) n j i ( ω ) ω ( ω 2 ω 2 ) 2 d ω .
n 0 r ( ω ) 1 = 2 π P 0 j c j ( ω , ω ) n j i ( ω ) ω ω 2 ω 2 d ω .
G ( ω ) = exp [ i β j ( ω ) L ] .
β j ( ω ) = 1 c ω 2 ω j , c 2 ,
n j r ( ω ) = { 0 if ω j , c > ω 0 1 ( ω j , c ω ) 2 if ω ω j , c
n j i ( ω ) = { ( ω j , c ω ) 2 1 if ω j , c > ω 0 0 if ω ω j , c .
F ( ω ) = [ n j ( ω ) 1 ] ω .
[ n j r ( ω ) 1 ] ω = 2 ω π P 0 n j i ( ω ) ω ω 2 ω 2 d ω
n j i ( ω ) ω = 2 π P 0 ω [ n j r ( ω ) 1 ] ω ω 2 ω 2 d ω .
d n j r ( ω ) d ω = 4 ω π 0 ω j , c n j i ( ω ) ω ( ω 2 ω 2 ) 2 d ω .
ε r ( ω ) = 1 + ω p 2 ω res 2 ω 2 iγω ,
ψ j ( ω ) = 1 π cos ( k t y ) ,
k t = ( ω c ) 2 + β j 2 ,
c ( k t ) c j ( ω , ω ) = 1 π cos ( k t y ) ψ 0 ( ω ) d y 2
= 1 π ψ 0 ( ω ) exp ( i k t y ) d y 2 ,
j c j ( ω , ω ) n j i ( ω ) ω = c ω c c ( k t ) k t 2 ( ω c ) 2 d k t ,
d n 0 r ( ω ) d ω = 4 ωc π P 0 ω c c ( k t ) k t 2 ( ω c ) 2 d k t ( ω 2 ω 2 ) 2 d ω .
E K ( y , z ) = E K ( y ) exp ( iKy ) exp ( iβz ) ,
K = m π Λ + i K i , m = 1,2
E c ( y , z ) = cos ( k 1 y ) exp ( iβz ) ,

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