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

Full Article  |  PDF Article
OSA Recommended Articles
Metallic and 3D-printed dielectric helical terahertz waveguides

Dominik Walter Vogt, Jessienta Anthony, and Rainer Leonhardt
Opt. Express 23(26) 33359-33369 (2015)

Real photonic waveguides: guiding light through imperfections

Daniele Melati, Andrea Melloni, and Francesco Morichetti
Adv. Opt. Photon. 6(2) 156-224 (2014)

Enhancement of dispersion modulation in nanoscale waveguides

Alexander A. Govyadinov and Viktor A. Podolskiy
J. Opt. Soc. Am. B 25(12) C127-C135 (2008)

References

  • View by:
  • |
  • |
  • |

  1. R. de L. Kronig , “ On the theory of dispersion of X-rays ,” J. Opt. Soc. Amer. Rev. Sci. Instrum.   12 , 547 – 557 ( 1926 ).
    [Crossref]
  2. H. A. Kramers , “ La diffusion de la lumiére 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]
  12. P. Yeh , A. Yariv , and C.-S. Hong , “ Electromagnetic propagation in periodic stratified media. I. General theory ,” J. Opt. Soc. Am.   67 , 423 – 438 ( 1977 ).
    [Crossref]
  13. F. Brechet , P. Leproux , P. Roy , J. Marcou , and D. Pagnoux , “ Analysis of bandpass filtering behaviour of single-mode 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 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-309 .
    [Crossref] [PubMed]

2005 (1)

2004 (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]

2003 (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 ).

2002 (1)

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

2000 (1)

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

1998 (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]

1982 (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]

1979 (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 ).

1977 (1)

1927 (1)

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

1926 (1)

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

Argyros, A.

Bassani, F.

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 ).

Birks, T. A.

Bjarklev, A.

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]

Brechet, F.

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

Broeng, J.

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]

Cheng, D. K.

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

Cordeiro, C. M. B.

de L. Kronig, R.

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

Hong, C.-S.

Jackson, J. D.

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

Kramers, H. A.

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

Lægsgaard, J.

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]

Leon-Saval, S. G.

Leproux, P.

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

Love, J. D.

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

Luan, F.

Lucarini, V.

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 ).

Marcou, J.

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

Milonni, P. W.

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

Nussenzveig, H. M.

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

Pagnoux, D.

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

Peiponen, K.-E.

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 ).

Riishede, J.

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]

Roy, P.

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

Russell, P. St. J.

Saarinen, J. J.

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 ).

Shevchenko, V. V.

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]

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 ).

Snyder, A. W.

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

Wang, Z. H.

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]

Yariv, A.

Yeh, P.

Atti Congr. Int. Fis. Como (1)

H. A. Kramers , “ La diffusion de la lumiére 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 single-mode 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. Amer. Rev. Sci. Instrum. (1)

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

J. Phys. B: At. Mol. Opt. Phys. (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 ).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


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)

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

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 ) ,

Metrics