Abstract

The transformation of radially traveling cylindrical waves between two cylindrical coordinate systems with skew (nonparallel) axes is derived for the first time to our knowledge. The analytical procedure is based on the complex integral representation of the Hankel function and appropriate contour deformation and change of variables to obtain a final Fourier transform expression of the cylindrical wave in the new system. Scalar and vector waves are considered. This new result is a powerful tool for the rigorous analysis of scattering and coupling in nonparallel optical fiber configurations.

© 2006 Optical Society of America

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  1. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, 1966).
  2. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).
  3. A. L. Jones, "Coupling of optical fibers and scattering in fibers," J. Opt. Soc. Am. 55, 261-271 (1965).
    [CrossRef]
  4. W. Wijngaard, "Guided normal modes of two parallel circular dielectric rods," J. Opt. Soc. Am. 63, 944-950 (1973).
    [CrossRef]
  5. A. Z. Elsherbeni and A. A. Kishk, "Modeling of cylindrical objects by circular dielectric and conducting cylinders," IEEE Trans. Antennas Propag. 40, 96-99 (1992).
    [CrossRef]
  6. A. A. Kishk, R. P. Parrikar, and A. Z. Elsherbeni, "Electromagnetic scattering from an eccentric multilayered circular cylinder," IEEE Trans. Antennas Propag. 40, 295-303 (1992).
    [CrossRef]
  7. W. C. Chew, L. Gurel, Y. M. Wang, G. Otto, R. L. Wagner, and Q. H. Liu, "A generalized recursive algorithm for wave-scattering solutions in two dimensions," IEEE Trans. Microwave Theory Tech. 40, 716-723 (1992).
    [CrossRef]
  8. D. Felbacq, G. Tayeb, and D. Maystre, "Scattering by a random set of parallel cylinders," J. Opt. Soc. Am. A 11, 2526-2538 (1994).
    [CrossRef]
  9. K. M. Lo, R. C. McPhedran, I. M. Bassett, and G. W. Milton, "An electromagnetic theory of dielectric waveguides with multiple embedded cylinders," J. Lightwave Technol. 12, 396-410 (1994).
    [CrossRef]
  10. C. Chang and H. Chang, "Theory of the circular harmonics expansion method for multiple-optical-fiber system," J. Lightwave Technol. 12, 415-417 (1994).
    [CrossRef]
  11. L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, and P. A. Robinson, "Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part I. Method," J. Opt. Soc. Am. A 17, 2165-2176 (2000).
    [CrossRef]
  12. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. M. de Sterke, and L. C. Botten, "Multipole method for microstructured optical fibers. I. Formulation," J. Opt. Soc. Am. B 19, 2322-2330 (2002).
    [CrossRef]
  13. E. G. Alivizatos, I. D. Chremmos, N. L. Tsitsas, and N. K. Uzunoglu, "Green's function method for the analysis of propagation in holey fibers," J. Opt. Soc. Am. A 21, 847-857 (2004).
    [CrossRef]
  14. N. K. Uzunoglu and P. A. Mandis, "Analysis of the coupling of two non-parallel dielectric rod waveguides," Int. J. Infrared Millim. Waves 8, 535-547 (1987).
    [CrossRef]
  15. T. A. Birks, J. C. Knight, and T. E. Dimmick, "High-resolution measurement of the fiber diameter variations using whispering gallery modes and no optical alignment," IEEE Photon. Technol. Lett. 12, 182-183 (2000).
    [CrossRef]
  16. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).
  17. C. T. Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE Press, 1994).

2004 (1)

2002 (1)

2000 (2)

1994 (3)

D. Felbacq, G. Tayeb, and D. Maystre, "Scattering by a random set of parallel cylinders," J. Opt. Soc. Am. A 11, 2526-2538 (1994).
[CrossRef]

K. M. Lo, R. C. McPhedran, I. M. Bassett, and G. W. Milton, "An electromagnetic theory of dielectric waveguides with multiple embedded cylinders," J. Lightwave Technol. 12, 396-410 (1994).
[CrossRef]

C. Chang and H. Chang, "Theory of the circular harmonics expansion method for multiple-optical-fiber system," J. Lightwave Technol. 12, 415-417 (1994).
[CrossRef]

1992 (3)

A. Z. Elsherbeni and A. A. Kishk, "Modeling of cylindrical objects by circular dielectric and conducting cylinders," IEEE Trans. Antennas Propag. 40, 96-99 (1992).
[CrossRef]

A. A. Kishk, R. P. Parrikar, and A. Z. Elsherbeni, "Electromagnetic scattering from an eccentric multilayered circular cylinder," IEEE Trans. Antennas Propag. 40, 295-303 (1992).
[CrossRef]

W. C. Chew, L. Gurel, Y. M. Wang, G. Otto, R. L. Wagner, and Q. H. Liu, "A generalized recursive algorithm for wave-scattering solutions in two dimensions," IEEE Trans. Microwave Theory Tech. 40, 716-723 (1992).
[CrossRef]

1987 (1)

N. K. Uzunoglu and P. A. Mandis, "Analysis of the coupling of two non-parallel dielectric rod waveguides," Int. J. Infrared Millim. Waves 8, 535-547 (1987).
[CrossRef]

1973 (1)

1965 (1)

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Alivizatos, E. G.

Asatryan, A. A.

Bassett, I. M.

K. M. Lo, R. C. McPhedran, I. M. Bassett, and G. W. Milton, "An electromagnetic theory of dielectric waveguides with multiple embedded cylinders," J. Lightwave Technol. 12, 396-410 (1994).
[CrossRef]

Birks, T. A.

T. A. Birks, J. C. Knight, and T. E. Dimmick, "High-resolution measurement of the fiber diameter variations using whispering gallery modes and no optical alignment," IEEE Photon. Technol. Lett. 12, 182-183 (2000).
[CrossRef]

Botten, L. C.

Chang, C.

C. Chang and H. Chang, "Theory of the circular harmonics expansion method for multiple-optical-fiber system," J. Lightwave Technol. 12, 415-417 (1994).
[CrossRef]

Chang, H.

C. Chang and H. Chang, "Theory of the circular harmonics expansion method for multiple-optical-fiber system," J. Lightwave Technol. 12, 415-417 (1994).
[CrossRef]

Chew, W. C.

W. C. Chew, L. Gurel, Y. M. Wang, G. Otto, R. L. Wagner, and Q. H. Liu, "A generalized recursive algorithm for wave-scattering solutions in two dimensions," IEEE Trans. Microwave Theory Tech. 40, 716-723 (1992).
[CrossRef]

Chremmos, I. D.

de Sterke, C. M.

Dimmick, T. E.

T. A. Birks, J. C. Knight, and T. E. Dimmick, "High-resolution measurement of the fiber diameter variations using whispering gallery modes and no optical alignment," IEEE Photon. Technol. Lett. 12, 182-183 (2000).
[CrossRef]

Elsherbeni, A. Z.

A. A. Kishk, R. P. Parrikar, and A. Z. Elsherbeni, "Electromagnetic scattering from an eccentric multilayered circular cylinder," IEEE Trans. Antennas Propag. 40, 295-303 (1992).
[CrossRef]

A. Z. Elsherbeni and A. A. Kishk, "Modeling of cylindrical objects by circular dielectric and conducting cylinders," IEEE Trans. Antennas Propag. 40, 96-99 (1992).
[CrossRef]

Felbacq, D.

Gurel, L.

W. C. Chew, L. Gurel, Y. M. Wang, G. Otto, R. L. Wagner, and Q. H. Liu, "A generalized recursive algorithm for wave-scattering solutions in two dimensions," IEEE Trans. Microwave Theory Tech. 40, 716-723 (1992).
[CrossRef]

Jones, A. L.

Kishk, A. A.

A. Z. Elsherbeni and A. A. Kishk, "Modeling of cylindrical objects by circular dielectric and conducting cylinders," IEEE Trans. Antennas Propag. 40, 96-99 (1992).
[CrossRef]

A. A. Kishk, R. P. Parrikar, and A. Z. Elsherbeni, "Electromagnetic scattering from an eccentric multilayered circular cylinder," IEEE Trans. Antennas Propag. 40, 295-303 (1992).
[CrossRef]

Knight, J. C.

T. A. Birks, J. C. Knight, and T. E. Dimmick, "High-resolution measurement of the fiber diameter variations using whispering gallery modes and no optical alignment," IEEE Photon. Technol. Lett. 12, 182-183 (2000).
[CrossRef]

Kuhlmey, B. T.

Liu, Q. H.

W. C. Chew, L. Gurel, Y. M. Wang, G. Otto, R. L. Wagner, and Q. H. Liu, "A generalized recursive algorithm for wave-scattering solutions in two dimensions," IEEE Trans. Microwave Theory Tech. 40, 716-723 (1992).
[CrossRef]

Lo, K. M.

K. M. Lo, R. C. McPhedran, I. M. Bassett, and G. W. Milton, "An electromagnetic theory of dielectric waveguides with multiple embedded cylinders," J. Lightwave Technol. 12, 396-410 (1994).
[CrossRef]

Mandis, P. A.

N. K. Uzunoglu and P. A. Mandis, "Analysis of the coupling of two non-parallel dielectric rod waveguides," Int. J. Infrared Millim. Waves 8, 535-547 (1987).
[CrossRef]

Maystre, D.

McPhedran, R. C.

Milton, G. W.

K. M. Lo, R. C. McPhedran, I. M. Bassett, and G. W. Milton, "An electromagnetic theory of dielectric waveguides with multiple embedded cylinders," J. Lightwave Technol. 12, 396-410 (1994).
[CrossRef]

Nicorovici, N. A.

Otto, G.

W. C. Chew, L. Gurel, Y. M. Wang, G. Otto, R. L. Wagner, and Q. H. Liu, "A generalized recursive algorithm for wave-scattering solutions in two dimensions," IEEE Trans. Microwave Theory Tech. 40, 716-723 (1992).
[CrossRef]

Parrikar, R. P.

A. A. Kishk, R. P. Parrikar, and A. Z. Elsherbeni, "Electromagnetic scattering from an eccentric multilayered circular cylinder," IEEE Trans. Antennas Propag. 40, 295-303 (1992).
[CrossRef]

Renversez, G.

Robinson, P. A.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

Tai, C. T.

C. T. Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE Press, 1994).

Tayeb, G.

Tsitsas, N. L.

Uzunoglu, N. K.

E. G. Alivizatos, I. D. Chremmos, N. L. Tsitsas, and N. K. Uzunoglu, "Green's function method for the analysis of propagation in holey fibers," J. Opt. Soc. Am. A 21, 847-857 (2004).
[CrossRef]

N. K. Uzunoglu and P. A. Mandis, "Analysis of the coupling of two non-parallel dielectric rod waveguides," Int. J. Infrared Millim. Waves 8, 535-547 (1987).
[CrossRef]

Wagner, R. L.

W. C. Chew, L. Gurel, Y. M. Wang, G. Otto, R. L. Wagner, and Q. H. Liu, "A generalized recursive algorithm for wave-scattering solutions in two dimensions," IEEE Trans. Microwave Theory Tech. 40, 716-723 (1992).
[CrossRef]

Wang, Y. M.

W. C. Chew, L. Gurel, Y. M. Wang, G. Otto, R. L. Wagner, and Q. H. Liu, "A generalized recursive algorithm for wave-scattering solutions in two dimensions," IEEE Trans. Microwave Theory Tech. 40, 716-723 (1992).
[CrossRef]

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, 1966).

White, T. P.

Wijngaard, W.

IEEE Photon. Technol. Lett. (1)

T. A. Birks, J. C. Knight, and T. E. Dimmick, "High-resolution measurement of the fiber diameter variations using whispering gallery modes and no optical alignment," IEEE Photon. Technol. Lett. 12, 182-183 (2000).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

A. Z. Elsherbeni and A. A. Kishk, "Modeling of cylindrical objects by circular dielectric and conducting cylinders," IEEE Trans. Antennas Propag. 40, 96-99 (1992).
[CrossRef]

A. A. Kishk, R. P. Parrikar, and A. Z. Elsherbeni, "Electromagnetic scattering from an eccentric multilayered circular cylinder," IEEE Trans. Antennas Propag. 40, 295-303 (1992).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

W. C. Chew, L. Gurel, Y. M. Wang, G. Otto, R. L. Wagner, and Q. H. Liu, "A generalized recursive algorithm for wave-scattering solutions in two dimensions," IEEE Trans. Microwave Theory Tech. 40, 716-723 (1992).
[CrossRef]

Int. J. Infrared Millim. Waves (1)

N. K. Uzunoglu and P. A. Mandis, "Analysis of the coupling of two non-parallel dielectric rod waveguides," Int. J. Infrared Millim. Waves 8, 535-547 (1987).
[CrossRef]

J. Lightwave Technol. (2)

K. M. Lo, R. C. McPhedran, I. M. Bassett, and G. W. Milton, "An electromagnetic theory of dielectric waveguides with multiple embedded cylinders," J. Lightwave Technol. 12, 396-410 (1994).
[CrossRef]

C. Chang and H. Chang, "Theory of the circular harmonics expansion method for multiple-optical-fiber system," J. Lightwave Technol. 12, 415-417 (1994).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (1)

Other (4)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

C. T. Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE Press, 1994).

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge U. Press, 1966).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941).

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Figures (2)

Fig. 1
Fig. 1

Geometry of two cylindrical coordinate systems with skew axes.

Fig. 2
Fig. 2

Contour of integration on the complex τ–plane ( φ 0 = arg ( a 0 ) , 0 φ 0 π 2 ) .

Equations (38)

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Ψ n , k z ( 2 ) ( r 1 ) = H n ( 2 ) ( a 0 ρ 1 ) exp [ j ( n ϕ 1 + k z z 1 ) ] , ρ 1 > 0 ,
a 0 = k 0 2 k z 2 , π 2 arg ( a 0 ) 0 .
Ψ m , k z ( 1 ) ( r 2 ) = J m ( a 0 ρ 2 ) exp [ j ( m ϕ 2 + k z z 2 ) ] , a 0 = k 0 2 k z 2 .
Ψ n , k z ( 2 ) ( r 1 ) = m = + j m n H m n ( 2 ) ( a 0 d ) Ψ m , k z ( 1 ) ( r 2 ) .
Ψ n , k z ( 2 ) ( r 1 ) = j n π exp [ j ( n ϕ 1 + k z z 1 ) ] C exp ( j a 0 ρ 1 cos ξ + j n ξ ) d ξ ,
Ψ n , k z ( 2 ) ( r 1 ) = 1 π exp ( j k z z 1 ) C exp [ j a 0 ρ 1 cos ( τ ϕ 1 + π 2 ) + j n τ ] d τ ,
x i = ρ i cos ϕ i , y i = ρ i sin ϕ i , 0 ϕ i < 2 π , i = 1 , 2 ,
( x 1 y 1 z 1 ) = [ cos γ 0 sin γ 0 1 0 sin γ 0 cos γ ] ( x 2 y 2 z 2 ) + ( 0 d 0 ) .
Ψ n , k z ( 2 ) ( r 1 ) = 1 π C exp [ j k x x 2 + j k y ( y 2 + d ) + j k z z 2 + j n τ ] d τ ,
k x = a 0 sin τ cos γ k z sin γ ,
k y = a 0 cos τ ,
k z = a 0 sin τ sin γ + k z cos γ
C : Im ( a 0 sin τ ) = 0 tanh ( τ 2 ) = tan [ arg ( a 0 ) ] tan ( τ 1 ) .
C [ Im ( a 0 ) = 0 ] : ( τ = π 2 + j τ 2 , τ 2 0 ) ( τ = τ 1 , π 2 τ 1 3 π 2 ) ( τ = 3 π 2 + j τ 2 , τ 2 0 ) ,
C [ Re ( a 0 ) = 0 ] : τ = π + j τ 2 , < τ 2 < + ,
k x = k z cot γ k z sin γ ,
k y = k 0 2 k x 2 k z 2 , Re ( k y ) < 0 , Im ( k y ) > 0 ,
sin τ = k z cos γ k z a 0 sin γ , cos τ = k y a 0 ,
k x = a 0 cos w , k y = a 0 sin w
exp ( j k x x 2 + j k y y 2 ) = exp [ j a 0 ρ 2 cos ( ϕ 2 w ) ] = m = + j m J m ( a 0 ρ 2 ) exp [ j m ( ϕ 2 w ) ] ,
Ψ n , k z ( 2 ) ( r 1 ) = + d k z m = + F γ , d ( m , k z ; n , k z ) Ψ m , k z ( 1 ) ( r 2 ) ,
F γ , d ( m , k z ; n , k z ) = 1 π k y sin γ exp [ j ( k y d + n τ m w + m π 2 ) ] .
F γ , d ( m , k z ; n , k z ) = F γ , d ( n , k z ; m , k z )
F π γ , d ( m , k z ; n , k z ) = F γ , d ( m , k z ; n , k z ) ,
F π γ , d ( m , k z ; n , k z ) = F γ , d ( m , k z ; n , k z ) .
F γ , d ( m , k z ; n , k z ) = ( 1 ) n π k y sin γ exp [ j ( k Y d n τ + m w + m π 2 ) ] .
M n , k z ( i ) ( r , k 0 ) = × [ z ̂ Ψ n , k z ( i ) ( r ) ] ,
N n , k z ( i ) ( r , k 0 ) = k 0 1 × M n , k z ( i ) ( r , k 0 ) , i = 1 , 2 .
( M n , k z ( 2 ) ( r 1 , k 0 ) N n , k z ( 2 ) ( r 1 , k 0 ) ) = m = + j m n H m n ( 2 ) ( a 0 d ) ( M m , k z ( 1 ) ( r 2 , k 0 ) N m , k z ( 1 ) ( r 2 , k 0 ) ) .
× [ x ̂ 2 Ψ m , k z ( 1 ) ( r 2 ) ] = j k z 2 a 0 [ M m 1 , k z ( 1 ) ( r 2 , k 0 ) M m + 1 , k z ( 1 ) ( r 2 , k 0 ) ] j k 0 2 a 0 [ N m 1 , k z ( 1 ) ( r 2 , k 0 ) N m + 1 , k z ( 1 ) ( r 2 , k 0 ) ] .
( M n , k z ( 2 ) ( r 1 , k 0 ) N n , k z ( 2 ) ( r 1 , k 0 ) ) = + d k z m = + T γ , d ( m , k z ; n , k z ) ( M m , k z ( 1 ) ( r 2 , k 0 ) N m , k z ( 1 ) ( r 2 , k 0 ) ) ,
T γ , d ( m , k z ; n , k z ) [ σ τ τ σ ] ,
{ σ τ } = exp [ j ( k y d + n τ m w + m π 2 ) ] π k y { cot γ + k z cos w a 0 j k 0 sin w a 0 } .
T γ , d ( m , k z ; n , k z ) = k 0 2 k z 2 k 0 2 k z 2 T γ , d ( n , k z ; m , k z )
( σ τ ) π γ , d ( m , k z ; n , k z ) = ( σ τ ) γ , d ( m , k z ; n , k z ) ,
( σ τ ) π γ , d ( m , k z ; n , k z ) = ( σ τ ) γ , d ( m , k z ; n , k z ) .
T γ , d ( m , k z ; n , k z ) [ σ τ τ σ ] ,
{ σ τ } = ( 1 ) n exp [ j ( k y d n τ + m w + m π 2 ) ] π k y { cot γ + k z cos w a 0 j k 0 sin w a 0 } .

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