Abstract

The mode structure and attenuation constants in parabolic hollow waveguides with arbitrary parabolic domains are investigated based on the exact vector field expressions and characteristic equations. Normalized attenuation charts are provided for a variety of mode numbers, parities, and polarizations. The analysis is not restricted to parabolic waveguides with a symmetric cross section.

© 2007 Optical Society of America

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References

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  1. G. F. Miner, Lines and Electromagnetic Fields for Engineers (Oxford U. Press, 1996).
  2. C. Someda, Electromagnetic Waves (Chapman & Hill, 1988).
  3. R. D. Spence and C. P. Wells, "The propagation of electromagnetic waves in parabolic pipes," Phys. Rev. 62, 58-62 (1942).
    [CrossRef]
  4. P. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), Vol. II.
  5. C. S. Kenney and P. L. Overfelt, "A simple approach to mode analysis for parabolic waveguides," IEEE Trans. Microwave Theory Tech. 39, 405-412 (1991).
    [CrossRef]
  6. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, "Parabolic nondiffracting optical wave fields," Opt. Lett. 29, 44-46 (2004).
    [CrossRef] [PubMed]
  7. C. López-Mariscal, M. A. Bandres, S. Chávez-Cerda, and J. C. Gutiérrez-Vega, "Observation of parabolic nondiffracting wave fields," Opt. Express 13, 2364-2369 (2005).
    [CrossRef] [PubMed]
  8. J. C. Gutiérrez-Vega and M. A. Bandres, "Helmholtz-Gauss waves," J. Opt. Soc. Am. A 22, 289-298 (2005).
    [CrossRef]
  9. M. A. Bandres and J. C. Gutiérrez-Vega, "Vector Helmholtz-Gauss and vector Laplace-Gauss beams," Opt. Lett. 30, 2155-2157 (2005).
    [CrossRef] [PubMed]
  10. C. López-Mariscal, M. A. Bandres, and J. C. Gutiérrez-Vega, "Observation of the experimental propagation properties of Helmholtz-Gauss beams," Opt. Eng. (Bellingham) 45, 068001 (2006).
    [CrossRef]
  11. M. Guizar-Sicairos and J. C. Gutiérrez-Vega, "Generalized Helmholtz-Gauss beam and its transformation by paraxial optical systems," Opt. Lett. 31, 2912-2914 (2006).
    [CrossRef] [PubMed]
  12. R. I. Hernandez-Aranda, Julio C. Gutiérrez-Vega, M. Guizar-Sicairos, and M. A. Bandres, "Propagation of generalized vector Helmholtz-Gauss beams through paraxial optical systems," Opt. Express 14, 8974-8988 (2006).
    [CrossRef] [PubMed]
  13. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1964).
  14. Julio C. Gutiérrez-Vega, R. M. Rodríguez-Dagnino, and S. Chávez-Cerda, "Attenuation characteristics in confocal annular elliptic waveguides and resonators," IEEE Trans. Microwave Theory Tech. 50, 1095-1100 (2002).
    [CrossRef]

2006

2005

2004

2002

Julio C. Gutiérrez-Vega, R. M. Rodríguez-Dagnino, and S. Chávez-Cerda, "Attenuation characteristics in confocal annular elliptic waveguides and resonators," IEEE Trans. Microwave Theory Tech. 50, 1095-1100 (2002).
[CrossRef]

1991

C. S. Kenney and P. L. Overfelt, "A simple approach to mode analysis for parabolic waveguides," IEEE Trans. Microwave Theory Tech. 39, 405-412 (1991).
[CrossRef]

1942

R. D. Spence and C. P. Wells, "The propagation of electromagnetic waves in parabolic pipes," Phys. Rev. 62, 58-62 (1942).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

C. S. Kenney and P. L. Overfelt, "A simple approach to mode analysis for parabolic waveguides," IEEE Trans. Microwave Theory Tech. 39, 405-412 (1991).
[CrossRef]

Julio C. Gutiérrez-Vega, R. M. Rodríguez-Dagnino, and S. Chávez-Cerda, "Attenuation characteristics in confocal annular elliptic waveguides and resonators," IEEE Trans. Microwave Theory Tech. 50, 1095-1100 (2002).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Eng. (Bellingham)

C. López-Mariscal, M. A. Bandres, and J. C. Gutiérrez-Vega, "Observation of the experimental propagation properties of Helmholtz-Gauss beams," Opt. Eng. (Bellingham) 45, 068001 (2006).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev.

R. D. Spence and C. P. Wells, "The propagation of electromagnetic waves in parabolic pipes," Phys. Rev. 62, 58-62 (1942).
[CrossRef]

Other

P. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953), Vol. II.

G. F. Miner, Lines and Electromagnetic Fields for Engineers (Oxford U. Press, 1996).

C. Someda, Electromagnetic Waves (Chapman & Hill, 1988).

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1964).

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Figures (6)

Fig. 1
Fig. 1

Geometry of the parabolic waveguide and the definition of the parabolic coordinates.

Fig. 2
Fig. 2

Absolute value of (a) E z in TM modes, and (b) H z in TE modes, for a variety of mode numbers, parities, and boundary ratios s = ξ 0 η 0 .

Fig. 3
Fig. 3

General behavior of the attenuation constants as a function of the normalized frequency W for both TM and TE modes.

Fig. 4
Fig. 4

Contour plots of log ( f m , n ) on the plane ( ξ 0 , η 0 ) for the lowest order even and odd modes. Labels corresponding to log ( f m , n ) = { 0.2 , 0 , 0.2 } are included in the plots; remaining labels have been omitted for clarity. For all plots, the separation between consecutive contour lines is 0.2, and 1 ξ 0 4 and 1 η 0 4 .

Fig. 5
Fig. 5

Contour plots of log ( g m , n ) on the plane ( ξ 0 , η 0 ) for the lowest order even and odd modes. Labels corresponding to log ( g m , n ) = { 0.2 , 0 , 0.2 } are included in the plots; remaining labels have been omitted for clarity. For all plots, the separation between consecutive contour lines is 0.2, and 1 ξ 0 4 and 1 η 0 4 .

Fig. 6
Fig. 6

Contour plots of log ( h m , n ) on the plane ( ξ 0 , η 0 ) for the lowest order even and odd modes. Labels corresponding to log ( h m , n ) = { 0.2 , 0 , 0.2 } are included in the plots; remaining labels have been omitted for clarity. For all plots, the separation between consecutive contour lines is 0.2, and 1 ξ 0 4 and 1 η 0 4 .

Tables (1)

Tables Icon

Table 1 Values of the Coefficients c 1 , c 2 , and c 3 for the Different Values of m and n

Equations (28)

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[ d 2 d ν 2 + ν 2 4 a ] { P e ( ν ; a ) P o ( ν ; a ) } = 0 ,
E z = P e ( 2 κ ξ ; a ) P e ( 2 κ η ; a ) ζ ( z , t ) ,
E t = i β κ 2 t E z ,
H t = i ω ϵ κ 2 z ̂ × t E z ,
P e ( 2 κ ξ 0 ; a ) = 0 , TM modes ,
P e ( 2 κ η 0 ; a ) = 0 , TM modes .
P e ( 2 κ ξ 0 ; a ) = 0 , TE modes ,
P e ( 2 κ η 0 ; a ) = 0 , TE modes ,
ω m , n c = κ m , n μ ϵ .
ψ m , n = P e ( 2 κ ξ ; a ) P e ( 2 κ η ; a ) .
α = P L 2 P T = 1 2 l Z wall H tan 2 d l S Z wave H t 2 d S ,
α TM 2 σ Z μ μ w = f m , n ( ξ 0 , η 0 ) W 3 W 2 1 ,
α TE 2 σ Z μ μ w = g m , n ( ξ 0 , η 0 ) W 2 1 W + h m , n ( ξ 0 , η 0 ) W 3 W ,
W = ω ω c = ω μ ϵ κ 1 ,
f m , n ( ξ 0 , η 0 ) = κ m , n ( A + B F ) ,
g m , n ( ξ 0 , η 0 ) = κ m , n ( K + L F ) ,
h m , n ( ξ 0 , η 0 ) = κ m , n κ m , n 2 ( I + J F ) ,
F = 0 ξ 0 0 η 0 ( U 2 V 2 + U 2 V 2 ) d ξ d η ,
A = V 2 ( η 0 ) 0 ξ 0 U 2 ( ξ 2 + η 0 2 ) 1 2 d ξ ,
B = U 2 ( ξ 0 ) 0 η 0 V 2 ( ξ 0 2 + η 2 ) 1 2 d η ,
I = V 2 ( η 0 ) 0 ξ 0 U 2 ( ξ 2 + η 0 2 ) 1 2 d ξ ,
J = U 2 ( ξ 0 ) 0 η 0 V 2 ( ξ 0 2 + η 2 ) 1 2 d η ,
K = V 2 ( η 0 ) 0 ξ 0 U 2 ( ξ 2 + η 0 2 ) 1 2 d ξ ,
L = U 2 ( ξ 0 ) 0 η 0 V 2 ( ξ 0 2 + η 2 ) 1 2 d η ,
W min TE ( ξ 0 , η 0 ) = 3 h 2 g + ( 3 h 2 g ) 2 h g + 1 ,
f m , n ( ξ 0 , ξ 0 ) = c 1 , m , n ξ 0 3 ,
g m , n ( ξ 0 , ξ 0 ) = c 2 , m , n ξ 0 3 ,
h m , n ( ξ 0 , ξ 0 ) = c 3 , m , n ξ 0 5 ,

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