Abstract

To obtain the coupling constant for core-mode–cladding-mode coupling, it is necessary to calculate the total power carried by the cladding mode, and the related formulas are given by T. Erdogan [J. Opt. Soc. Am. A 14, 1773 (1997)] and [J. Opt. Soc. Am. A 17, 2113 (2000) ]. I found that my formulas for P1 and P3 are the same as Eqs. (B2) and (B16) in the 1997 paper, but Eqs. (B3)–(B15) for P2 are worth commenting on, and this comment gives our derivations.

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References

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  1. T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760-1773 (1997).
    [CrossRef]
  2. T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters: errata,” J. Opt. Soc. Am. A 14, 2113 (2000).
    [CrossRef]

2000 (1)

T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters: errata,” J. Opt. Soc. Am. A 14, 2113 (2000).
[CrossRef]

1997 (1)

Erdogan, T.

T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters: errata,” J. Opt. Soc. Am. A 14, 2113 (2000).
[CrossRef]

T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760-1773 (1997).
[CrossRef]

J. Opt. Soc. Am. A (2)

T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters: errata,” J. Opt. Soc. Am. A 14, 2113 (2000).
[CrossRef]

T. Erdogan, “Cladding-mode resonances in short- and long-period fiber grating filters,” J. Opt. Soc. Am. A 14, 1760-1773 (1997).
[CrossRef]

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Equations (29)

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P 1 = 1 2 Re 0 2 π d ϕ 0 a 1 r d r ( E r cl H ϕ cl * H r cl * E ϕ cl ) = ( E 1 ν cl ) 2 π u 1 2 4 { [ n cl , eff Z 0 n cl , eff Z 0 ζ 0 2 n 1 2 + ( 1 + n cl , eff 2 n 1 2 ) Im ( ζ 0 ) ] 0 a 1 2 J 2 2 ( u 1 r ) r d r + [ n cl , eff Z 0 n cl , eff Z 0 ζ 0 2 n 1 2 ( 1 + n cl , eff 2 n 1 2 ) Im ( ζ 0 ) ] 0 a 1 2 J 0 2 ( u 1 r ) r d r } = ( E 1 ν cl ) 2 π u 1 2 a 1 2 4 { [ n cl , eff Z 0 n cl , eff Z 0 ζ 0 2 n 1 2 + ( 1 + n cl , eff 2 n 1 2 ) Im ( ζ 0 ) ] [ J 2 2 ( u 1 a 1 ) J 1 ( u 1 a 1 ) J 3 ( u 1 a 1 ) ] + [ n cl , eff Z 0 n cl , eff Z 0 ζ 0 2 n 1 2 ( 1 + n cl , eff 2 n 1 2 ) Im ( ζ 0 ) ] [ J 0 2 ( u 1 a 1 ) + J 1 2 ( u 1 a 1 ) ] } .
P 3 = 1 2 Re 0 2 π d ϕ a 2 r d r ( E r cl H ϕ cl * H r cl * E ϕ cl ) = ( E 1 ν cl ) 2 π 3 u 1 4 u 2 4 a 1 2 a 2 2 J 1 2 ( u 1 a 1 ) 16 w 3 2 K 1 2 ( w 3 a 2 ) { [ n cl , eff Z 0 n 3 2 G 3 2 n cl , eff Z 0 F 3 2 ( 1 + n cl , eff 2 n 3 2 ) F 3 Im ( G 3 ) ] [ K 2 2 ( w 3 a 2 ) K 1 ( w 3 a 2 ) K 3 ( w 3 a 2 ) ] + [ n cl , eff Z 0 n 3 2 G 3 2 n cl , eff Z 0 F 3 2 + ( 1 + n cl , eff 2 n 3 2 ) F 3 Im ( G 3 ) ] [ K 0 2 ( w 3 a 2 ) K 1 2 ( w 3 a 2 ) ] } .
J 1 ( u 2 a 1 ) = 1 2 [ J 0 ( u 2 a 1 ) J 2 ( u 2 a 1 ) ] ,
N 1 ( u 2 a 1 ) = 1 2 [ N 0 ( u 2 a 1 ) N 2 ( u 2 a 1 ) ] .
P 2 = 1 2 Re 0 2 π d ϕ a 1 a 2 r d r ( E r cl H ϕ cl * H r cl * E ϕ cl ) = ( E 1 ν cl ) 2 π 3 u 1 4 a 1 2 J 1 2 ( u 1 a 1 ) 4 { ( n cl , eff Z 0 F 2 2 n cl , eff Z 0 n 2 2 G 2 2 ) ( Q + Q ̃ ) + 1 u 2 2 ( n cl , eff Z 0 n cl , eff Z 0 n 2 2 ζ 0 2 n 1 4 ) ( R + R ̃ ) + 2 n cl , eff u 2 ( Z 0 ζ 0 n 1 2 G 2 1 Z 0 F 2 ) ( S + S ̃ ) + ( 1 + n cl , eff 2 n 2 2 ) F 2 Im ( G 2 ) Q 0 + ( 1 + n cl , eff 2 n 2 2 ) n 2 2 n 1 2 u 2 2 Im ( ζ 0 ) R 0 ( 1 + n cl , eff 2 n 2 2 ) [ n 2 2 n 1 2 u 2 F 2 Im ( ζ 0 ) + 1 u 2 Im ( G 2 ) ] S 0 } ,
Q = a 1 a 2 ( u 2 r ) r 1 2 ( r ) d ( u 2 r ) ,
Q ̃ = a 1 a 2 1 u 2 r p 1 2 ( r ) d ( u 2 r ) ,
R = a 1 a 2 ( u 2 r ) s 1 2 ( r ) d ( u 2 r ) ,
R ̃ = a 1 a 2 1 u 2 r q 1 2 ( r ) d ( u 2 r ) ,
S = a 1 a 2 ( u 2 r ) r 1 ( r ) s 1 ( r ) d ( u 2 r ) ,
S ̃ = a 1 a 2 1 u 2 r p 1 ( r ) q 1 ( r ) d ( u 2 r ) ,
Q 0 = 2 a 1 a 2 d ( u 2 r ) p 1 ( r ) r 1 ( r ) ,
R 0 = 2 a 1 a 2 d ( u 2 r ) s 1 ( r ) q 1 ( r ) ,
S 0 = a 1 a 2 d ( u 2 r ) [ p 1 ( r ) s 1 ( r ) + r 1 ( r ) q 1 ( r ) ] .
Q = a 1 a 2 d ( u 2 r ) u 2 r r 1 2 ( r ) = a 1 a 2 d ( u 2 r ) u 2 r [ J 1 ( u 2 r ) N 1 ( u 2 a 1 ) J 1 ( u 2 a 1 ) N 1 ( u 2 r ) ] 2 = θ J N 1 2 ( u 2 a 1 ) + θ N J 1 2 ( u 2 a 1 ) 2 θ J N J 1 ( u 2 a 1 ) N 1 ( u 2 a 1 ) ,
Q ̃ = a 1 a 2 d ( u 2 r ) 1 u 2 r p 1 2 ( r ) = θ ̃ J N 1 2 ( u 2 a 1 ) + θ ̃ N J 1 2 ( u 2 a 1 ) 2 θ ̃ J N J 1 ( u 2 a 1 ) N 1 ( u 2 a 1 ) ,
R = a 1 a 2 d ( u 2 r ) u 2 r s 1 2 ( r ) , = a 1 a 2 d ( u 2 r ) u 2 r [ J 1 ( u 2 r ) N 1 ( u 2 a 1 ) J 1 ( u 2 a 1 ) N 1 ( u 2 r ) ] 2 = θ J [ N 1 ( u 2 a 1 ) ] 2 + θ N [ J 1 ( u 2 a 1 ) ] 2 2 θ J N J 1 ( u 2 a 1 ) N 1 ( u 2 a 1 ) ,
R ̃ = a 1 a 2 d ( u 2 r ) 1 u 2 r q 1 2 ( r ) = a 1 a 2 d ( u 2 r ) 1 u 2 r [ J 1 ( u 2 r ) N 1 ( u 2 a 1 ) J 1 ( u 2 a 1 ) N 1 ( u 2 r ) ] 2 = θ ̃ J [ N 1 ( u 2 a 1 ) ] 2 + θ ̃ N [ J 1 ( u 2 a 1 ) ] 2 2 θ ̃ J N J 1 ( u 2 a 1 ) N 1 ( u 2 a 1 ) ,
S = a 1 a 2 d ( u 2 r ) u 2 r r 1 ( r ) s 1 ( r ) = a 1 a 2 d ( u 2 r ) u 2 r [ J 1 ( u 2 r ) N 1 ( u 2 a 1 ) J 1 ( u 2 a 1 ) N 1 ( u 2 r ) ] × [ J 1 ( u 2 r ) N 1 ( u 2 a 1 ) J 1 ( u 2 a 1 ) N 1 ( u 2 r ) ] = θ J N 1 ( u 2 a 1 ) N 1 ( u 2 a 1 ) + θ N J 1 ( u 2 a 1 ) J 1 ( u 2 a 1 ) θ J N [ J 1 ( u 2 a 1 ) N 1 ( u 2 a 1 ) + J 1 ( u 2 a 1 ) N 1 ( u 2 a 1 ) ]
S ̃ = a 1 a 2 d ( u 2 r ) 1 u 2 r p 1 ( r ) q 1 ( r ) = a 1 a 2 d ( u 2 r ) 1 u 2 r [ J 1 ( u 2 r ) N 1 ( u 2 a 1 ) J 1 ( u 2 a 1 ) N 1 ( u 2 r ) ] × [ J 1 ( u 2 r ) N 1 ( u 2 a 1 ) J 1 ( u 2 a 1 ) N 1 ( u 2 r ) ] = θ ̃ J N 1 ( u 2 a 1 ) N 1 ( u 2 a 1 ) + θ ̃ N J 1 ( u 2 a 1 ) J 1 ( u 2 a 1 ) θ ̃ J N [ J 1 ( u 2 a 1 ) N 1 ( u 2 a 1 ) + J 1 ( u 2 a 1 ) N 1 ( u 2 a 1 ) ] .
θ J = a 1 a 2 d ( u 2 r ) u 2 r [ J 1 ( u 2 r ) ] 2 = [ u 2 2 r 2 + 1 2 J 0 2 ( u 2 r ) + u 2 2 r 2 1 2 J 1 2 ( u 2 r ) ] a 1 a 2 ,
θ N = a 1 a 2 d ( u 2 r ) u 2 r [ N 1 ( u 2 r ) ] 2 = [ u 2 2 r 2 + 1 2 N 0 2 ( u 2 r ) + u 2 2 r 2 1 2 N 1 2 ( u 2 r ) ] a 1 a 2 ,
θ J N = a 1 a 2 d ( u 2 r ) u 2 r J 1 ( u 2 r ) N 1 ( u 2 r ) = [ u 2 2 r 2 + 1 2 J 0 ( u 2 r ) N 0 ( u 2 r ) + u 2 2 r 2 1 2 J 1 ( u 2 r ) N 1 ( u 2 r ) ] a 1 a 2 ,
θ ̃ J = a 1 a 2 d ( u 2 r ) 1 u 2 r J 1 2 ( u 2 r ) = 1 2 [ J 0 2 ( u 2 r ) + J 1 2 ( u 2 r ) ] a 1 a 2 ,
θ ̃ N = a 1 a 2 d ( u 2 r ) 1 u 2 r N 1 2 ( u 2 r ) = 1 2 [ N 0 2 ( u 2 r ) + N 1 2 ( u 2 r ) ] a 1 a 2 ,
θ ̃ J N = a 1 a 2 d ( u 2 r ) 1 u 2 r J 1 ( u 2 r ) N 1 ( u 2 r ) = 1 2 [ J 0 ( u 2 r ) N 0 ( u 2 r ) + J 1 ( u 2 r ) N 1 ( u 2 r ) ] a 1 a 2
Q 0 = 2 a 1 a 2 d ( u 2 r ) p 1 ( r ) r 1 ( r ) = [ p 1 2 ( r ) ] a 1 a 2 = [ N 1 2 ( u 2 a 1 ) J 1 2 ( u 2 r ) 2 J 1 ( u 2 a 1 ) N 1 ( u 2 a 1 ) × J 1 ( u 2 r ) N 1 ( u 2 r ) + J 1 2 ( u 2 a 1 ) N 1 2 ( u 2 r ) ] a 1 a 2 ,
R 0 = 2 a 1 a 2 d ( u 2 r ) s 1 ( r ) q 1 ( r ) = [ q 1 2 ( r ) ] a 1 a 2 = [ N 1 2 ( u 2 a 1 ) J 1 2 ( u 2 r ) 2 J 1 ( u 2 a 1 ) N 1 ( u 2 a 1 ) × J 1 ( u 2 r ) N 1 ( u 2 r ) + J 1 2 ( u 2 a 1 ) N 1 2 ( u 2 r ) ] a 1 a 2 ,
S 0 = a 1 a 2 d ( u 2 r ) [ p 1 ( r ) s 1 ( r ) + r 1 ( r ) q 1 ( r ) ] = [ p 1 ( r ) q 1 ( r ) ] a 1 a 2 = [ N 1 ( u 2 a 1 ) N 1 ( u 2 a 1 ) J 1 2 ( u 2 r ) [ N 1 ( u 2 a 1 ) J 1 ( u 2 a 1 ) + N 1 ( u 2 a 1 ) J 1 ( u 2 a 1 ) ] J 1 ( u 2 r ) N 1 ( u 2 r ) + J 1 ( u 2 a 1 ) J 1 ( u 2 a 1 ) N 1 2 ( u 2 r ) ] a 1 a 2 .

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