Abstract

Highly localized fiber Bragg gratings can be inscribed point-by-point with focused ultrashort pulses. The transverse localization of the resonant grating causes strong coupling to cladding modes of high azimuthal and radial order. In this paper, we show how the reflected cladding modes can be fully analyzed, taking their vectorial nature, orientation and degeneracies into account. The observed modes’ polarization and intensity distributions are directly tied to the dispersive properties and show abrupt transitions in nature, strongly correlated with changes in the coupling strengths.

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2011

2010

Y. Shevchenko, C. Chen, M. Dakka, and J. Albert, “Polarization-selective grating excitation of plasmons in cylindrical optical fibers,” Opt. Letters35, 637–639 (2010).
[CrossRef]

G. Marshall, R. Williams, N. Jovanovic, M. J. Steel, and M. J. Withford, “Point-by-point written fiber-bragg gratings and their application in complex grating designs,” Opt. Express18, 19844–19859 (2010).
[CrossRef] [PubMed]

2009

2008

S. Ramachandran, J. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser & Photon. Rev.2, 429–447 (2008).
[CrossRef] [PubMed]

2006

2005

A. Martinez, Y. Lai, M. Dubov, and I. Khrushchev, “Vector bending sensors based on fibre bragg gratings inscribed by infrared femtosecond laser,” Electron. Lett.41, 472–474 (2005).
[CrossRef]

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett.94, 143902 (2005).
[CrossRef] [PubMed]

2001

2000

1997

1996

S. Hewlett, J. D. Love, G. Meltz, T. Bailey, and W. Morey, “Coupling characteristics of photo-induced bragg gratings in depressed-and matched-cladding fibre,” Opt. Quant. Electron.28, 1641–1654 (1996).
[CrossRef]

1989

1978

1961

Abouraddy, A. F.

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett.94, 143902 (2005).
[CrossRef] [PubMed]

Albert, J.

L.-Y. Shao, J. P. Coyle, S. T. Barry, and J. Albert, “Anomalous permittivity and plasmon resonances of copper nanoparticle conformal coatings on optical fibers,” Opt. Mater. Express1, 128–137 (2011).
[CrossRef]

Y. Shevchenko, C. Chen, M. Dakka, and J. Albert, “Polarization-selective grating excitation of plasmons in cylindrical optical fibers,” Opt. Letters35, 637–639 (2010).
[CrossRef]

T. Guo, C. Chen, and J. Albert, “Non-uniform-tilt-modulated fiber bragg grating for temperature-immune micro-displacement measurement,” Meas. Sci. Technol.20, 034007 (2009).
[CrossRef]

T. Guo, L. Shao, H.-Y. Tam, P. A. Krug, and J. Albert, “Tilted fiber grating accelerometer incorporating an abrupt biconical taper for cladding to core recoupling,” Opt. Express17, 20651–20660 (2009).
[CrossRef] [PubMed]

Andrés, M. V.

Bailey, T.

S. Hewlett, J. D. Love, G. Meltz, T. Bailey, and W. Morey, “Coupling characteristics of photo-induced bragg gratings in depressed-and matched-cladding fibre,” Opt. Quant. Electron.28, 1641–1654 (1996).
[CrossRef]

Barry, S. T.

Becker, R. G.

Burdge, G.

Chen, C.

Y. Shevchenko, C. Chen, M. Dakka, and J. Albert, “Polarization-selective grating excitation of plasmons in cylindrical optical fibers,” Opt. Letters35, 637–639 (2010).
[CrossRef]

T. Guo, C. Chen, and J. Albert, “Non-uniform-tilt-modulated fiber bragg grating for temperature-immune micro-displacement measurement,” Meas. Sci. Technol.20, 034007 (2009).
[CrossRef]

Coyle, J. P.

Cruz, J. L.

Cui, Y.

Dakka, M.

Y. Shevchenko, C. Chen, M. Dakka, and J. Albert, “Polarization-selective grating excitation of plasmons in cylindrical optical fibers,” Opt. Letters35, 637–639 (2010).
[CrossRef]

Díez, A.

Dubov, M.

A. Martinez, Y. Lai, M. Dubov, and I. Khrushchev, “Vector bending sensors based on fibre bragg gratings inscribed by infrared femtosecond laser,” Electron. Lett.41, 472–474 (2005).
[CrossRef]

Duparré, M.

Eggleton, B.

Erdogan, T.

Fini, J.

S. Ramachandran, J. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser & Photon. Rev.2, 429–447 (2008).
[CrossRef] [PubMed]

Fink, Y.

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett.94, 143902 (2005).
[CrossRef] [PubMed]

Flamm, D.

Fuerbach, A.

Gambling, W.

Ghalmi, S.

S. Ramachandran, J. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser & Photon. Rev.2, 429–447 (2008).
[CrossRef] [PubMed]

Guo, T.

T. Guo, C. Chen, and J. Albert, “Non-uniform-tilt-modulated fiber bragg grating for temperature-immune micro-displacement measurement,” Meas. Sci. Technol.20, 034007 (2009).
[CrossRef]

T. Guo, L. Shao, H.-Y. Tam, P. A. Krug, and J. Albert, “Tilted fiber grating accelerometer incorporating an abrupt biconical taper for cladding to core recoupling,” Opt. Express17, 20651–20660 (2009).
[CrossRef] [PubMed]

Hewlett, S.

S. Hewlett, J. D. Love, G. Meltz, T. Bailey, and W. Morey, “Coupling characteristics of photo-induced bragg gratings in depressed-and matched-cladding fibre,” Opt. Quant. Electron.28, 1641–1654 (1996).
[CrossRef]

Joannopoulos, J. D.

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett.94, 143902 (2005).
[CrossRef] [PubMed]

Jovanovic, N.

Kaiser, T.

Kerbage, C.

Khrushchev, I.

A. Martinez, Y. Lai, M. Dubov, and I. Khrushchev, “Vector bending sensors based on fibre bragg gratings inscribed by infrared femtosecond laser,” Electron. Lett.41, 472–474 (2005).
[CrossRef]

Krug, P. A.

Lai, Y.

A. Martinez, Y. Lai, M. Dubov, and I. Khrushchev, “Vector bending sensors based on fibre bragg gratings inscribed by infrared femtosecond laser,” Electron. Lett.41, 472–474 (2005).
[CrossRef]

Lee, K.

Love, J. D.

S. Hewlett, J. D. Love, G. Meltz, T. Bailey, and W. Morey, “Coupling characteristics of photo-induced bragg gratings in depressed-and matched-cladding fibre,” Opt. Quant. Electron.28, 1641–1654 (1996).
[CrossRef]

Lu, C.

Marshall, G.

Marshall, G. D.

Martinez, A.

A. Martinez, Y. Lai, M. Dubov, and I. Khrushchev, “Vector bending sensors based on fibre bragg gratings inscribed by infrared femtosecond laser,” Electron. Lett.41, 472–474 (2005).
[CrossRef]

Meltz, G.

S. Hewlett, J. D. Love, G. Meltz, T. Bailey, and W. Morey, “Coupling characteristics of photo-induced bragg gratings in depressed-and matched-cladding fibre,” Opt. Quant. Electron.28, 1641–1654 (1996).
[CrossRef]

Mermelstein, M.

S. Ramachandran, J. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser & Photon. Rev.2, 429–447 (2008).
[CrossRef] [PubMed]

Morey, W.

S. Hewlett, J. D. Love, G. Meltz, T. Bailey, and W. Morey, “Coupling characteristics of photo-induced bragg gratings in depressed-and matched-cladding fibre,” Opt. Quant. Electron.28, 1641–1654 (1996).
[CrossRef]

Nicholson, J. W.

S. Ramachandran, J. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser & Photon. Rev.2, 429–447 (2008).
[CrossRef] [PubMed]

Nolte, S.

Payne, D.

Ramachandran, S.

S. Ramachandran, J. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser & Photon. Rev.2, 429–447 (2008).
[CrossRef] [PubMed]

Sáez-Rodriguez, D.

Schröter, S.

Shao, L.

Shao, L.-Y.

Shapira, O.

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett.94, 143902 (2005).
[CrossRef] [PubMed]

Shevchenko, Y.

Y. Shevchenko, C. Chen, M. Dakka, and J. Albert, “Polarization-selective grating excitation of plasmons in cylindrical optical fibers,” Opt. Letters35, 637–639 (2010).
[CrossRef]

Snitzer, E.

Snyder, A. W.

Steel, M. J.

Tam, H.-Y.

Thomas, J. U.

Tsao, C.

Tünnermann, A.

Voigtländer, C.

Westbrook, P.

White, C.

Williams, R.

Williams, R. J.

Windeler, R.

Withford, M. J.

Yan, M. F.

S. Ramachandran, J. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser & Photon. Rev.2, 429–447 (2008).
[CrossRef] [PubMed]

Young, W.

Electron. Lett.

A. Martinez, Y. Lai, M. Dubov, and I. Khrushchev, “Vector bending sensors based on fibre bragg gratings inscribed by infrared femtosecond laser,” Electron. Lett.41, 472–474 (2005).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Laser & Photon. Rev.

S. Ramachandran, J. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser & Photon. Rev.2, 429–447 (2008).
[CrossRef] [PubMed]

Meas. Sci. Technol.

T. Guo, C. Chen, and J. Albert, “Non-uniform-tilt-modulated fiber bragg grating for temperature-immune micro-displacement measurement,” Meas. Sci. Technol.20, 034007 (2009).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Letters

Y. Shevchenko, C. Chen, M. Dakka, and J. Albert, “Polarization-selective grating excitation of plasmons in cylindrical optical fibers,” Opt. Letters35, 637–639 (2010).
[CrossRef]

Opt. Mater. Express

Opt. Quant. Electron.

S. Hewlett, J. D. Love, G. Meltz, T. Bailey, and W. Morey, “Coupling characteristics of photo-induced bragg gratings in depressed-and matched-cladding fibre,” Opt. Quant. Electron.28, 1641–1654 (1996).
[CrossRef]

Phys. Rev. Lett.

O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete modal decomposition for optical waveguides,” Phys. Rev. Lett.94, 143902 (2005).
[CrossRef] [PubMed]

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Figures (7)

Fig. 1
Fig. 1

Spectrally sorted resonances of the investigated FBG. Coupling constants have been evaluated for the transverse geometry shown in the inset. The polarization direction of the launched fundamental mode is indicated by the double arrow. In the graph, the height of the red lines represents the coupling strength of the fundamental mode to TE or HE modes, the blue lines for TM or EH modes. For HE and EH modes, the line represents the sum of the coupling constants for even and odd modes. The horizontal dash separates their individual contributions, with the lower part representing coupling to the odd mode. The virtual cutoff wavelengths separating the different coupling regimes (see [9]) are shown as vertical dashed lines. The ℓ = 1 and ℓ = 2 degeneracies are highlighted at 1540 nm by two vertical dotted lines.

Fig. 2
Fig. 2

Setup for imaging the cladding modes for PbP FBGs. Transmission spectra were also obtained by recording the signal of the photodiode while sweeping the wavelength of the SWS.

Fig. 3
Fig. 3

Typical transmission spectrum with lowest azimuthal order vectorial labels (l, m). All numbers refer to the HE resonances, which are always at the longer wavelength of the EH/HE doublets. Red labels denote ℓ = 1 modes, blue labels denote ℓ = 2 modes, the vertical dotted lines correspond to the computed l = 1 and l = 2 resonances of the hybrid modes (see also Fig. 1). Horizontal lines below the spectrum indicate the range over which mode patterns of various forms were observed. The virtual cutoffs are also labeled. The lower plots are simply magnifications of the main plot.

Fig. 4
Fig. 4

Typical mode patterns observed of each class: (a) ring, (b) bow tie, (c) and (d) quad tie. Labels indicate the mode indices and the wavelength at which they were observed.

Fig. 5
Fig. 5

Linearly polarized reflections at selected resonances of the highly localized FBG, top row (a)–(e). The black arrows indicate the polarization of the reflected patterns, which coincides with launched fundamental mode. The second row ((f)–(j)) displays the computed patterns with arrows indicating the direction of the electric field. The inset (k) shows how in the case of the ℓ = 2 resonances, the linearly polarized superpositions can be constructed from equal contributions of TE, TM and HE2m modes.

Fig. 6
Fig. 6

Measured (top row, (a)–(d)) and computed (bottom row, (e)–(f)) mode patterns in the quasi-TE/TM regime. The patterns (a) and (c) are predominantly azimuthally polarized, while (b) and (d) exhibited radial polarization.

Fig. 7
Fig. 7

Reflected patterns with more complex polarization (a) and (b). The bottom row displays the superposition with the next higher degenerate azimuthal mode ((c) and (d)).

Equations (30)

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λ = ( n ¯ 11 ( λ ) + n ¯ l m ( λ ) ) Λ / ν
κ l m = 2 π c 4 λ 0 2 π d ϕ 0 a 1 d r r Δ ε ( x , y , z ) E 11 T E l m T * ,
Δ ε ( x , y , z ) 2 ε 0 n ( x , y ) Δ n ( x , y , z ) = ε 0 n ( x , y ) ( Δ n 0 ( x , y ) + ν = 1 Δ n ν ( x , y ) 2 cos ( 2 π ν Λ z ) ) ,
Δ n 0 ( x , y ) = 2 w Λ Δ n max ( x , y ) for the DC portion and
Δ n ν ( x , y ) = 2 π ν sin ( π ν w Λ ) Δ n max ( x , y )
d A 11 ( z ) d z = l , m i κ l m B l m ( z )
d B l m ( z ) d z = i κ l m A 11 ( z ) .
LP 1 x even = TM 0 m + HE 2 m even and LP 1 x odd = TE 0 m + HE 2 m odd
LP 1 y even = TE 0 m + HE 2 m odd and LP 1 y odd = TM 0 m HE 2 m eve ,
J ( K p l ( a 2 ) + r l ( a 2 ) u 2 ) 1 u 2 ( K q l ( a 2 ) + s l ( a 2 ) u 2 ) = 0 ,
J ( K p l ( a 2 ) + n 2 2 n 3 2 r l ( a 2 ) u 2 ) n 2 2 n 1 2 1 u 2 ( K q l ( a 2 ) + n 2 2 n 3 2 s l ( a 2 ) u 2 ) = 0 .
u l m 2 = ( 2 π / λ ) 2 ( n 1 2 n ¯ 2 ) , u 2 2 = ( 2 π / λ ) 2 ( n 2 2 n ¯ 2 ) , w 3 2 = ( 2 π / λ ) 2 ( n ¯ 2 n 3 2 ) ,
σ = i l n ¯ , v 21 = 1 u 2 2 1 u l m 2 , v 32 = 1 w 3 2 + 1 u 2 2 , J = J l ( u l m a 1 ) u l m J l ( u l m a 1 ) , K = K l ( w 3 a 2 ) w 3 K l ( w 3 a 2 ) .
p l ( r ) = J l ( u 2 r ) N l ( u 2 a 1 ) J l ( u 2 a 1 ) N l ( u 2 r ) ,
q l ( r ) = J l ( u 2 r ) N l ( u 2 a 1 ) J l ( u 2 a 1 ) N l ( u 2 r ) ,
r l ( r ) = J l ( u 2 r ) N l ( u 2 a 1 ) J l ( u 2 a 1 ) N l ( u 2 r ) ,
s l ( r ) = J l ( u 2 r ) N l ( u 2 a 1 ) J l ( u 2 a 1 ) N l ( u 2 r ) ,
C l m = π a 1 u l m 2 J l ( u l m a 1 ) 2
E ϕ = i E l m u 1 J 0 ( u 1 r ) e i ( β z ω t )
H z = E l m n ¯ Z 0 u 1 2 β J 0 ( u 1 r ) e i ( β z ω t )
H r = i E l m n ¯ Z 0 u 1 J 0 ( u 1 r ) e i ( β z ω t ) ,
E ϕ cl = i E l m C l m u 2 ( J r 0 ( r ) + s 0 ( r ) u 2 ) e i ( β z ω t )
H z cl = E l m C l m n ¯ Z 0 u 2 2 β ( J p 0 ( r ) + q 0 ( r ) u 2 ) e i ( β z ω t )
H r cl = i E l m C l m n ¯ Z 0 u 2 ( J r 0 ( r ) + s 0 ( r ) u 2 ) e i ( β z ω t ) .
E z = i E l m n ¯ u i 2 β J 0 ( u 1 r ) e i ( β z ω t )
E r = E l m n ¯ u 1 J 0 ( u 1 r ) e i ( β z ω t )
H ϕ = E l m 1 Z 0 u 1 J 0 ( u 1 r ) e i ( β z ω t )
E z cl = i E l m C l m n ¯ u 2 β ( J p 0 ( r ) + n 2 2 n 1 2 q 0 ( r ) u 2 ) e i ( β z ω t )
E r cl = { u 2 2 n 1 2 u 1 n 2 2 } E l m C l m n ¯ u 1 u 2 ( J r 0 ( r ) + n 2 2 n 1 2 s 0 ( r ) u 2 ) e i ( β z ω t )
H ϕ cl = { u 2 } E l m C l m 1 Z 0 ( J r 0 ( r ) + n 2 2 n 1 2 s 0 ( r ) u 2 ) e i ( β z ω t )

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