Abstract

We report an analysis of the transmission characteristics of long period gratings written in a linearly tapered optical fiber. Such gratings are best analyzed by a coupled mode theory that includes several cladding modes simultaneously. However, the numerical solution needs to be modified to take into account the changing propagation constants along the length of the fiber. The transmission spectra show certain distinctive features that depend on the grating period and choice of the phase-matched coupled cladding modes. For low-order modes, the resonances broaden and tend to vanish as the taper of the angle increases. For higher-order modes, the resonances remain well defined but shift toward higher or lower wavelengths and broaden. The transmission spectrum is also highly sensitive to a change in the ambient atmosphere.

© 2009 Optical Society of America

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  1. A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long period fiber gratings as band rejection filters,” J. Lightwave Technol. 14, 58-65 (1996).
    [CrossRef]
  2. T. Erdogan, “Fiber Grating Spectra,” J. Lightwave Technol. 15, 1277-1294 (1997).
    [CrossRef]
  3. R. Kashyap, Fiber Bragg Gratings (Academic, 1999).
  4. A. J. Fielding, K. Edinger, and C. C. Davis, “Experimental observation of mode evolution in single mode tapered optical fibers,” J. Lightwave Technol. 17, 1649-1656 (1999).
    [CrossRef]
  5. T. Allsop, F. Floreani, K. P. Jedrzejewski, P. V. S. Marques, R. Romero, and D. J. Webb, “Spectral characteristics of tapered LPG device as sensing element for refractive index and temperature,” J. Lightwave Technol. 24, 870-878 (2006).
    [CrossRef]
  6. T. Allsop, F. Floreani, K. P. Jedrzejewski, P. V. S. Marques, R. Romero, D. J. Webband, and I. Bennion, “Refractive index sensing with long-period grating fabricated in biconical tapered fiber,” Electron. Lett. 41, 471-472 (2005).
    [CrossRef]
  7. J.-F. Ding, A. P. Zhang, L.-Y. Shao, J.-H. Yan, and S. He, “Fiber-taper seeded long-period grating pair as a highly sensitive refractive index sensor,” IEEE Photon. Technol. Lett. 17, 1247-1249 (2005).
    [CrossRef]
  8. A. K. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge U. Press, 1998).
  9. R. Singh, H. Kumar, and E. K. Sharma, “Necessity of three-layer fiber geometry for characterisation of cladding mode characteristics,” Microwave Opt. Technol. Lett. 37, 45-49(2003).
    [CrossRef]
  10. H. Kim, J. Bae, J. Lee, J. Chun, and S. B. Lee, “Analysis of concatenated long period gratings having phase shifted and cascaded effects,” Jpn. J. Appl. Phys. 42, 5098-5101 (2003).
    [CrossRef]

2006

2005

T. Allsop, F. Floreani, K. P. Jedrzejewski, P. V. S. Marques, R. Romero, D. J. Webband, and I. Bennion, “Refractive index sensing with long-period grating fabricated in biconical tapered fiber,” Electron. Lett. 41, 471-472 (2005).
[CrossRef]

J.-F. Ding, A. P. Zhang, L.-Y. Shao, J.-H. Yan, and S. He, “Fiber-taper seeded long-period grating pair as a highly sensitive refractive index sensor,” IEEE Photon. Technol. Lett. 17, 1247-1249 (2005).
[CrossRef]

2003

R. Singh, H. Kumar, and E. K. Sharma, “Necessity of three-layer fiber geometry for characterisation of cladding mode characteristics,” Microwave Opt. Technol. Lett. 37, 45-49(2003).
[CrossRef]

H. Kim, J. Bae, J. Lee, J. Chun, and S. B. Lee, “Analysis of concatenated long period gratings having phase shifted and cascaded effects,” Jpn. J. Appl. Phys. 42, 5098-5101 (2003).
[CrossRef]

1999

1997

T. Erdogan, “Fiber Grating Spectra,” J. Lightwave Technol. 15, 1277-1294 (1997).
[CrossRef]

1996

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long period fiber gratings as band rejection filters,” J. Lightwave Technol. 14, 58-65 (1996).
[CrossRef]

Allsop, T.

T. Allsop, F. Floreani, K. P. Jedrzejewski, P. V. S. Marques, R. Romero, and D. J. Webb, “Spectral characteristics of tapered LPG device as sensing element for refractive index and temperature,” J. Lightwave Technol. 24, 870-878 (2006).
[CrossRef]

T. Allsop, F. Floreani, K. P. Jedrzejewski, P. V. S. Marques, R. Romero, D. J. Webband, and I. Bennion, “Refractive index sensing with long-period grating fabricated in biconical tapered fiber,” Electron. Lett. 41, 471-472 (2005).
[CrossRef]

Bae, J.

H. Kim, J. Bae, J. Lee, J. Chun, and S. B. Lee, “Analysis of concatenated long period gratings having phase shifted and cascaded effects,” Jpn. J. Appl. Phys. 42, 5098-5101 (2003).
[CrossRef]

Bennion, I.

T. Allsop, F. Floreani, K. P. Jedrzejewski, P. V. S. Marques, R. Romero, D. J. Webband, and I. Bennion, “Refractive index sensing with long-period grating fabricated in biconical tapered fiber,” Electron. Lett. 41, 471-472 (2005).
[CrossRef]

Bhatia, V.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long period fiber gratings as band rejection filters,” J. Lightwave Technol. 14, 58-65 (1996).
[CrossRef]

Chun, J.

H. Kim, J. Bae, J. Lee, J. Chun, and S. B. Lee, “Analysis of concatenated long period gratings having phase shifted and cascaded effects,” Jpn. J. Appl. Phys. 42, 5098-5101 (2003).
[CrossRef]

Davis, C. C.

Ding, J.-F.

J.-F. Ding, A. P. Zhang, L.-Y. Shao, J.-H. Yan, and S. He, “Fiber-taper seeded long-period grating pair as a highly sensitive refractive index sensor,” IEEE Photon. Technol. Lett. 17, 1247-1249 (2005).
[CrossRef]

Edinger, K.

Erdogan, T.

T. Erdogan, “Fiber Grating Spectra,” J. Lightwave Technol. 15, 1277-1294 (1997).
[CrossRef]

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long period fiber gratings as band rejection filters,” J. Lightwave Technol. 14, 58-65 (1996).
[CrossRef]

Fielding, A. J.

Floreani, F.

T. Allsop, F. Floreani, K. P. Jedrzejewski, P. V. S. Marques, R. Romero, and D. J. Webb, “Spectral characteristics of tapered LPG device as sensing element for refractive index and temperature,” J. Lightwave Technol. 24, 870-878 (2006).
[CrossRef]

T. Allsop, F. Floreani, K. P. Jedrzejewski, P. V. S. Marques, R. Romero, D. J. Webband, and I. Bennion, “Refractive index sensing with long-period grating fabricated in biconical tapered fiber,” Electron. Lett. 41, 471-472 (2005).
[CrossRef]

Ghatak, A. K.

A. K. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge U. Press, 1998).

He, S.

J.-F. Ding, A. P. Zhang, L.-Y. Shao, J.-H. Yan, and S. He, “Fiber-taper seeded long-period grating pair as a highly sensitive refractive index sensor,” IEEE Photon. Technol. Lett. 17, 1247-1249 (2005).
[CrossRef]

Jedrzejewski, K. P.

T. Allsop, F. Floreani, K. P. Jedrzejewski, P. V. S. Marques, R. Romero, and D. J. Webb, “Spectral characteristics of tapered LPG device as sensing element for refractive index and temperature,” J. Lightwave Technol. 24, 870-878 (2006).
[CrossRef]

T. Allsop, F. Floreani, K. P. Jedrzejewski, P. V. S. Marques, R. Romero, D. J. Webband, and I. Bennion, “Refractive index sensing with long-period grating fabricated in biconical tapered fiber,” Electron. Lett. 41, 471-472 (2005).
[CrossRef]

Judkins, J. B.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long period fiber gratings as band rejection filters,” J. Lightwave Technol. 14, 58-65 (1996).
[CrossRef]

Kashyap, R.

R. Kashyap, Fiber Bragg Gratings (Academic, 1999).

Kim, H.

H. Kim, J. Bae, J. Lee, J. Chun, and S. B. Lee, “Analysis of concatenated long period gratings having phase shifted and cascaded effects,” Jpn. J. Appl. Phys. 42, 5098-5101 (2003).
[CrossRef]

Kumar, H.

R. Singh, H. Kumar, and E. K. Sharma, “Necessity of three-layer fiber geometry for characterisation of cladding mode characteristics,” Microwave Opt. Technol. Lett. 37, 45-49(2003).
[CrossRef]

Lee, J.

H. Kim, J. Bae, J. Lee, J. Chun, and S. B. Lee, “Analysis of concatenated long period gratings having phase shifted and cascaded effects,” Jpn. J. Appl. Phys. 42, 5098-5101 (2003).
[CrossRef]

Lee, S. B.

H. Kim, J. Bae, J. Lee, J. Chun, and S. B. Lee, “Analysis of concatenated long period gratings having phase shifted and cascaded effects,” Jpn. J. Appl. Phys. 42, 5098-5101 (2003).
[CrossRef]

Lemaire, P. J.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long period fiber gratings as band rejection filters,” J. Lightwave Technol. 14, 58-65 (1996).
[CrossRef]

Marques, P. V. S.

T. Allsop, F. Floreani, K. P. Jedrzejewski, P. V. S. Marques, R. Romero, and D. J. Webb, “Spectral characteristics of tapered LPG device as sensing element for refractive index and temperature,” J. Lightwave Technol. 24, 870-878 (2006).
[CrossRef]

T. Allsop, F. Floreani, K. P. Jedrzejewski, P. V. S. Marques, R. Romero, D. J. Webband, and I. Bennion, “Refractive index sensing with long-period grating fabricated in biconical tapered fiber,” Electron. Lett. 41, 471-472 (2005).
[CrossRef]

Romero, R.

T. Allsop, F. Floreani, K. P. Jedrzejewski, P. V. S. Marques, R. Romero, and D. J. Webb, “Spectral characteristics of tapered LPG device as sensing element for refractive index and temperature,” J. Lightwave Technol. 24, 870-878 (2006).
[CrossRef]

T. Allsop, F. Floreani, K. P. Jedrzejewski, P. V. S. Marques, R. Romero, D. J. Webband, and I. Bennion, “Refractive index sensing with long-period grating fabricated in biconical tapered fiber,” Electron. Lett. 41, 471-472 (2005).
[CrossRef]

Shao, L.-Y.

J.-F. Ding, A. P. Zhang, L.-Y. Shao, J.-H. Yan, and S. He, “Fiber-taper seeded long-period grating pair as a highly sensitive refractive index sensor,” IEEE Photon. Technol. Lett. 17, 1247-1249 (2005).
[CrossRef]

Sharma, E. K.

R. Singh, H. Kumar, and E. K. Sharma, “Necessity of three-layer fiber geometry for characterisation of cladding mode characteristics,” Microwave Opt. Technol. Lett. 37, 45-49(2003).
[CrossRef]

Singh, R.

R. Singh, H. Kumar, and E. K. Sharma, “Necessity of three-layer fiber geometry for characterisation of cladding mode characteristics,” Microwave Opt. Technol. Lett. 37, 45-49(2003).
[CrossRef]

Sipe, J. E.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long period fiber gratings as band rejection filters,” J. Lightwave Technol. 14, 58-65 (1996).
[CrossRef]

Thyagarajan, K.

A. K. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge U. Press, 1998).

Vengsarkar, A. M.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, and J. E. Sipe, “Long period fiber gratings as band rejection filters,” J. Lightwave Technol. 14, 58-65 (1996).
[CrossRef]

Webb, D. J.

Webband, D. J.

T. Allsop, F. Floreani, K. P. Jedrzejewski, P. V. S. Marques, R. Romero, D. J. Webband, and I. Bennion, “Refractive index sensing with long-period grating fabricated in biconical tapered fiber,” Electron. Lett. 41, 471-472 (2005).
[CrossRef]

Yan, J.-H.

J.-F. Ding, A. P. Zhang, L.-Y. Shao, J.-H. Yan, and S. He, “Fiber-taper seeded long-period grating pair as a highly sensitive refractive index sensor,” IEEE Photon. Technol. Lett. 17, 1247-1249 (2005).
[CrossRef]

Zhang, A. P.

J.-F. Ding, A. P. Zhang, L.-Y. Shao, J.-H. Yan, and S. He, “Fiber-taper seeded long-period grating pair as a highly sensitive refractive index sensor,” IEEE Photon. Technol. Lett. 17, 1247-1249 (2005).
[CrossRef]

Electron. Lett.

T. Allsop, F. Floreani, K. P. Jedrzejewski, P. V. S. Marques, R. Romero, D. J. Webband, and I. Bennion, “Refractive index sensing with long-period grating fabricated in biconical tapered fiber,” Electron. Lett. 41, 471-472 (2005).
[CrossRef]

IEEE Photon. Technol. Lett.

J.-F. Ding, A. P. Zhang, L.-Y. Shao, J.-H. Yan, and S. He, “Fiber-taper seeded long-period grating pair as a highly sensitive refractive index sensor,” IEEE Photon. Technol. Lett. 17, 1247-1249 (2005).
[CrossRef]

J. Lightwave Technol.

Jpn. J. Appl. Phys.

H. Kim, J. Bae, J. Lee, J. Chun, and S. B. Lee, “Analysis of concatenated long period gratings having phase shifted and cascaded effects,” Jpn. J. Appl. Phys. 42, 5098-5101 (2003).
[CrossRef]

Microwave Opt. Technol. Lett.

R. Singh, H. Kumar, and E. K. Sharma, “Necessity of three-layer fiber geometry for characterisation of cladding mode characteristics,” Microwave Opt. Technol. Lett. 37, 45-49(2003).
[CrossRef]

Other

R. Kashyap, Fiber Bragg Gratings (Academic, 1999).

A. K. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge U. Press, 1998).

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Figures (12)

Fig. 1
Fig. 1

LPG in a linearly tapered fiber with tapered angle θ.

Fig. 2
Fig. 2

Phase-matching curves for different LP 0 m modes in the untapered fiber of 4.6 μm core radius and 62.5 μm cladding radius, with m marked on the curves.

Fig. 3
Fig. 3

Phase-matching curves for mode coupling to low-order modes. The core radius changes from 4.6 to 3.6 μm from the lowest to highest curves in each group.

Fig. 4
Fig. 4

Phase-matching curves for mode coupling to higher-order modes for core radii varying from 4.6 to 3.6 μm (marked on each group of curves).

Fig. 5
Fig. 5

Variation of coupling coefficient κ with core radius in the wavelength span from 1.3 to 1.6 μm for coupling to different modes (marked on the curves).

Fig. 6
Fig. 6

Transmission spectra for a 12.7 cm long grating of period 700 μm (solid line) and for a 7.5 cm long grating of period 565 μm (dotted line) in an untapered fiber.

Fig. 7
Fig. 7

Transmission spectra for a 12.7 cm long grating with period 700 μm in a tapered fiber for different taper angles defined by values of z 0 ( cm / μm ) marked on the curves.

Fig. 8
Fig. 8

Transmission spectra for the 7.5 cm long LPG with a grating period of 565 μm in a fiber without taper (WOT) and for different tapered angles defined by values of z 0 ( cm / μm ) marked on the curves.

Fig. 9
Fig. 9

Phase-matching curves around grating period 565 μm for different core radii marked on the curves.

Fig. 10
Fig. 10

Power in core mode with propagation distance for a grating period of 565 μm for different wavelengths (marked on the curves in micrometers) in a tapered fiber with tapered angle defined by z 0 = 10 cm / μm .

Fig. 11
Fig. 11

Transmission spectra for a grating period of 565 μm for a grating length of 15 cm in a tapered fiber with z 0 = 10 cm / μm showing the effect of changing the ambient refractive index from 1.0 to 1.2 and 1.4 (marked on the curves).

Fig. 12
Fig. 12

Transmission spectra for the LPG of 6.0 cm length with a grating period of 425 μm in a fiber without taper (WOT) and for different tapered angles defined by values of z 0 ( cm / μm ) marked on the curves.

Equations (18)

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λ m = ( n e f f m n e f f m ) Λ ,
n = n 1 , r < a c o , n = n 2 , a c o < r < a c l n = n a , r > a c l , ,
a c o ( z ) = a 0 z z 0 , a c l ( z ) = b 0 a 0 a c o ,
ψ ( r ) = A J 0 ( U r / a c o ) , r a c o , ψ ( r ) = B K 0 ( W r / a c o ) + C I 0 ( W r / a c o ) , a c o r a c l ψ ( r ) = D K 0 ( W 1 r / a c o ) , r a c l , ,
U = a c o k 0 ( n 1 2 n eff 2 ) , W = a c o k 0 ( n eff 2 n 2 2 ) ,         W 1 = a c o k 0 ( n eff 2 n a 2 ) ;
[ J ¯ 0 ( U ) K ¯ 0 ( W ) ] [ I ¯ 0 ( W c ) + K ¯ 0 ( W 1 c ) ] [ I ¯ 0 ( W ) + J ¯ 0 ( U ) ] [ K ¯ 0 ( W 1 C ) K ¯ 0 ( W c ) ] = I 1 ( W ) K 1 ( W c ) K 1 ( W ) I 1 ( W c ) ,
ψ ( r ) = A J 0 ( U r / a c o ) , r a c o , ψ ( r ) = B J 0 ( U ¯ r / a c o ) + CY 0 ( U ¯ r / a c o ) , a c o r a c l , ψ ( r ) = D K 0 ( W 1 r / a c o ) , r a c l ,
U = a c o k 0 ( n 1 2 n eff 2 ) ,     U ¯ = a c o k 0 ( n 2 2 n eff 2 ) ,     W 1 = a c o k 0 ( n eff 2 n a 2 ) ;
[ J ¯ 0 ( U ) J ¯ 0 ( U ¯ ) ] [ K ¯ 0 ( W 1 C ) Y ¯ 0 ( U ¯ c ) ] [ J ¯ 0 ( U ) Y ¯ 0 ( U ¯ ) ] [ K ¯ 0 ( W 1 C ) J ¯ 0 ( U ¯ c ) ] = Y 1 ( U ¯ ) J 1 ( U ¯ c ) Y 1 ( U ¯ c ) J 1 ( U ¯ ) .
ψ = m = 1 m = N B m ( z ) ψ m ( r ) e i β m z ,
d d z [ B 1 B 2 B N ] = [ D 11 D 12 D 1 N D 21 D 22 D 2 N D N 1 D N 2 D N N ] [ B 1 B 2 B N ] ,
D m n = 2 i κ m n ( z ) e i Δ β mn ( z ) z sin K z ,
κ m n = k 0 4 0 a ψ m Δ n 2 ψ n r d r ,
κ m n = k 0 4 A m A n Δ n 2 a c o 2 U m 2 ( U n c ) 2 [ U m J 1 ( U m ) J 0 ( U n c ) U n c J 0 ( U m ) J 1 ( U n c ) ] for     m n ,
κ m m = k 0 8 A m 2 Δ n 2 a c o 2 [ J 0 2 ( U m ) + J 1 2 ( U m ) ] for     m = n ,
d d z B = D B .
D 1 n = κ 1 n ( z ) e j Γ n ( z ) , D n 1 = κ n 1 ( z ) e j Γ n ( z ) ,
Γ n ( z ) = 2 π λ [ n eff 1 ( z ) n eff n ( z ) ] 2 π Λ ,

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