Abstract

We demonstrate how to convert several arbitrary optical fiber modes into a single mode and vice versa using superimposed long period gratings (SLPG). As an example, we theoretically consider SLPG consisting of five gratings, which couple first six LP0j modes of a single mode fiber. We optimize the SLPG output to form light beams that are focused at a distance 0.5 mm and 1 mm from the fiber. In addition, we optimize the SLPG output to generate a beam with an amplitude that is uniform inside a 40 angle with a ±0.2 % accuracy. In the latter case, the refractive index profile of the SLPG is calculated for an SMF-28 fiber. The proposed SLPG devices can be used as efficient all-fiber mode focusers and beam shapers.

©2008 Optical Society of America

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References

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  1. R. Kashyap, Fiber Bragg gratings (Academic Press, 1999).
  2. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
    [Crossref]
  3. S. Ramachandran, “Dispersion-Tailored Few-Mode Fibers: A Versatile Platform for In-Fiber Photonic Devices,” J. Lightwave Technol. 23, 3426–3443 (2005).
    [Crossref]
  4. M. Sumetsky and B. J. Eggleton, “Fiber Bragg gratings for dispersion compensation in optical communication systems,” J. Opt. Fiber. Commun. Rep. 2, 256–278 (2005).
    [Crossref]
  5. A. V. Buryak, K. Y. Kolossovski, and D. Y. Stepanov, “Optimization of refractive index sampling for multichannel fiber Bragg gratings,” IEEE J. Quantum Electron. 39, 91–98 (2003).
    [Crossref]
  6. J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, “Dammann Fiber Bragg Gratings and Phase-Only Sampling for High Channel Counts,” IEEE Photon. Technol. Lett. 14,1309–1311 (2002).
    [Crossref]
  7. S. Ramachandran, J. W. Nicholson, S. Ghalmi, M. F. Yan, P. Wisk, E. Monberg, and F. V. Dimarcello, “Light propagation with ultralarge modal areas in optical fibers,” Opt. Lett. 31, 1797–1799 (2006).
    [Crossref] [PubMed]
  8. T. Erdogan, D. Stegall, and A. Heaney, “Direct single-mode fiber to free space coupling assisted by a cladding mode”, in Proceedings of Optical Fiber Communication Conference, vol. 4, Paper FK4 (San Diego, CA, 1999).
  9. Y. Li and T. Erdogan, “Cladding-mode assisted fiber-to-fiber and fiber-to-free-space coupling,” Opt. Commun. 183, 377–388 (2000).
    [Crossref]
  10. W. T. Chen and L. A. Wang, “Optical coupling method utilizing a lensed fiber integrated with a long-period fiber grating,” Appl. Opt. 39, 4490–4500 (2000).
    [Crossref]
  11. W. Y. Su, G. W. Chern, and L. A. Wang, “Analysis of cladding-mode couplings for a lensed fiber integrated with a long-period fiber grating by use of the beam-propagation method,” Appl. Opt. 41, 6576–6584 (2002).
    [Crossref] [PubMed]
  12. M. J. Kim, T. J. Eom, U.C. Paek, and B. H. Lee, “Lens-free optical fiber connector having a long working distance assisted by matched long-period fiber gratings,” J. Lightwave Technol. 23, 588–596 (2005).
    [Crossref]
  13. G. Nemova and R. Kashyap, “Highly efficient lens couplers for laser-diodes based on long period grating in special graded-index fiber,” Opt. Commun. 261, 249–257 (2006).
    [Crossref]
  14. F. M. Dickey and S. C. Holswade, eds., Laser Beam Shaping: Theory and Techniques (Marcel Dekker, 2000).
    [Crossref]
  15. F.M. Dickey, S.C. Holswade, and D.L. Shealy, eds. Laser Beam Shaping Applications, (CRC Press, 2005).
    [Crossref]
  16. D. L. Shealy and J. A. Hoffnagle, “Laser beam shaping profiles and propagation,” Appl. Opt. 45, 5118–5131 (2006).
    [Crossref] [PubMed]

2006 (3)

2005 (3)

2003 (1)

A. V. Buryak, K. Y. Kolossovski, and D. Y. Stepanov, “Optimization of refractive index sampling for multichannel fiber Bragg gratings,” IEEE J. Quantum Electron. 39, 91–98 (2003).
[Crossref]

2002 (2)

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, “Dammann Fiber Bragg Gratings and Phase-Only Sampling for High Channel Counts,” IEEE Photon. Technol. Lett. 14,1309–1311 (2002).
[Crossref]

W. Y. Su, G. W. Chern, and L. A. Wang, “Analysis of cladding-mode couplings for a lensed fiber integrated with a long-period fiber grating by use of the beam-propagation method,” Appl. Opt. 41, 6576–6584 (2002).
[Crossref] [PubMed]

2000 (2)

Y. Li and T. Erdogan, “Cladding-mode assisted fiber-to-fiber and fiber-to-free-space coupling,” Opt. Commun. 183, 377–388 (2000).
[Crossref]

W. T. Chen and L. A. Wang, “Optical coupling method utilizing a lensed fiber integrated with a long-period fiber grating,” Appl. Opt. 39, 4490–4500 (2000).
[Crossref]

1997 (1)

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[Crossref]

Buryak, A. V.

A. V. Buryak, K. Y. Kolossovski, and D. Y. Stepanov, “Optimization of refractive index sampling for multichannel fiber Bragg gratings,” IEEE J. Quantum Electron. 39, 91–98 (2003).
[Crossref]

Chen, W. T.

Chern, G. W.

Dimarcello, F. V.

Eggleton, B. J.

M. Sumetsky and B. J. Eggleton, “Fiber Bragg gratings for dispersion compensation in optical communication systems,” J. Opt. Fiber. Commun. Rep. 2, 256–278 (2005).
[Crossref]

Eom, T. J.

Erdogan, T.

Y. Li and T. Erdogan, “Cladding-mode assisted fiber-to-fiber and fiber-to-free-space coupling,” Opt. Commun. 183, 377–388 (2000).
[Crossref]

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[Crossref]

T. Erdogan, D. Stegall, and A. Heaney, “Direct single-mode fiber to free space coupling assisted by a cladding mode”, in Proceedings of Optical Fiber Communication Conference, vol. 4, Paper FK4 (San Diego, CA, 1999).

Ghalmi, S.

Heaney, A.

T. Erdogan, D. Stegall, and A. Heaney, “Direct single-mode fiber to free space coupling assisted by a cladding mode”, in Proceedings of Optical Fiber Communication Conference, vol. 4, Paper FK4 (San Diego, CA, 1999).

Hoffnagle, J. A.

Kashyap, R.

G. Nemova and R. Kashyap, “Highly efficient lens couplers for laser-diodes based on long period grating in special graded-index fiber,” Opt. Commun. 261, 249–257 (2006).
[Crossref]

R. Kashyap, Fiber Bragg gratings (Academic Press, 1999).

Kim, M. J.

Kolossovski, K. Y.

A. V. Buryak, K. Y. Kolossovski, and D. Y. Stepanov, “Optimization of refractive index sampling for multichannel fiber Bragg gratings,” IEEE J. Quantum Electron. 39, 91–98 (2003).
[Crossref]

Lee, B. H.

Li, H.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, “Dammann Fiber Bragg Gratings and Phase-Only Sampling for High Channel Counts,” IEEE Photon. Technol. Lett. 14,1309–1311 (2002).
[Crossref]

Li, Y.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, “Dammann Fiber Bragg Gratings and Phase-Only Sampling for High Channel Counts,” IEEE Photon. Technol. Lett. 14,1309–1311 (2002).
[Crossref]

Y. Li and T. Erdogan, “Cladding-mode assisted fiber-to-fiber and fiber-to-free-space coupling,” Opt. Commun. 183, 377–388 (2000).
[Crossref]

Monberg, E.

Nemova, G.

G. Nemova and R. Kashyap, “Highly efficient lens couplers for laser-diodes based on long period grating in special graded-index fiber,” Opt. Commun. 261, 249–257 (2006).
[Crossref]

Nicholson, J. W.

Paek, U.C.

Popelek, J.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, “Dammann Fiber Bragg Gratings and Phase-Only Sampling for High Channel Counts,” IEEE Photon. Technol. Lett. 14,1309–1311 (2002).
[Crossref]

Ramachandran, S.

Rothenberg, J. E.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, “Dammann Fiber Bragg Gratings and Phase-Only Sampling for High Channel Counts,” IEEE Photon. Technol. Lett. 14,1309–1311 (2002).
[Crossref]

Shealy, D. L.

Sheng, Y.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, “Dammann Fiber Bragg Gratings and Phase-Only Sampling for High Channel Counts,” IEEE Photon. Technol. Lett. 14,1309–1311 (2002).
[Crossref]

Stegall, D.

T. Erdogan, D. Stegall, and A. Heaney, “Direct single-mode fiber to free space coupling assisted by a cladding mode”, in Proceedings of Optical Fiber Communication Conference, vol. 4, Paper FK4 (San Diego, CA, 1999).

Stepanov, D. Y.

A. V. Buryak, K. Y. Kolossovski, and D. Y. Stepanov, “Optimization of refractive index sampling for multichannel fiber Bragg gratings,” IEEE J. Quantum Electron. 39, 91–98 (2003).
[Crossref]

Su, W. Y.

Sumetsky, M.

M. Sumetsky and B. J. Eggleton, “Fiber Bragg gratings for dispersion compensation in optical communication systems,” J. Opt. Fiber. Commun. Rep. 2, 256–278 (2005).
[Crossref]

Wang, L. A.

Wang, Y.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, “Dammann Fiber Bragg Gratings and Phase-Only Sampling for High Channel Counts,” IEEE Photon. Technol. Lett. 14,1309–1311 (2002).
[Crossref]

Wilcox, R. B.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, “Dammann Fiber Bragg Gratings and Phase-Only Sampling for High Channel Counts,” IEEE Photon. Technol. Lett. 14,1309–1311 (2002).
[Crossref]

Wisk, P.

Yan, M. F.

Zweiback, J.

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, “Dammann Fiber Bragg Gratings and Phase-Only Sampling for High Channel Counts,” IEEE Photon. Technol. Lett. 14,1309–1311 (2002).
[Crossref]

Appl. Opt. (3)

IEEE J. Quantum Electron. (1)

A. V. Buryak, K. Y. Kolossovski, and D. Y. Stepanov, “Optimization of refractive index sampling for multichannel fiber Bragg gratings,” IEEE J. Quantum Electron. 39, 91–98 (2003).
[Crossref]

IEEE Photon. Technol. Lett. (1)

J. E. Rothenberg, H. Li, Y. Li, J. Popelek, Y. Sheng, Y. Wang, R. B. Wilcox, and J. Zweiback, “Dammann Fiber Bragg Gratings and Phase-Only Sampling for High Channel Counts,” IEEE Photon. Technol. Lett. 14,1309–1311 (2002).
[Crossref]

J. Lightwave Technol. (3)

J. Opt. Fiber. Commun. Rep. (1)

M. Sumetsky and B. J. Eggleton, “Fiber Bragg gratings for dispersion compensation in optical communication systems,” J. Opt. Fiber. Commun. Rep. 2, 256–278 (2005).
[Crossref]

Opt. Commun. (2)

Y. Li and T. Erdogan, “Cladding-mode assisted fiber-to-fiber and fiber-to-free-space coupling,” Opt. Commun. 183, 377–388 (2000).
[Crossref]

G. Nemova and R. Kashyap, “Highly efficient lens couplers for laser-diodes based on long period grating in special graded-index fiber,” Opt. Commun. 261, 249–257 (2006).
[Crossref]

Opt. Lett. (1)

Other (4)

F. M. Dickey and S. C. Holswade, eds., Laser Beam Shaping: Theory and Techniques (Marcel Dekker, 2000).
[Crossref]

F.M. Dickey, S.C. Holswade, and D.L. Shealy, eds. Laser Beam Shaping Applications, (CRC Press, 2005).
[Crossref]

R. Kashyap, Fiber Bragg gratings (Academic Press, 1999).

T. Erdogan, D. Stegall, and A. Heaney, “Direct single-mode fiber to free space coupling assisted by a cladding mode”, in Proceedings of Optical Fiber Communication Conference, vol. 4, Paper FK4 (San Diego, CA, 1999).

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Figures (6)

Fig. 1.
Fig. 1. Illustration of the superimposed LPG mode converters.

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Fig. 2.
Fig. 2. Diagram of mode conversion.

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Fig. 3.
Fig. 3. Distribution of the field amplitude. Top plots - along the end-wall of the fiber. Bottom plots - at distances 0.5 mm (red solid curves) and 1 mm (blue dashed curves) from the fiber. All fields are normalized to the same input power.

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Fig. 4.
Fig. 4. Distribution of the field amplitude of the beams generated by the optimized linear combinations of the LP0j modes and by the individual LP0j modes. All fields are normalized to the same input power.

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Fig. 5.
Fig. 5. Distribution of the field amplitudes in the far-field region for the beams generated by the individual LP0j modes (blue curves) and by the homogenized linear combinations of the LP0j modes (red curve). All fields are normalized to the same input power.

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Fig. 6.
Fig. 6. The refractive index profile of the SLPG generating the homogenized beam considered in Section 6.

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Tables (3)

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Table 1. Optimum coefficients in the linear combination of LPoj modes, Eq. (14), for the beams focused at distances 0.5 and 1 mm from the fiber end and the homogenized beam.

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Table 2. Fraction of the total beam power inside the 15 µm radius circle at 0.5 mm from the fiber end (second row) and inside the 25 µm radius circle at 1 mm from the fiber end for the beams generated by the LP oj modes and for the optimized beams.

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Table 3. Design of the SLPG generating the homogenized beam

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Equations (24)

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δ n ( x , y , z ) = [ δ n 0 + ∑ j > k = 1 N δ n jk cos ( 2 π z Λ jk + ϕ jk ) ] θ ( ρ core − x 2 + y 2 ) ,
dA j dz = i ∑ k = 1 N κ jk exp [ i ( β j − β k + 2 π Λ jk ) z + i ϕ jk ] A j ,
κ jj = π δ n 0 λ I jj ,
κ jk = π δ n jk λ I jk , j ≠ k ,
I jk = ∫ x 2 + y 2 < ρ core dxdye j ( x , y ) e k ( x , y ) .
d A ¯ j d z = i ∑ k = 1 k ≠ j N κ jk exp [ i ( β j − β k − κ jj + κ kk + 2 π Λ jk ) z + i ϕ jk ] A ¯ j , A j ( z ) = A ¯ j ( z ) exp ( i κ jj z )
2 π Λ jk = β k − β j − κ kk + κ jj ,
d A ¯ j dz = i ∑ k = 1 k ≠ j N κ jk e i ϕ jk A ¯ j , A j ( z ) = A ¯ j ( z ) exp ( i κ jj z ) .
κ jk = 0 for j , k > 1 .
A 1 ( z ) = exp ( i κ 11 z ) [ cos ( μ z ) A 1 ( 0 ) + i μ sin ( μ z ) ∑ j = 2 N κ 1 j e i ϕ 1 j A j ( 0 ) ] ,
A k ( z ) = exp ( i κ kk z ) [ i κ 1 k μ sin ( μ z ) A 1 ( 0 ) e − i ϕ 1 k + κ 1 k μ 2 ( cos ( μ z ) − 1 ) ∑ j = 2 N κ 1 j e i ( ϕ 1 j − ϕ 1 k ) A j ( 0 ) + A k ( 0 ) ] ,
μ = ∑ j = 2 N κ 1 j 2 , k = 2 , 3 , . . . , N .
ϕ 1 j = α 1 − α j ± π 2 , κ 1 j = κ 12 ∣ A j ( 0 ) ∣ ∣ A 2 ( 0 ) ∣ , tan ( μ L ) = ∓ ∣ A 2 ( 0 ) ∣ μ ∣ A 1 ( 0 ) ∣ κ 12 .
ϕ 1 j = γ 1 − γ j − κ 11 L + κ jj L ± π 2 , κ 1 j = κ 12 ∣ A j ( L ) ∣ ∣ A 2 ( L ) ∣ , tan ( μ L ) = ± ∣ A 2 ( L ) ∣ μ ∣ A 1 ( L ) ∣ κ 12 .
E 0 ( x , y , z ) = ∑ j A j ( 0 ) exp ( i β j z − i ω t ) e j ( x , y )
P = ∑ j ∣ A j ( 0 ) ∣ 2
E out ( ρ , z ) = 2 π i λ ( z − L ) exp [ 2 π i λ ( z − L ) ] ∫ 0 R E out ( ρ 1 , L ) exp [ π i ( ρ 1 2 + ρ 2 ) λ ( z − L ) ] J 0 [ 2 π ρ 1 ρ λ ( z − L ) ] ρ 1 d ρ 1 ,
ρ = x 2 + y 2 .
E out ( ρ , L ) = ∑ j = 1 6 A j e 0 j LP ( ρ )
F ( A 2 , A 3 , . . . A 6 ) = ∫ ρ m ∞ ∣ E out ( ρ , z 0 ) ∣ d ρ
E far ( θ , r ) = 2 π i λ r exp { 2 π i λ r } f ( θ ) , r = ( z − L ) 2 + ρ 2 ,
f ( θ ) = ∫ 0 R E out ( ρ 1 , L ) J 0 [ 2 π θ ρ 1 λ ] ρ 1 d ρ 1 , θ = ρ z − L ≪ 1 .
F ( A 1 , A 2 , A 3 , . . . A 6 ) = ∫ 0 θ m ∣ E far ( θ , r 0 ) − E 0 ∣ d θ
δ n ( x , y , z ) = [ δ n 0 + ∑ k = 2 6 δ n 1 k cos ( 2 π z Λ 1 k + ϕ 1 k ) ] θ ( ρ core − x 2 + y 2 )

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