Abstract

The objective of the present paper is to introduce and numerically demonstrate the operation of a novel band-pass filter based on the phenomenon of resonant tunneling in multi-core photonic crystal fibers (PCFs). The proposed PCF consists of two identical cores separated by a third one which acts as a resonator. With a fine adjustment of the design parameters associated with the resonant-core, phase matching at a single wavelength can be achieved, thus enabling very narrow-band resonant directional coupling between the input and the output cores. The validation of the design is ensured with an accurate PCF analysis based on finite element and beam propagation algorithms. The proposed narrow band-pass filter can be employed in various applications such as all fiber bandpass/bandstop filtering.

© 2005 Optical Society of America

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  1. P.St.J. Russell , “ Photonic crystal fibers ,” Science   299 , 358 – 362 ( 2003 ).
    [Crossref] [PubMed]
  2. S. Kawanishi , T. Yamamoto , H. Kubota , M. Tanaka , and S. Yamaguchi , “ Dispersion controlled and polarization maintaining photonic crystal fibers for high performance network systems ,” IEICE Trans. Electron.   E87-C , 336 – 342 ( 2004 ).
  3. B.J. Mangan , J.C. Knight , T.A. Birks , P.St.J. Russell , and A.H. Greenaway , “ Experimental study of dual-core photonic crystal fibre ,” Electron. Lett.   36 , 1358 – 1359 ( 2000 ).
    [Crossref]
  4. W.N. MacPherson , J.D.C. Jones , B.J. Mangan , J.C. Knight , and P.St.J. Russell , “ Two-core photonic crystal fiber for Doppler difference velocimetry ,” Opt. Commun.   233 , 375 – 380 ( 2003 ).
    [Crossref]
  5. K. Kitayama and Y. Ishida , “ Wavelength-selective coupling of two-core optical fiber: application and design ,” J. Opt. Soc. Am. A   2 , 90 – 94 ( 1985 ).
    [Crossref]
  6. R. Zengerle and O.G. Leminger , “ Narrow-band wavelength-selective directional couplers made of dissimilar single-mode fibers ,” J. Lightwave Technol.   LT-5 , 1196 – 1198 ( 1987 ).
    [Crossref]
  7. E. Eisenmann and E. Weidel , “ Single-mode fused biconical couplers for wavelength division multiplexing with channel spacing between 100-300 nm ,” J. Lightwave Technol.   LT-6 , 113 – 119 ( 1988 ).
    [Crossref]
  8. B. Malo , F. Bilodeau , K.O. Hill , D.C. Johnson , and J. Albert , “ Unbalanced dissimilar-fiber Mach-Zehnder interferometer: application as filter ,” Electron. Lett.   25 , 1416 – 1417 ( 1989 ).
    [Crossref]
  9. K. Thyagarajan , S.D. Seshadri , and A.K. Ghatak , “ Waveguide polarizer based on resonant tunneling ,” J. Lightwave Technol.   9 , 315 – 317 ( 1991 ).
    [Crossref]
  10. M. Skorobogatiy , K. Saitoh , and M. Koshiba “ Resonant directional coupling of hollow Bragg fibers ,” Opt. Lett.   29 , 2112 – 2114 ( 2004 ).
    [Crossref] [PubMed]
  11. K. Saitoh and M. Koshiba , “ Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides ,” J. Lightwave Technol.   19 , 405 – 413 ( 2001 ).
    [Crossref]
  12. K. Saitoh , Y. Sato , and M. Koshiba , “ Coupling characteristics of dual-core photonic crystal fiber couplers ,” Opt. Express   11 , 3188 – 3195 ( 2003 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-24-3188 .
    [Crossref] [PubMed]
  13. J.P. Donnelly , H.A. Haus , and N. Whitaker , “ Symmetric three-guide optical coupler with nonidentical center and outside guides ,” IEEE J. Quantum Electron.   QE-23 , 401 – 406 ( 1987 ).
    [Crossref]
  14. K. Saitoh and M. Koshiba , “ Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers ,” IEEE J. Quantum Electron.   38 , 927 – 933 ( 2002 ).
    [Crossref]
  15. B. Ortega and L. Dong , “ Characteristics of mismatched twin-core fiber spectral filters ,” IEEE Photon. Technol. Lett.   10 , 991 – 993 ( 1998 ).
    [Crossref]
  16. Y. Tsuchida , K. Saitoh , and M. Koshiba , “ Design and characterization of single-mode holey fibers with low bending losses ”, Opt. Express   13 , 4770 – 4779 ( 2005 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-12-4770 .
    [Crossref] [PubMed]
  17. T. Tjugiarto , G.D. Peng , and P.L. Chu , “ Bandpass filtering effect in tapered asymmetrical twin-core optical fibers ,” Electron. Lett.   29 , 1077 – 1078 ( 1993 ).
    [Crossref]
  18. B. Wu and P.L. Chu , “ Narrow-bandpass filter with gain by use of twin-core rare-earth-doped fiber ,” Opt. Lett.   18 , 1913 – 1915 ( 1993 ).
    [Crossref] [PubMed]
  19. B. Ortega and L. Dong , “ Accurate tuning of mismatched twin-core fiber filters ,” Opt. Lett.   23 , 1277 – 1279 ( 1998 ).
    [Crossref]

2005 (1)

2004 (2)

M. Skorobogatiy , K. Saitoh , and M. Koshiba “ Resonant directional coupling of hollow Bragg fibers ,” Opt. Lett.   29 , 2112 – 2114 ( 2004 ).
[Crossref] [PubMed]

S. Kawanishi , T. Yamamoto , H. Kubota , M. Tanaka , and S. Yamaguchi , “ Dispersion controlled and polarization maintaining photonic crystal fibers for high performance network systems ,” IEICE Trans. Electron.   E87-C , 336 – 342 ( 2004 ).

2003 (3)

W.N. MacPherson , J.D.C. Jones , B.J. Mangan , J.C. Knight , and P.St.J. Russell , “ Two-core photonic crystal fiber for Doppler difference velocimetry ,” Opt. Commun.   233 , 375 – 380 ( 2003 ).
[Crossref]

P.St.J. Russell , “ Photonic crystal fibers ,” Science   299 , 358 – 362 ( 2003 ).
[Crossref] [PubMed]

K. Saitoh , Y. Sato , and M. Koshiba , “ Coupling characteristics of dual-core photonic crystal fiber couplers ,” Opt. Express   11 , 3188 – 3195 ( 2003 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-24-3188 .
[Crossref] [PubMed]

2002 (1)

K. Saitoh and M. Koshiba , “ Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers ,” IEEE J. Quantum Electron.   38 , 927 – 933 ( 2002 ).
[Crossref]

2001 (1)

2000 (1)

B.J. Mangan , J.C. Knight , T.A. Birks , P.St.J. Russell , and A.H. Greenaway , “ Experimental study of dual-core photonic crystal fibre ,” Electron. Lett.   36 , 1358 – 1359 ( 2000 ).
[Crossref]

1998 (2)

B. Ortega and L. Dong , “ Characteristics of mismatched twin-core fiber spectral filters ,” IEEE Photon. Technol. Lett.   10 , 991 – 993 ( 1998 ).
[Crossref]

B. Ortega and L. Dong , “ Accurate tuning of mismatched twin-core fiber filters ,” Opt. Lett.   23 , 1277 – 1279 ( 1998 ).
[Crossref]

1993 (2)

T. Tjugiarto , G.D. Peng , and P.L. Chu , “ Bandpass filtering effect in tapered asymmetrical twin-core optical fibers ,” Electron. Lett.   29 , 1077 – 1078 ( 1993 ).
[Crossref]

B. Wu and P.L. Chu , “ Narrow-bandpass filter with gain by use of twin-core rare-earth-doped fiber ,” Opt. Lett.   18 , 1913 – 1915 ( 1993 ).
[Crossref] [PubMed]

1991 (1)

K. Thyagarajan , S.D. Seshadri , and A.K. Ghatak , “ Waveguide polarizer based on resonant tunneling ,” J. Lightwave Technol.   9 , 315 – 317 ( 1991 ).
[Crossref]

1989 (1)

B. Malo , F. Bilodeau , K.O. Hill , D.C. Johnson , and J. Albert , “ Unbalanced dissimilar-fiber Mach-Zehnder interferometer: application as filter ,” Electron. Lett.   25 , 1416 – 1417 ( 1989 ).
[Crossref]

1988 (1)

E. Eisenmann and E. Weidel , “ Single-mode fused biconical couplers for wavelength division multiplexing with channel spacing between 100-300 nm ,” J. Lightwave Technol.   LT-6 , 113 – 119 ( 1988 ).
[Crossref]

1987 (2)

R. Zengerle and O.G. Leminger , “ Narrow-band wavelength-selective directional couplers made of dissimilar single-mode fibers ,” J. Lightwave Technol.   LT-5 , 1196 – 1198 ( 1987 ).
[Crossref]

J.P. Donnelly , H.A. Haus , and N. Whitaker , “ Symmetric three-guide optical coupler with nonidentical center and outside guides ,” IEEE J. Quantum Electron.   QE-23 , 401 – 406 ( 1987 ).
[Crossref]

1985 (1)

Albert, J.

B. Malo , F. Bilodeau , K.O. Hill , D.C. Johnson , and J. Albert , “ Unbalanced dissimilar-fiber Mach-Zehnder interferometer: application as filter ,” Electron. Lett.   25 , 1416 – 1417 ( 1989 ).
[Crossref]

Bilodeau, F.

B. Malo , F. Bilodeau , K.O. Hill , D.C. Johnson , and J. Albert , “ Unbalanced dissimilar-fiber Mach-Zehnder interferometer: application as filter ,” Electron. Lett.   25 , 1416 – 1417 ( 1989 ).
[Crossref]

Birks, T.A.

B.J. Mangan , J.C. Knight , T.A. Birks , P.St.J. Russell , and A.H. Greenaway , “ Experimental study of dual-core photonic crystal fibre ,” Electron. Lett.   36 , 1358 – 1359 ( 2000 ).
[Crossref]

Chu, P.L.

T. Tjugiarto , G.D. Peng , and P.L. Chu , “ Bandpass filtering effect in tapered asymmetrical twin-core optical fibers ,” Electron. Lett.   29 , 1077 – 1078 ( 1993 ).
[Crossref]

B. Wu and P.L. Chu , “ Narrow-bandpass filter with gain by use of twin-core rare-earth-doped fiber ,” Opt. Lett.   18 , 1913 – 1915 ( 1993 ).
[Crossref] [PubMed]

Dong, L.

B. Ortega and L. Dong , “ Accurate tuning of mismatched twin-core fiber filters ,” Opt. Lett.   23 , 1277 – 1279 ( 1998 ).
[Crossref]

B. Ortega and L. Dong , “ Characteristics of mismatched twin-core fiber spectral filters ,” IEEE Photon. Technol. Lett.   10 , 991 – 993 ( 1998 ).
[Crossref]

Donnelly, J.P.

J.P. Donnelly , H.A. Haus , and N. Whitaker , “ Symmetric three-guide optical coupler with nonidentical center and outside guides ,” IEEE J. Quantum Electron.   QE-23 , 401 – 406 ( 1987 ).
[Crossref]

Eisenmann, E.

E. Eisenmann and E. Weidel , “ Single-mode fused biconical couplers for wavelength division multiplexing with channel spacing between 100-300 nm ,” J. Lightwave Technol.   LT-6 , 113 – 119 ( 1988 ).
[Crossref]

Ghatak, A.K.

K. Thyagarajan , S.D. Seshadri , and A.K. Ghatak , “ Waveguide polarizer based on resonant tunneling ,” J. Lightwave Technol.   9 , 315 – 317 ( 1991 ).
[Crossref]

Greenaway, A.H.

B.J. Mangan , J.C. Knight , T.A. Birks , P.St.J. Russell , and A.H. Greenaway , “ Experimental study of dual-core photonic crystal fibre ,” Electron. Lett.   36 , 1358 – 1359 ( 2000 ).
[Crossref]

Haus, H.A.

J.P. Donnelly , H.A. Haus , and N. Whitaker , “ Symmetric three-guide optical coupler with nonidentical center and outside guides ,” IEEE J. Quantum Electron.   QE-23 , 401 – 406 ( 1987 ).
[Crossref]

Hill, K.O.

B. Malo , F. Bilodeau , K.O. Hill , D.C. Johnson , and J. Albert , “ Unbalanced dissimilar-fiber Mach-Zehnder interferometer: application as filter ,” Electron. Lett.   25 , 1416 – 1417 ( 1989 ).
[Crossref]

Ishida, Y.

Johnson, D.C.

B. Malo , F. Bilodeau , K.O. Hill , D.C. Johnson , and J. Albert , “ Unbalanced dissimilar-fiber Mach-Zehnder interferometer: application as filter ,” Electron. Lett.   25 , 1416 – 1417 ( 1989 ).
[Crossref]

Jones, J.D.C.

W.N. MacPherson , J.D.C. Jones , B.J. Mangan , J.C. Knight , and P.St.J. Russell , “ Two-core photonic crystal fiber for Doppler difference velocimetry ,” Opt. Commun.   233 , 375 – 380 ( 2003 ).
[Crossref]

Kawanishi, S.

S. Kawanishi , T. Yamamoto , H. Kubota , M. Tanaka , and S. Yamaguchi , “ Dispersion controlled and polarization maintaining photonic crystal fibers for high performance network systems ,” IEICE Trans. Electron.   E87-C , 336 – 342 ( 2004 ).

Kitayama, K.

Knight, J.C.

W.N. MacPherson , J.D.C. Jones , B.J. Mangan , J.C. Knight , and P.St.J. Russell , “ Two-core photonic crystal fiber for Doppler difference velocimetry ,” Opt. Commun.   233 , 375 – 380 ( 2003 ).
[Crossref]

B.J. Mangan , J.C. Knight , T.A. Birks , P.St.J. Russell , and A.H. Greenaway , “ Experimental study of dual-core photonic crystal fibre ,” Electron. Lett.   36 , 1358 – 1359 ( 2000 ).
[Crossref]

Koshiba, M.

Kubota, H.

S. Kawanishi , T. Yamamoto , H. Kubota , M. Tanaka , and S. Yamaguchi , “ Dispersion controlled and polarization maintaining photonic crystal fibers for high performance network systems ,” IEICE Trans. Electron.   E87-C , 336 – 342 ( 2004 ).

Leminger, O.G.

R. Zengerle and O.G. Leminger , “ Narrow-band wavelength-selective directional couplers made of dissimilar single-mode fibers ,” J. Lightwave Technol.   LT-5 , 1196 – 1198 ( 1987 ).
[Crossref]

MacPherson, W.N.

W.N. MacPherson , J.D.C. Jones , B.J. Mangan , J.C. Knight , and P.St.J. Russell , “ Two-core photonic crystal fiber for Doppler difference velocimetry ,” Opt. Commun.   233 , 375 – 380 ( 2003 ).
[Crossref]

Malo, B.

B. Malo , F. Bilodeau , K.O. Hill , D.C. Johnson , and J. Albert , “ Unbalanced dissimilar-fiber Mach-Zehnder interferometer: application as filter ,” Electron. Lett.   25 , 1416 – 1417 ( 1989 ).
[Crossref]

Mangan, B.J.

W.N. MacPherson , J.D.C. Jones , B.J. Mangan , J.C. Knight , and P.St.J. Russell , “ Two-core photonic crystal fiber for Doppler difference velocimetry ,” Opt. Commun.   233 , 375 – 380 ( 2003 ).
[Crossref]

B.J. Mangan , J.C. Knight , T.A. Birks , P.St.J. Russell , and A.H. Greenaway , “ Experimental study of dual-core photonic crystal fibre ,” Electron. Lett.   36 , 1358 – 1359 ( 2000 ).
[Crossref]

Ortega, B.

B. Ortega and L. Dong , “ Characteristics of mismatched twin-core fiber spectral filters ,” IEEE Photon. Technol. Lett.   10 , 991 – 993 ( 1998 ).
[Crossref]

B. Ortega and L. Dong , “ Accurate tuning of mismatched twin-core fiber filters ,” Opt. Lett.   23 , 1277 – 1279 ( 1998 ).
[Crossref]

Peng, G.D.

T. Tjugiarto , G.D. Peng , and P.L. Chu , “ Bandpass filtering effect in tapered asymmetrical twin-core optical fibers ,” Electron. Lett.   29 , 1077 – 1078 ( 1993 ).
[Crossref]

Russell, P.St.J.

P.St.J. Russell , “ Photonic crystal fibers ,” Science   299 , 358 – 362 ( 2003 ).
[Crossref] [PubMed]

W.N. MacPherson , J.D.C. Jones , B.J. Mangan , J.C. Knight , and P.St.J. Russell , “ Two-core photonic crystal fiber for Doppler difference velocimetry ,” Opt. Commun.   233 , 375 – 380 ( 2003 ).
[Crossref]

B.J. Mangan , J.C. Knight , T.A. Birks , P.St.J. Russell , and A.H. Greenaway , “ Experimental study of dual-core photonic crystal fibre ,” Electron. Lett.   36 , 1358 – 1359 ( 2000 ).
[Crossref]

Saitoh, K.

Sato, Y.

Seshadri, S.D.

K. Thyagarajan , S.D. Seshadri , and A.K. Ghatak , “ Waveguide polarizer based on resonant tunneling ,” J. Lightwave Technol.   9 , 315 – 317 ( 1991 ).
[Crossref]

Skorobogatiy, M.

Tanaka, M.

S. Kawanishi , T. Yamamoto , H. Kubota , M. Tanaka , and S. Yamaguchi , “ Dispersion controlled and polarization maintaining photonic crystal fibers for high performance network systems ,” IEICE Trans. Electron.   E87-C , 336 – 342 ( 2004 ).

Thyagarajan, K.

K. Thyagarajan , S.D. Seshadri , and A.K. Ghatak , “ Waveguide polarizer based on resonant tunneling ,” J. Lightwave Technol.   9 , 315 – 317 ( 1991 ).
[Crossref]

Tjugiarto, T.

T. Tjugiarto , G.D. Peng , and P.L. Chu , “ Bandpass filtering effect in tapered asymmetrical twin-core optical fibers ,” Electron. Lett.   29 , 1077 – 1078 ( 1993 ).
[Crossref]

Tsuchida, Y.

Weidel, E.

E. Eisenmann and E. Weidel , “ Single-mode fused biconical couplers for wavelength division multiplexing with channel spacing between 100-300 nm ,” J. Lightwave Technol.   LT-6 , 113 – 119 ( 1988 ).
[Crossref]

Whitaker, N.

J.P. Donnelly , H.A. Haus , and N. Whitaker , “ Symmetric three-guide optical coupler with nonidentical center and outside guides ,” IEEE J. Quantum Electron.   QE-23 , 401 – 406 ( 1987 ).
[Crossref]

Wu, B.

Yamaguchi, S.

S. Kawanishi , T. Yamamoto , H. Kubota , M. Tanaka , and S. Yamaguchi , “ Dispersion controlled and polarization maintaining photonic crystal fibers for high performance network systems ,” IEICE Trans. Electron.   E87-C , 336 – 342 ( 2004 ).

Yamamoto, T.

S. Kawanishi , T. Yamamoto , H. Kubota , M. Tanaka , and S. Yamaguchi , “ Dispersion controlled and polarization maintaining photonic crystal fibers for high performance network systems ,” IEICE Trans. Electron.   E87-C , 336 – 342 ( 2004 ).

Zengerle, R.

R. Zengerle and O.G. Leminger , “ Narrow-band wavelength-selective directional couplers made of dissimilar single-mode fibers ,” J. Lightwave Technol.   LT-5 , 1196 – 1198 ( 1987 ).
[Crossref]

Electron. Lett. (3)

B.J. Mangan , J.C. Knight , T.A. Birks , P.St.J. Russell , and A.H. Greenaway , “ Experimental study of dual-core photonic crystal fibre ,” Electron. Lett.   36 , 1358 – 1359 ( 2000 ).
[Crossref]

B. Malo , F. Bilodeau , K.O. Hill , D.C. Johnson , and J. Albert , “ Unbalanced dissimilar-fiber Mach-Zehnder interferometer: application as filter ,” Electron. Lett.   25 , 1416 – 1417 ( 1989 ).
[Crossref]

T. Tjugiarto , G.D. Peng , and P.L. Chu , “ Bandpass filtering effect in tapered asymmetrical twin-core optical fibers ,” Electron. Lett.   29 , 1077 – 1078 ( 1993 ).
[Crossref]

IEEE J. Quantum Electron. (2)

J.P. Donnelly , H.A. Haus , and N. Whitaker , “ Symmetric three-guide optical coupler with nonidentical center and outside guides ,” IEEE J. Quantum Electron.   QE-23 , 401 – 406 ( 1987 ).
[Crossref]

K. Saitoh and M. Koshiba , “ Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers ,” IEEE J. Quantum Electron.   38 , 927 – 933 ( 2002 ).
[Crossref]

IEEE Photon. Technol. Lett. (1)

B. Ortega and L. Dong , “ Characteristics of mismatched twin-core fiber spectral filters ,” IEEE Photon. Technol. Lett.   10 , 991 – 993 ( 1998 ).
[Crossref]

IEICE Trans. Electron. (1)

S. Kawanishi , T. Yamamoto , H. Kubota , M. Tanaka , and S. Yamaguchi , “ Dispersion controlled and polarization maintaining photonic crystal fibers for high performance network systems ,” IEICE Trans. Electron.   E87-C , 336 – 342 ( 2004 ).

J. Lightwave Technol. (4)

K. Thyagarajan , S.D. Seshadri , and A.K. Ghatak , “ Waveguide polarizer based on resonant tunneling ,” J. Lightwave Technol.   9 , 315 – 317 ( 1991 ).
[Crossref]

R. Zengerle and O.G. Leminger , “ Narrow-band wavelength-selective directional couplers made of dissimilar single-mode fibers ,” J. Lightwave Technol.   LT-5 , 1196 – 1198 ( 1987 ).
[Crossref]

E. Eisenmann and E. Weidel , “ Single-mode fused biconical couplers for wavelength division multiplexing with channel spacing between 100-300 nm ,” J. Lightwave Technol.   LT-6 , 113 – 119 ( 1988 ).
[Crossref]

K. Saitoh and M. Koshiba , “ Full-vectorial finite element beam propagation method with perfectly matched layers for anisotropic optical waveguides ,” J. Lightwave Technol.   19 , 405 – 413 ( 2001 ).
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

W.N. MacPherson , J.D.C. Jones , B.J. Mangan , J.C. Knight , and P.St.J. Russell , “ Two-core photonic crystal fiber for Doppler difference velocimetry ,” Opt. Commun.   233 , 375 – 380 ( 2003 ).
[Crossref]

Opt. Express (2)

Opt. Lett. (3)

Science (1)

P.St.J. Russell , “ Photonic crystal fibers ,” Science   299 , 358 – 362 ( 2003 ).
[Crossref] [PubMed]

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Figures (8)

Fig.1.
Fig.1.

Schematic cross-section of the proposed bandpass filter in three-core PCF.

Fig. 2.
Fig. 2.

Magnified areas around (a) the input core-A and (b) the resonator core-B of the structure in Fig. 1. Bottom figures represent their effective refractive index profiles.

Fig. 3.
Fig. 3.

Schematic of 3 supermodes of a 3 core PCF filter.

Fig. 4.
Fig. 4.

Variation of the value of 2n eff,3-n eff,1-n eff,2 as a function of dc /Λ at 1.55 μm.

Fig. 5.
Fig. 5.

Bandpass filtering characteristics of three-core PCF filter.

Fig. 6.
Fig. 6.

Snapshots of electric field distribution at (a) z=0, (b) z=Lc /3 (c) z=Lc /2, (d) z=2Lc /3, and (e) z=Lc .

Fig. 7.
Fig. 7.

Impact of the design parameters on the central wavelength of the filter. The variance of the central resonance wavelength-λ0 is plotted as a function of the fabrication tolerance for various design parameters, that is d (green line), dc (red line), and Λ (blue line). It is evident that the filter’s response is insensitive to variations of the lattice constant-Λ due to the non-proximity feature of the coupling mechanism between the input and the output cores.

Fig. 8.
Fig. 8.

Total coupling length as a function of the bending radius. The high impact of the small bending radii to the coupling length is evident. As the bending radius tends to large values the convergence to the nominal coupling length of Lc = 7.58 cm is clear.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

( ϕ 1 + ϕ 2 ) 2 + ϕ 3 .
( ϕ 1 exp ( j β 1 z ) + ϕ 2 exp ( j β 2 z ) ) 2 + ϕ 3 exp ( j β 3 z )
n eff , 1 n eff , 3 = n eff , 3 n eff , 2
2 n eff , 3 n eff , 1 n eff , 2 = 0
L c = λ 0 2 ( n eff , 1 n eff , 3 ) .

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