Abstract

A rigorous modal three-dimensional calculation method is used to analyze the diffraction and absorption losses in Fabry–Perot (FP) multilayer interferential filters between both single-mode input and output fibers. With an ordinary personal computer, this method affords accurate calculation of the absorption losses and the diffraction into the cladding modes of both fibers as well as the reflectance into the input fiber. The calculation results show that absorption losses and diffraction are troublesome for high-finesse FP filters.

© 2008 Optical Society of America

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  1. J. Bittebierre and B. Lazaridès, "Narrow-bandpass hybrid filters with broad rejection band for single-mode waveguides," Appl. Opt. 40, 11-19 (2001).
    [CrossRef]
  2. S. Peng and A. Oliner, "Guidance and leakage properties of open dielectric waveguides. I. Mathematical formulations," IEEE Trans. Microwave Theory Tech. 29, 843-855 (1981).
    [CrossRef]
  3. A. Sudbo, "Numerically stable formulation of the transverse resonance method for vector mode-field calculations in dielectric waveguides," IEEE Photon. Technol. Lett. 5, 342-344 (1993).
    [CrossRef]
  4. G. Sztefka and H. Nolting, "Bidirectional eigenmode propagation for large refractive index steps," IEEE Photon. Technol. Lett. 5, 554-557 (1993).
    [CrossRef]
  5. A. W. Snyder and J. D. Lowe, Optical Waveguide Theory (Chapman & Hall, 1983).
  6. C. Yeh and G. Lindgren, "Computing the propagation characteristics of radially stratified fibers: an efficient method," Appl. Opt. 16, 483-493 (1977).
    [CrossRef] [PubMed]
  7. P. Biensman, "Rigourous and efficient modelling of wavelength scale photonic components," Ph.D. thesis (University of Gent, Belgium, 2001).
  8. D. F. G. Gallagher and T. P. Felici, "Eigenmode expansion methods for simulation of optical propagation in photonics: pros and cons," Proc. SPIE 4987, 69-82 (2003).
    [CrossRef]
  9. H. A. MacLeod, Thin Film Optical Filters (Institute of Physics Publishing, 2001).
    [CrossRef]
  10. J. Lumeau, M. Cathelinaud, J. Bittebierre, and M. Lequime, "Ultranarrow bandpass hybrid filter with wide rejection band," Appl. Opt. 45, 1328-1332 (2006).
    [CrossRef] [PubMed]
  11. M. B. Alsous, J. Bittebierre, R. Richier, and H. Ahmad, "Construction of all-fiber Nd3+ fibre lasers with multidielectric mirrors," Pure Appl. Opt. 5, 777-790 (1996).
    [CrossRef]
  12. H. Yanagawa, T. Ochiai, H. Hayakawa, and H. Miyazawa, "Filter-embedded design and its applications to passive components," J. Lightwave Technol. 7, 1646-1653 (1989).
    [CrossRef]

2006 (1)

2003 (1)

D. F. G. Gallagher and T. P. Felici, "Eigenmode expansion methods for simulation of optical propagation in photonics: pros and cons," Proc. SPIE 4987, 69-82 (2003).
[CrossRef]

2001 (3)

H. A. MacLeod, Thin Film Optical Filters (Institute of Physics Publishing, 2001).
[CrossRef]

P. Biensman, "Rigourous and efficient modelling of wavelength scale photonic components," Ph.D. thesis (University of Gent, Belgium, 2001).

J. Bittebierre and B. Lazaridès, "Narrow-bandpass hybrid filters with broad rejection band for single-mode waveguides," Appl. Opt. 40, 11-19 (2001).
[CrossRef]

1996 (1)

M. B. Alsous, J. Bittebierre, R. Richier, and H. Ahmad, "Construction of all-fiber Nd3+ fibre lasers with multidielectric mirrors," Pure Appl. Opt. 5, 777-790 (1996).
[CrossRef]

1993 (2)

A. Sudbo, "Numerically stable formulation of the transverse resonance method for vector mode-field calculations in dielectric waveguides," IEEE Photon. Technol. Lett. 5, 342-344 (1993).
[CrossRef]

G. Sztefka and H. Nolting, "Bidirectional eigenmode propagation for large refractive index steps," IEEE Photon. Technol. Lett. 5, 554-557 (1993).
[CrossRef]

1989 (1)

H. Yanagawa, T. Ochiai, H. Hayakawa, and H. Miyazawa, "Filter-embedded design and its applications to passive components," J. Lightwave Technol. 7, 1646-1653 (1989).
[CrossRef]

1983 (1)

A. W. Snyder and J. D. Lowe, Optical Waveguide Theory (Chapman & Hall, 1983).

1981 (1)

S. Peng and A. Oliner, "Guidance and leakage properties of open dielectric waveguides. I. Mathematical formulations," IEEE Trans. Microwave Theory Tech. 29, 843-855 (1981).
[CrossRef]

1977 (1)

Ahmad, H.

M. B. Alsous, J. Bittebierre, R. Richier, and H. Ahmad, "Construction of all-fiber Nd3+ fibre lasers with multidielectric mirrors," Pure Appl. Opt. 5, 777-790 (1996).
[CrossRef]

Alsous, M. B.

M. B. Alsous, J. Bittebierre, R. Richier, and H. Ahmad, "Construction of all-fiber Nd3+ fibre lasers with multidielectric mirrors," Pure Appl. Opt. 5, 777-790 (1996).
[CrossRef]

Biensman, P.

P. Biensman, "Rigourous and efficient modelling of wavelength scale photonic components," Ph.D. thesis (University of Gent, Belgium, 2001).

Bittebierre, J.

Cathelinaud, M.

Felici, T. P.

D. F. G. Gallagher and T. P. Felici, "Eigenmode expansion methods for simulation of optical propagation in photonics: pros and cons," Proc. SPIE 4987, 69-82 (2003).
[CrossRef]

Gallagher, D. F. G.

D. F. G. Gallagher and T. P. Felici, "Eigenmode expansion methods for simulation of optical propagation in photonics: pros and cons," Proc. SPIE 4987, 69-82 (2003).
[CrossRef]

Hayakawa, H.

H. Yanagawa, T. Ochiai, H. Hayakawa, and H. Miyazawa, "Filter-embedded design and its applications to passive components," J. Lightwave Technol. 7, 1646-1653 (1989).
[CrossRef]

Lazaridès, B.

Lequime, M.

Lindgren, G.

Lowe, J. D.

A. W. Snyder and J. D. Lowe, Optical Waveguide Theory (Chapman & Hall, 1983).

Lumeau, J.

MacLeod, H. A.

H. A. MacLeod, Thin Film Optical Filters (Institute of Physics Publishing, 2001).
[CrossRef]

Miyazawa, H.

H. Yanagawa, T. Ochiai, H. Hayakawa, and H. Miyazawa, "Filter-embedded design and its applications to passive components," J. Lightwave Technol. 7, 1646-1653 (1989).
[CrossRef]

Nolting, H.

G. Sztefka and H. Nolting, "Bidirectional eigenmode propagation for large refractive index steps," IEEE Photon. Technol. Lett. 5, 554-557 (1993).
[CrossRef]

Ochiai, T.

H. Yanagawa, T. Ochiai, H. Hayakawa, and H. Miyazawa, "Filter-embedded design and its applications to passive components," J. Lightwave Technol. 7, 1646-1653 (1989).
[CrossRef]

Oliner, A.

S. Peng and A. Oliner, "Guidance and leakage properties of open dielectric waveguides. I. Mathematical formulations," IEEE Trans. Microwave Theory Tech. 29, 843-855 (1981).
[CrossRef]

Peng, S.

S. Peng and A. Oliner, "Guidance and leakage properties of open dielectric waveguides. I. Mathematical formulations," IEEE Trans. Microwave Theory Tech. 29, 843-855 (1981).
[CrossRef]

Richier, R.

M. B. Alsous, J. Bittebierre, R. Richier, and H. Ahmad, "Construction of all-fiber Nd3+ fibre lasers with multidielectric mirrors," Pure Appl. Opt. 5, 777-790 (1996).
[CrossRef]

Snyder, A. W.

A. W. Snyder and J. D. Lowe, Optical Waveguide Theory (Chapman & Hall, 1983).

Sudbo, A.

A. Sudbo, "Numerically stable formulation of the transverse resonance method for vector mode-field calculations in dielectric waveguides," IEEE Photon. Technol. Lett. 5, 342-344 (1993).
[CrossRef]

Sztefka, G.

G. Sztefka and H. Nolting, "Bidirectional eigenmode propagation for large refractive index steps," IEEE Photon. Technol. Lett. 5, 554-557 (1993).
[CrossRef]

Yanagawa, H.

H. Yanagawa, T. Ochiai, H. Hayakawa, and H. Miyazawa, "Filter-embedded design and its applications to passive components," J. Lightwave Technol. 7, 1646-1653 (1989).
[CrossRef]

Yeh, C.

Appl. Opt. (3)

IEEE Photon. Technol. Lett. (2)

A. Sudbo, "Numerically stable formulation of the transverse resonance method for vector mode-field calculations in dielectric waveguides," IEEE Photon. Technol. Lett. 5, 342-344 (1993).
[CrossRef]

G. Sztefka and H. Nolting, "Bidirectional eigenmode propagation for large refractive index steps," IEEE Photon. Technol. Lett. 5, 554-557 (1993).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

S. Peng and A. Oliner, "Guidance and leakage properties of open dielectric waveguides. I. Mathematical formulations," IEEE Trans. Microwave Theory Tech. 29, 843-855 (1981).
[CrossRef]

J. Lightwave Technol. (1)

H. Yanagawa, T. Ochiai, H. Hayakawa, and H. Miyazawa, "Filter-embedded design and its applications to passive components," J. Lightwave Technol. 7, 1646-1653 (1989).
[CrossRef]

Proc. SPIE (1)

D. F. G. Gallagher and T. P. Felici, "Eigenmode expansion methods for simulation of optical propagation in photonics: pros and cons," Proc. SPIE 4987, 69-82 (2003).
[CrossRef]

Pure Appl. Opt. (1)

M. B. Alsous, J. Bittebierre, R. Richier, and H. Ahmad, "Construction of all-fiber Nd3+ fibre lasers with multidielectric mirrors," Pure Appl. Opt. 5, 777-790 (1996).
[CrossRef]

Other (3)

H. A. MacLeod, Thin Film Optical Filters (Institute of Physics Publishing, 2001).
[CrossRef]

A. W. Snyder and J. D. Lowe, Optical Waveguide Theory (Chapman & Hall, 1983).

P. Biensman, "Rigourous and efficient modelling of wavelength scale photonic components," Ph.D. thesis (University of Gent, Belgium, 2001).

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Figures (9)

Fig. 1
Fig. 1

Multilayer interferential filter inserted between an input and an output fiber. The structure of the filter is composed of H high-refractive-index layers and L low-refractive-index layers (the represented structure is simulated in Subsections 5A and 5B1). (a) Real filter: the filter is deposited directly on the cleaved input fiber end. The output fiber is glued at the other side of the filter. (b) Simulated filter: the lengths represented in this diagram and in the last one are not to scale. The diameters of the input and output fibers, of the fictitious fibers, and of the embedding air layer are reduced to accelerate the simulation. (c) Simulated filter with light: the illumination intensity is simulated at 1494.4   nm for the parameters corresponding to Subsection 5.B.1. The horizontal scale is illustrated by the 1 μ m long input and output fiber represented. The vertical scale is illustrated by the 40 μ m diameter of the input, output, and fictitious fibers.

Fig. 2
Fig. 2

T 1 , 1 , R 1 , 1 , T 1 , and R 1 for the filter ( H L ) 5 ( L H ) 5 at λ = 1495   nm , for two fiber diameters ( D F = 40 μ m and D F = 80 μ m ), and for layers deposited on the fibers with various diameters D H = D L . F40∕HL12 means D H = D L = 12 μ m diameter layers centered on a D F = 40 μ m diameter fiber. F80∕H80∕L76 means D H = 80 μ m diameter H layers and D L = 76 μ m diameter L layers centered on a D F = 80 μ m diameter fiber. The index n corresponds to calculations with power normalization at the joints.

Fig. 3
Fig. 3

Spectra of the filter F IN ( H L ) 5 ( L H ) 5 F OUT . The vertex of T 1 , 1 coincides exactly with a step of the vertical grid of 0.4   nm . The points for λ = 1495   nm are the same as in Figs. 3 and 2 for F 40 / H L 40 .

Fig. 4
Fig. 4

Power in the filter F IN ( H L ) 5 ( L H ) 5 F OUT at 1494.4   nm as a function of z. The forward power is the solid line, and the backward power is the dashed line. They are sampled only one time in each fiber (input, output, and fictitious fiber). Thus, the power in the H and L layers can be easily distinguished in the figure.

Fig. 5
Fig. 5

Spectra of the filter F IN ( H L ) 2 ( L H ) 2 F OUT .

Fig. 6
Fig. 6

Spectra of the filter F IN L [ ( H L ) 2 ( L H ) 2 L ] 3 F OUT .

Fig. 7
Fig. 7

Spectra of the filter F IN ( H L ) 5 ( L H ) 5 F OUT with absorption losses in the layers.

Fig. 8
Fig. 8

Response of the filters F IN ( H B ) n ( B H ) n F OUT for n = 2 , 3, 4, 5 at λ = 1494.4   nm and with absorption losses. The simulated points have finesses increasing with n.

Fig. 9
Fig. 9

Response of the filters F IN [ ( H B ) n ( B H ) n ] 3 L F OUT for n = 2 , 3 , 4 at λ = 1494.4   nm and with absorption losses.

Tables (2)

Tables Icon

Table 1 Number of Guided Modes ( Ng ) in the Basis for the L Layers of Weak Diameters

Tables Icon

Table 2 Response of the Filter F IN( HL )5( LH )5 F OUT at λ = 1494.4 nm with Absorption Losses as a Function of the Glue Optical Thickness

Equations (8)

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

w ( z ) = w 0 1 + z 2 / z R 2 , z R = π w 0 2 λ ,
s I ( x , y ) r s I ( x , y ) | System | s O ( x , y ) r s O ( x , y ) .
s I = ( s 1 I s 2 I s N I I ) , s O = ( s 1 O s 2 O s N O O ) , r s I = ( r s 1 I r s 2 I r s N I I ) , r s O = ( r s 1 O r s 2 O r s N O O ) .
( r s I r s O ) = ( R I T I | T O R O ) ( s I s O ) .
mode   m | mode   n = S E m , t × H n , t d S ,
s I = ( 1 0 0 0 ) , s O = 0 ( 1 0 0 )   ( r 1 , 1 r N I , 1 ) | System | ( 0 0 0 )   ( t 1 , 1 t N I , 1 ) .
1 R 1 = i = 1 N I R i , 1 | System | 0 T 1 = i = 1 N I T i , 1 ,
R ( z ) = z ( 1 + z R 2 z 2 ) .

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