Abstract

We investigate numerically the optical field in the region immediately behind the input facets of dielectric step-index single-mode slab and fiber waveguides. Visualization of the intensity distributions gives insight into the formation of the fundamental mode and of radiation modes. For a more quantitative characterization we determine the amount of optical power and mode purity of the field in core vicinity as a function of propagation distance. The investigation assists in designing and optimizing waveguides being employed as modal filters, e.g. for astronomical interferometers.

© 2007 Optical Society of America

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References

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  1. B. Mennesson, M. Ollivier, and C. Ruilier, "Use of single-mode waveguides to correct the optical defects of a nulling interferometer," J. Opt. Soc. Am. A 19, 596-602 (2002).
    [CrossRef]
  2. O. Wallner, W. R. Leeb, and R. Flatscher, "Design of spatial and modal filters for nulling interferometry," in Interferometry for Optical Astronomy II, Wesley A. Traub, ed., Proc. SPIE 4838, 668-679 (2003).
    [CrossRef]
  3. J. C. Flanagan, D. J. Richardson, M. J. Foster, and I. Bakalski, "Microstructured fibers for broadband wavefront filtering in the mid-IR," Opt. Express 14, 11773-11786 (2006).
    [CrossRef] [PubMed]
  4. C. V. M. Fridlund, "DARWIN - The Infrared Space Interferometry Mission," ESA Bulletin 103, 20-25 (2000).
  5. O. Wallner,  et al., "Minimum length of a single-mode fiber spatial filter," J. Opt. Soc. Am. A 19, 2445-2448 (2002).
    [CrossRef]
  6. P. Cheben, D. -X. Xu, S. Janz, and A. Densmore, "Subwavelength waveguide grating for mode conversion and light coupling in integrated optics," Opt. Express 14, 4695-4702 (2006).
    [CrossRef] [PubMed]
  7. L , Li, "New formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. Soc. Am. A 14, 2758-2767 (1997).
    [CrossRef]
  8. A. W. Snyder and J. D. Love, Optical waveguide theory (Chapman & Hall, 1983), p. 259 ff.
  9. G. Grau and W. Freude, Optische Nachrichtentechnik (Springer, 1991), pp. 51.
  10. E. Silberstein et al., "Use of grating theories in integrated optics," J. Opt. Soc. Am. A 18, 2865-2875 (2001).
    [CrossRef]
  11. C. Zhou and L. Li, "Formulation of the Fourier modal method for symmetric crossed gratings in symmetric mountings," J. Opt. A: Pure Appl. Opt. 6, 43-50 (2004).
    [CrossRef]

2006 (2)

J. C. Flanagan, D. J. Richardson, M. J. Foster, and I. Bakalski, "Microstructured fibers for broadband wavefront filtering in the mid-IR," Opt. Express 14, 11773-11786 (2006).
[CrossRef] [PubMed]

P. Cheben, D. -X. Xu, S. Janz, and A. Densmore, "Subwavelength waveguide grating for mode conversion and light coupling in integrated optics," Opt. Express 14, 4695-4702 (2006).
[CrossRef] [PubMed]

2004 (1)

C. Zhou and L. Li, "Formulation of the Fourier modal method for symmetric crossed gratings in symmetric mountings," J. Opt. A: Pure Appl. Opt. 6, 43-50 (2004).
[CrossRef]

2002 (2)

2001 (1)

2000 (1)

C. V. M. Fridlund, "DARWIN - The Infrared Space Interferometry Mission," ESA Bulletin 103, 20-25 (2000).

1997 (1)

ESA Bulletin (1)

C. V. M. Fridlund, "DARWIN - The Infrared Space Interferometry Mission," ESA Bulletin 103, 20-25 (2000).

J. Opt. A: Pure Appl. Opt. (1)

C. Zhou and L. Li, "Formulation of the Fourier modal method for symmetric crossed gratings in symmetric mountings," J. Opt. A: Pure Appl. Opt. 6, 43-50 (2004).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Express (2)

P. Cheben, D. -X. Xu, S. Janz, and A. Densmore, "Subwavelength waveguide grating for mode conversion and light coupling in integrated optics," Opt. Express 14, 4695-4702 (2006).
[CrossRef] [PubMed]

J. C. Flanagan, D. J. Richardson, M. J. Foster, and I. Bakalski, "Microstructured fibers for broadband wavefront filtering in the mid-IR," Opt. Express 14, 11773-11786 (2006).
[CrossRef] [PubMed]

Other (3)

O. Wallner, W. R. Leeb, and R. Flatscher, "Design of spatial and modal filters for nulling interferometry," in Interferometry for Optical Astronomy II, Wesley A. Traub, ed., Proc. SPIE 4838, 668-679 (2003).
[CrossRef]

A. W. Snyder and J. D. Love, Optical waveguide theory (Chapman & Hall, 1983), p. 259 ff.

G. Grau and W. Freude, Optische Nachrichtentechnik (Springer, 1991), pp. 51.

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Figures (16)

Fig. 1.
Fig. 1.

System model. Pin .. input power, 2w .. beam width (defined by first zeros in the focal plane), 2a .. core width, DA .. input pupil width, DB .. output pupil width, DC .. near-core power width, p(z) .. near-core power, EW .. waveguide field just before output facet, Ef .. field of fundamental mode. The constructive and destructive superposition of field Ew with the fundamental mode Ef leading to output power P+ and P- serves only to illustrate the definition of mode purity MP. The right part shows the cross section in case of a fiber waveguide.

Fig. 2.
Fig. 2.

Slab waveguide operated at λ=λ0: Intensity distributions without input pupil for a beam width of 2w=11 λ0 and perfect alignment (Δz=0, ε=0, Δx=0). (a) extension in x-direction is 500 λ0, (b) extension in x-direction is 100 λ0.

Fig. 3.
Fig. 3.

Slab waveguide operated at λ=λ0: Intensity distributions with input pupil (DA=11 λ0) for a beam width of 2w=11 λ0. (a) perfect alignment (Δz=0, ε=0, Δx=0) (b) defocus Δz=50 λ0 (c) angular misalignment ε=1° (d) lateral misalignment Δx=1 λ0.

Fig. 4.
Fig. 4.

Slab waveguide: Near-core power p(z) as a function of distance z from the input facet for the cases of Fig. 2 and Figs. 3(a) to 3(d). The numbers at the right give the coupling efficiencies η.

Fig. 5.
Fig. 5.

Slab waveguide operated at λ=λ0: Mode purity MP as a function of waveguide length L for a beam width of 2w=11 λ0 and for input pupil diameters of DA=11 λ0 and DA=100 λ0. The latter case is practically identical to the case without input pupil (DA=∞).

Fig. 6.
Fig. 6.

Slab waveguide: Mode purity MP as a function of waveguide length L for the four cases of alignment specified in Fig. 3. (a) for operation at λ=λ0, (b) for operation at λ=1.5 λ0.

Fig. 7.
Fig. 7.

Schematic of inhomogeneous waveguide.

Fig. 8.
Fig. 8.

Slab waveguide operated at λ=λ0: Mode purity as a function of L for a waveguide with (blue) and without (red) an inhomogeneous core with a length of about 2000 λ0, starting at z=2000 λ0. The input beam parameters are 2w=11 λ0 and perfect alignment.

Fig. 9.
Fig. 9.

Slab waveguide operated at λ=λ0: Near-core power p(z) as a function of distance z for a waveguide with (blue) and without (red) inhomogeneities. The input beam parameters are 2w=11 λ0 and perfect alignment. The values at the right of the diagram represent the coupling efficiencies η to the fundamental mode in the third section.

Fig. 10.
Fig. 10.

Fiber operated at λ=λ0: Intensity distributions without input pupil, input beam diameter 2w=8.8 λ0. (a) perfect alignment (b) defocus Δz=50 λ0,.

Fig. 11.
Fig. 11.

Fiber operated at λ=λ0: Intensity distributions with input pupil, input beam diameter 2w=8.8 λ0. (a) perfect alignment (b) defocus Δz=50 λ0,.

Fig. 12.
Fig. 12.

Fiber operated at λ=λ0: Near-core power p(z) as a function of distance z for 2w=8.8 λ0. The numbers at the right give the coupling efficiencies η. Dotted red line: perfect alignment without input pupil (DA=∞) Solid red line: perfect alignment with input pupil (DA=8.8 λ0) Blue line:defocus Δz=50 λ0,, with input pupil (DA=8.8 λ0).

Fig. 13.
Fig. 13.

Fiber operated at λ=λ0 with input beam diameter 2w=8.8 λ0: Mode purity MP as a function of waveguide length L for input pupil diameters of DA=8.8 λ0 and DA=100 λ0. The latter case is practically identical to the case without input pupil (DA=∞).

Fig. 14.
Fig. 14.

Fiber: Mode purity MP as a function of waveguide length L for perfect alignment (red line) and for a defocus of Δz=50 λ0, (blue line) when operated at λ=λ0. The dotted lines give MP for operation at λ=1.5 λ0 (green line: perfect alignment, magenta line: defocus of Δz=50 λ0,).

Fig. 15.
Fig. 15.

Comparison of near-core power p(z) for slab (dotted lines) and fiber (solid lines) with input pupils, operated at λ=λ0. The red lines are for perfect alignment, the blue lines for a defocus of Δz=50 λ0.

Fig. 16.
Fig. 16.

Comparison of mode purity MP for slab (dotted lines) and fiber (solid lines) with input pupils, operated at λ=λ0. The red lines are for perfect alignment, the blue lines for a defocus of Δz=50 λ0.

Tables (2)

Tables Icon

Table 1. Waveguide parameters used for the numerical calculations (Δ .. relative index difference, V .. normalized frequency, VC .. cut-off frequency). Also given is the input pupil width DA (if not explicitly set to infinity) and the output pupil width DB.

Tables Icon

Table 2: Comparison of results (perfect alignment).

Equations (9)

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f ( x ' ) = sin ( π x ' w ) x '
f ( r ) = J 1 ( 1.22 π r w ) r with r = x 2 + y 2 ,
p ( z ) = P DC ( z ) P in
η = P DC , z P in
P slab ± D B 2 D B 2 I slab ± ( x ) dx and P fiber ± 0 2 π 0 D B 2 I fiber ± ( r , φ ) rdrd φ
I slab ± A w ( x , z = L ) exp [ j φ w ( x , z = L ) ] ± A f ( x ) A w ( x = 0 ) A f ( x = 0 ) exp [ j φ w ( x = 0 , z = L ) ] 2 ,
MP = P + P
E ( x , y , z ) = S m ( x , y ) e j b m z
E 1 ( x , z ) = m = M M [ ( A m 1 exp ( j b m 1 z ) + B m 1 exp ( j b m 1 z ) ) p = P P q = Q Q S pqm 1 exp ( jKpx ) exp ( jKqy ) ]

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