Abstract

The transition from the near to the far field of the fundamental mode radiating out of a photonic crystal fiber is investigated experimentally and theoretically. It is observed that the hexagonal shape of the near field rotates two times by π/6 when moving into the far field, and eventually six satellites form around a nearly gaussian far-field pattern. A semi-empirical model is proposed, based on describing the near field as a sum of seven gaussian distributions, which qualitatively explains all the observed phenomena and quantitatively predicts the relative intensity of the six satellites in the far field.

© 2002 Optical Society of America

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References

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    [CrossRef]

J. Opt. A: Pure Appl. Opt.

"Special Issue on Photonic Bandgaps,�?? J. Opt. A: Pure Appl. Opt. 3, S103�??S207 (2001).

Opt. Express

Opt. Fiber Technol.

J. Broeng, D. Mogilevstev, S. E. Barkou, and A. Bjarklev, �??Photonic crystal fibers: A new class of optical waveguides,�?? Opt. Fiber Technol. 5, 305�??330 (1999).
[CrossRef]

Opt. Lett.

Science

J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, �??Photonic band gap guidance in optical fibers,�?? Science 282, 1476�??1478 (1998).
[CrossRef] [PubMed]

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. S. J. Russell, P. J. Roberts, and D. C. Allan, �??Single-mode photonic band gap guidance of light in air,�?? Science 285, 1537�??1539 (1999).
[CrossRef] [PubMed]

J. C. Knight and P. S. J. Russell, �??Applied optics: New ways to guide light,�?? Science 296, 276�??277 (2002).
[CrossRef] [PubMed]

Other

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

A. K. Ghatak and K. Thyagarajan, Optical Electronics (Cambridge, Cambridge University Press, 1989).
[CrossRef]

Supplementary Material (1)

» Media 1: GIF (3032 KB)     

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Figures (6)

Fig. 1.
Fig. 1.

Schematic of a single-mode PCF (z < 0) with an end-facet from where light is radiated into free space (z > 0).

Fig. 2.
Fig. 2.

Experimentally observed near-field intensity distributions for a PCF with ʌ ≃ 3.5 μm and d/ʌ ≃ 0.5 (micro-graph in panel a) at a free-space wavelength λ = 635 nm. The distance from the end-facet varies from z = 0 to z ~ 10 μm (panels b to f). At a further distance the six low-intensity satellite spots develop (panels g and h, logarithmic scale).

Fig. 3.
Fig. 3.

Experimentally observed far-field intensity distribution showing an overall gaussian profile with six additional low-intensity satellite spots along one of the two principal directions (line 2). Angles are given in radians.

Fig. 4.
Fig. 4.

Panel a shows the experimentally observed near-field intensity along the two principal directions 1 and 2 (see insert of panel b). Panel b shows the numerically calculated intensity distribution in a corresponding ideal PCF with the solid lines showing the intensity along the principal directions and the difference. The blue and red dashed lines show gaussian fits to I 2 and I 2 - I 1 and the dashed green line shows their difference.

Fig. 5.
Fig. 5.

Near-field intensity distribution calculated from Eq. (5) with values of wh , wc , and γ determined from the intensity in the PCF obtained by a fully-vectorial calculation, see Fig. 4. The distance varies from z = 0 to z = 8ʌ (panels a to i) in steps of Δz = ʌ (see also animation with Δz = ʌ/4, 3 Mbyte). [Media 1]

Fig. 6.
Fig. 6.

Far-field intensity distribution (z = 1000ʌ ⊫ λ) corresponding to the near field in Fig. 5. The intensity distribution has an overall gaussian profile with six additional low-intensity satellite spots along one of the two principal directions (line 2).

Equations (5)

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H ( x , y , z ) = h ( x , y ) e ± ( ω ) z ,
I ( s ) = h ( s ) 2 = j A j u ( s s j , w j ) 2 , u ( s , w ) = exp ( s 2 w 2 ) .
u ( s , w ) u ( s , z , w ) = ( 1 i 2 z k w 2 ) 1 exp [ ik ( z + s 2 2 R ( z ) ) s 2 w 2 ( z ) ] ,
I ( s ) = A 2 u ( s , w c ) γ j = 1 6 u ( s s j , w h ) 2 ,
I ( s ) I ( s , z ) = A 2 u ( s , z , w c ) γ j = 1 6 u ( s s j , z , w h ) 2 .

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