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Near-field to far-field transition of photonic crystal fibers: symmetries and interference phenomena

Niels Asger Mortensen and Jacob Riis Folkenberg

Open Access Open Access

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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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).

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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).

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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.

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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.

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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]

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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).

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Equations (5)

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H ( x , y , z ) = h ( x , y ) e ± iβ ( 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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