Abstract

The six-fold rotational symmetry of photonic crystal fibers has important manifestations in the radiated fields in terms of i) a focusing phenomena at a finite distance from the end-facet and ii) the formation of low-intensity satellite peaks in the asymptotic far field. For our study, we employ a surface equivalence principle which allows us to rigorously calculate radiated fields starting from fully-vectorial simulations of the near field. Our simulations show that the focusing is maximal at a characteristic distance from the end-facet. For large-mode area fibers the typical distance is of the order 10×Λ with Λ being the pitch of the triangular air-hole lattice of the photonic crystal fiber.

© 2005 Optical Society of America

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References

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IEEE J. Quantum Electron. (1)

J. Vu�?kovi�?, M. Lon�?ar, H. Mabuchi, and A. Scherer, �??Optimization of the Q Factor in photonic crystal microcavities,�?? IEEE J. Quantum Electron. 38, 850 (2002).
[CrossRef]

J. Opt. Soc. Am. A (2)

Laser Focus World (1)

T. F. Johnston, �??M2 concept characterizes beam quality,�?? Laser Focus World 26, 173 (1990).

Opt. Express (4)

Opt. Lett. (2)

Other (2)

C. A. Balanis, Advanced Engineering Electromagnetic (John Wiley & Sons, New York, 1989).

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

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Figures (4)

Fig. 1.
Fig. 1.

Schematics of the fiber geometry and the imaginary closed surface S=S 1+S 2, where S 1 is a circle parallel and close to the end-facet and S 2 is a semi-sphere concentric with S 1.

Fig. 2.
Fig. 2.

Variation of the electric field intensity with distance from the end-facet (z=0) at the center of fiber.

Fig. 3.
Fig. 3.

Electric field amplitudes at distances z from the end-facet varying from z=0 (Panel A) to z=12×Λ (Panel H).

Fig. 4.
Fig. 4.

Asymptotic far-field intensities calculated by the rigorous approach as well as the approximate expression, Eq. (6), along the high-symmetry directions ϕ=0 and ϕ=π/2. The inset shows the corresponding full contour plot obtained with the rigorous approach.

Equations (8)

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π e ( r ) = j 4 π ω ε 0 S J ( r ) e j k r r r r d S ,
π m ( r ) = j 4 π ω μ 0 S M ( r ) e j k r r r r d S .
E ( r ) = ( k 2 + 2 ) π e ( r ) j ω μ 0 × π m ( r ) ,
H ( r ) = j ω ε 0 × π e ( r ) + ( k 2 + 2 ) π m ( r ) ,
π m ( r ) j 4 π ω μ 0 exp ( jkr ) r N ( θ , ϕ ) ,
N ( θ , ϕ ) = S M ( r ) e jk r ̂ · r dS ,
E ( r ) 1 4 π e jkr r ( jk ) ( r ̂ × N ) .
E ( r ) jk 2 π e jkr r S E r ( x , y , z = 0 ) × exp [ jk ( x sin θ cos ϕ + y sin θ sin ϕ ) ] dS ,

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