Abstract

We investigate the influence of reflection symmetry on the properties of the modes of microstructured optical fibers. It is found that structures with reflection symmetry tend to support non-degenerate modes which are closer in nature to the analogous TE and TM modes of circular step-index fibers, as compared with fibers with only rotational symmetry. Reflection symmetry induces modes to exhibit smaller longitudinal components and transverse fields which are more strongly reminiscent of the radial and azimuthal modes of circular fibers. The tendency towards “transversality” can be viewed as a result of the interaction of group theoretical restrictions on the mode profiles and minimization of the Maxwell Hamiltonian.

© 2004 Optical Society of America

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References

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IEEE J. Quantum Electron.

S.-H. Kim and Y.-H. Lee, �??Symmetry relations of two-dimensional photonic crystal cavity modes,�?? IEEE J. Quantum Electron. 39, 1081�??1085 (2003).
[CrossRef]

IEEE Transactions on MTT

P. R. McIsaac, �??Symmetry-induced modal characteristics of uniform waveguides,�?? IEEE Transactions on Microwave Theory and Techniques MTT-23(5), 421�??433 (1975).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. B

Opt. Express

Opt. Lett.

Other

RSoft Design Group, Inc. <a href="http://www.rsoftdesign.com">http://www.rsoftdesign.com</a>.

C. Vassallo, Optical Waveguide Concepts (Elsevier, Amsterdam, 1991).

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Figures (10)

Fig. 1.
Fig. 1.

Schematic of three “satellite” fibers with symmetry groups C 6v [(a) and (b)] and C 6 [(c)]. All three fibers support degenerate modes, but only (a) and (b) possess reflection symmetry. The fibers are referred to in the text as A6v , B6v and C 6.

Fig. 2.
Fig. 2.

(a) hz , (b) h t and (c) e t for the TE0 mode of a circular fiber. The fields of the TM0 mode are qualitatively similar with the roles of E and H reversed.

Fig. 3.
Fig. 3.

Mode spectra for (a) a glass-air circular step-index fiber (core diameter 5µm, nco =1.45), and (b) fiber A6v at l=1.55µm.

Fig. 4.
Fig. 4.

Mode profiles for the lowest p=1 mode of the fiber in Fig. 1b. (a) hz , (b) ez , (c) h t and (d) e t .

Fig. 5.
Fig. 5.

Boundary effects for the p=1 mode in fiber B6v: transverse curl of h t . For the p=2 mode, the profile looks similar with the roles of the fields swapped.

Fig. 6.
Fig. 6.

Mode profiles for the second lowest p=1 mode of the fiber in Fig. 1c. (a) hz , (b) ez , (c) h t and (d) e t .

Fig. 7.
Fig. 7.

Transversality of first 6 modes for the three fibers in Fig. 1. The two fibers with reflection symmetry exhibit larger contrast between the major and minor longitudinal components.

Fig. 8.
Fig. 8.

Examples of systematic fibers with C 3v and C 3 symmetry. (a) and (b) hz and ez for C 3v . (c) and (d) hz and ez for C 3.

Fig. 9.
Fig. 9.

Degree of transversality for TE and TM-like modes in matched C 4 and C 4v fibers. Crosses denote the C 4 results.

Fig. 10.
Fig. 10.

(a) Transversality µ TE=fz [E]/fz [H] and µ TM=fz [H]/fz [E] as a function of rotational order for single ring fibers. Small µ indicates strong transversality. Also profiles of H t for (b) n=3 and (c) n=9.

Tables (1)

Tables Icon

Table 1. First two symmetry classes (all containing non-degenerate modes) for several symmetry groups.

Equations (16)

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E ( r , t ) = [ e r , e ϕ , e z ] exp ( i [ β z ω t ] ) = [ e t ( r t ) , e z ( r t ) ] exp ( i [ β z ω t ] ) ,
H ( r , t ) = [ h r , h ϕ , h z ] exp ( i [ β z ω t ] ) = [ h t ( r t ) , h z ( r t ) ] exp ( i [ β z ω t ] ) ,
e t = i k 0 2 n ( r t ) 2 β 2 [ β t e z ( k 0 Z 0 ) z ̂ × t h z ] ,
h t = i k 0 2 n ( r t ) 2 β 2 [ β t h z ( k 0 Z 0 ) z ̂ × t e z ] ,
t · e t = i β k 0 2 n ( r t ) 2 β 2 t 2 e z ,
t × e t = ik 0 Z 0 k 0 2 n ( r t ) 2 β 2 t 2 h z ,
t · h t = i β k 0 2 n ( r t ) 2 β 2 t 2 h z ,
t × h t = ik 0 Z 0 k 0 2 n ( r t ) 2 β 2 t 2 e z .
f z [ E ] = e z 2 E 2
f z [ H ] = h z 2 H 2
g ( r t ) = d 2 r t g ( r t ) .
f = t f + f z ,
· F = t · F t + F z z ,
× F = t × F t z ̂ × ( t F z F t z ) .
t × ( t f ) = 0 ,
t · z ̂ × ( t f ) = 0 ,

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