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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 22 — Nov. 3, 2003
  • pp: 2838–2847
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Quantitative characterization of higher-order mode converters in weakly multimoded fibers

M. Skorobogatiy, Charalambos Anastassiou, Steven G. Johnson, O. Weisberg, Torkel D. Engeness, Steven A. Jacobs, Rokan U. Ahmad, and Yoel Fink  »View Author Affiliations


Optics Express, Vol. 11, Issue 22, pp. 2838-2847 (2003)
http://dx.doi.org/10.1364/OE.11.002838


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Abstract

We present a rigorous analysis methodology of fundamental to higher order mode converters in step index few mode optical fibers. We demonstrate experimental conversion from a fundamental LP01 mode to the higher order LP11 mode utilizing a multiple mechanical bend mode converter.We perform a quantitative analysis of the measured light intensity, and demonstrate a modal decomposition algorithm to characterize the modal content excited in the fiber. Theoretical modelling of the current mode converter is then performed and compared with experimental findings.

© 2003 Optical Society of America

1. Introduction

2. Serpentine mode converter, design and experimental setup

Fig. 1. Serpentine mode converter geometry. Optical fiber is in-between the two sets of tightly wound wires of diameter Dw , chosen to satisfy phase matching condition between the converted modes. Fiber diameter Df is generally comparable to the Dw .
Fig. 2. Schematic of an experimental setup. Light from He:Ne laser, is coupled into the fiber through the coupling lens. A mode stripper is then applied consisting of several loops of tightly wound fiber. The wire wrapped mandrel on xyz stage was used as a mode converter with fiber squeezed between two wire sets. The far field images were captured with a CCD camera placed 13mm from the fiber end.

A schematic of the experimental setup is shown in Fig. 2. Light with λ=632.8 nm from a He:Ne laser is coupled into the fiber, exciting mostly the LP 01 mode of the fiber. This is achieved by first collimating the laser beam to 1 cm and then focusing it into the fiber with a lens of focal length 8.3 cm and 3.8 cm for the SMF-28 and the SM-750 respectively. A mode stripper is then applied to ensure that any higher order modes excited are stripped and only the LP 01 mode is sent to the mode converter. The mode stripper consisted of three loops of 6 mm diameter for the SMF-28 and only one loop of 12.7 mm diameter for the SM-750. The LP 01 to LP 11 mode converter, shown in Fig. 2, was built by wrapping two cylinders (12.7 mm diameter) with a copper wire and squeezing the fiber in between them. The copper wire diameter was chosen to be as close to the beat length of the two modes as possible.We used 0.512 mm and 0.254 mm wire diameters for the SMF-28 and the SM-750, respectively. The wire-wrapped mandrels were put on the xyz stages to adjust the amount of separation between the mandrels and their relative alignment and pressure applied to the fiber in order to achieve maximal conversion. Care was also taken to ensure there were no additional sharp bends after the mode converter, this was particularly critical for the SM-750 due to the very high bending losses of its LP 11 mode.

Fig. 3. Far field images taken 13 mm from exit of SMF-28 (a) and (b) and SM-750 (c) and (d) being excited with He:Ne. (a) No mode converter applied for SMF-28 showing the LP 01. (b) Mode converter applied with N=35 turns of 0.512 mm copper wires showing 65% conversion to LP 11 (established by modal characterization algorithm). (c) No mode converter applied for SM-750 showing the LP 01 mode. (d) Mode converter applied with N=14 turns of 0.254 mm copper wires showing 55% conversion to LP 11 (established by modal characterization algorithm).

Table 1. Experimental parameters of fibers and wires including core diameter, core and cladding refractive indexes, wire diameter and LP 01 to LP 11 conversion efficiency

table-icon
View This Table

3. Modal characterization algorithm

We tested several function minimization algorithms to reduce the value of objective function. As we mentioned earlier, fitting phases is generally more problematic than fitting weights as phase information is lost in the intensity plots. However, we found that standard multidimensional conjugate gradient (CG) minimization algorithm where weights and phases are treated on equal footing works very well in this problem. Another very efficient minimization method that worked somewhat faster than CG was multidimensional Newton method. Weights and phases were treated on equal footing, however, not all dimensions were used to find a new searched direction. We used SVD decomposition of the Hessian of an objective function to reduce degenerate subspaces.

To ensure consistency in the calculations, we always normalize both the observed and calculated modes so that their integrated intensity on the screen is unity. In order to compare the calculated modes to the real modes of the fiber, it is also important that the calculated modes and the real modes be centered at the same location, and we therefore need an accurate measure of the center of the fiber. We locate the center experimentally by first sending a white light signal through the fiber and determining the “center of mass” of this image. This “center of mass” gives a correct center determination, because white source is incoherent, and the image that the white light creates is cylindrically symmetrical without any finer structure to it.

We now describe in more detail the computation of the objective function, the sum of the squares of the difference in predicted and measured intensity, a calculation that we simplify somewhat by the use of Fourier series. The predicted intensity is the z component of the real part of the Poynting vector of the summed modes. We define (Eρmj (ρ),Eθmj (ρ))exp(imθ) and (Hρmj (ρ),Hθmj (ρ))exp(imθ) to be the transverse components of the electromagnetic fields corresponding to the propagated eigenmode j of a fiber characterized by an angular index m and propagation constant in the fiber βj . The total field is then

Fρ,θ=jFρ,θmj(ρ)eimθ·wj·eiϕj
(1)

I(ρ,θ)=14(H*×E+H×E*)z=
Δm(i,j,mimj=Δmfij(ρ)·ωi·ωj·ei(ϕiϕl))·eiΔmθΔmI˜(Δm,p)·eiΔmθ,
(2)

where we introduced a cross-flux as

fij(ρ)=14(Eρmi(ρ)·Hθ*mj(ρ)Eθmi(ρ)·Hρ*mj(ρ))z.
(3)

We now introduce the Fourier transform of the measured intensity field Imes (ρ,θ) as

I˜mes(Δm,ρ)=12π02πImes(ρ,θ)·eiΔmθdθ.
(4)

Objective function is defined as an integral of the square of the difference between the experimental intensity and fitted intensity, and can be further expressed as

O=ρdρdθ(I(ρ,θ)Imes(ρ,θ))2
=ρdρΔm(I˜(Δm,ρ)I˜mes(Δm,ρ))·(I˜(Δm,ρ)I˜mes(Δm,ρ))
(5)

Substitution of the Ĩm,ρ) into the above equation gives

O=Δm(i,j,mimj=Δmi,j,mimj=Δm[ρdρfij(ρ)fij*(ρ)]ωiωjωiωjei(ϕi+ϕjϕjϕi)2·Rei,j,mimj=Δm[ρdρfij(ρ)I˜mes(Δm,ρ)]ωiωjej(ϕiϕj)+ρdρI˜mes(Δm,ρ)2).
(6)

4. Theoretical modelling

Fig. 4. (a) Dielectric profile of a cylindrically symmetric fiber. Concentric dielectric interfaces are characterized by their radii ρi . (b) Perturbations of a fiber center line in a 2D plane. Cross-section perpendicular to the fiber center line is assumed cylindrically symmetric, while the position of the fiber center is described by an analytic curve (X(s), s) confined to a plane.

Let (x,y,z) be the coordinates in a Cartesian coordinate system. In unperturbed, cylindrically symmetric, step-index fibers, the position of the dielectric interfaces can be described in cylindrical coordinates by a set of points (x=ρ cos(θ), y=ρ sin(θ), z=s), where for the i-th dielectric interface ρ=ρi ,θ∈(0,2π) (Fig. 4(a). Now, consider perturbations of a fiber center line in a 2D plane. For such a perturbation, each cross-section perpendicular to the fiber center line remains cylindrically symmetric, while the position of the fiber center is described by an analytic curve confined to a plane. If (X(s), s) is an analytic curve in a 2D plane (see Fig. 4(b), then the coordinate mapping

x=X(s)+ρcos(θ)11+(X(s)s)2
y=ρsin(θ)
z=sρcos(θ)X(s)s1+(X(s)s)2,
(7)

will describe such a fiber center line perturbation. Here, as before, ρ=ρi , θ∈(0,2π) define the points on the i-th unperturbed dielectric interface, while the corresponding (x(ρ,θ,s),y(ρ,θ,s), z(ρ,θ,s)) describe the points on the perturbed interfaces, and s is chosen along one of the axes in a Cartesian coordinate system.

We choose a fiber center curve (X(s),s) to describe the features of a mode converter shown on Fig. 1 as follows. Input and output regions of the length L are described by elastic beam equations with a solution Xi,o (s)=j=03 Aji,o ·sj , where the coefficients Ai,o are determined by zero-slope boundary conditions at the fixed points. In the converter region, the fiber center-line is described by a sinusoidal variation X(s)=Dw+Df2+δ(sin(2πss0Dw)1) , where δ characterizes the strength of the perturbation. Parameters Aji,o and s0 are chosen to produce a continuous curve with continuous 1st and 2nd derivatives. For a fixed number of wire turns N and a transition length L, the parameter δ is chosen to maximize the LP 01 to LP 11 conversion for each of the polarizations.

iBC(s)s=MC(s),
(8)

where Bβ*,β=ββδβ,β is a normalization matrix, and M is a matrix of coupling elements given by

Mβ*,m;β,m=ωcρdρdθexpi(mm)θ×
(Eρ0(ρ)Eθ0(ρ)Es0(ρ)Hρ0(ρ)Hθ0(ρ)Hs0(ρ))β*m(εdρρεdρθεdρs000εdθρεdθθεdθs000εdsρεdsθεdss000000dρρdρθdρs000dθρdθθdθs000dsρdsθdss)(Eρ0(ρ)Eθ0(ρ)ES0(ρ)Hρ0(ρ)Hθ0(ρ)HS0(ρ))βm,
(9)

where the integration is performed over the unperturbed fields of a straight fiber. To the second order in δ, the only non-zero elements of the coupling matrix are the ones corresponding to the |m′-m|=1 and |m′-m|=0 transitions. From Ref. [15

15. M. Skorobogatiy, Steven A. Jacobs, Steven G. Johnson, and Yoel Fink, “Geometric variations in high index-contrast waveguides, coupled mode theory in curvilinear coordinates,” Opt. Express 10, 1227 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-21-1227 [CrossRef] [PubMed]

] they are the following. For |m′-m|=0 non-zero coupling elements are calculated from 9 using dss=dρρ=dθθ=ωδ22(X(s)s)2 . For |m′-m|=1 non-zero coupling elements are calculated from 9 using dss=dρρ=dθθ=ωρδ22X(s)s2 . Equations (8) present a system of first order linear coupled differential equations with respect to a vector of expansion coefficients C⃗(s). The boundary condition of a single incoming HE 11 mode at the input of a converter defines a boundary value problem that can be solved numerically.

Fig. 5. Theoretical simulation of modal conversion as a function of a wire diameter in SMF-28 fiber undergoing N=35 consecutive bends. Fiber centerline is described by a sinusoid with an amplitude of δ=49 nm and a pitch equal to the wire diameter Dw . Solid and dashed lines correspond to the perpendicular and parallel to the plane of the converter polarizations of an incoming LP 01 mode. Upper two curves correspond to the modal weights of the higher order LP 11 group after conversion, while lower two curves correspond to the remaining weights of the original LP 01 mode.

5. Discussion

On the basis of theoretical simulations and experiments on mode conversion we conclude that knowing fiber and wire parameters as precisely as possible (ideally to better than a fraction of a percent accuracy) is crucial in establishing a good quantitative agreement between experimental and theoretical results for “serpentine” bend converters. Particularly, from simulations in low index-contrast systems we observe that phase matching condition should be observed with better than 0.5% accuracy to achieve a 10% uncertainty in the value of the conversion amplitudes near the resonance. This translates into a sub percent accuracy in the assumed values of the fiber parameters such as fiber core radius and core-clad index contrast, and converter parameters such as wire diameter. Because of such tight tolerances, some level of tunability has to be built into the converter design to make it practical.

Fig. 6. Theoretical simulation of modal conversion as a function of a wire diameter in SM-750 fiber undergoing N=14 consecutive bends. Fiber centerline is described by a sinusoid with an amplitude of δ=110 nm and a pitch equal to the wire diameter Dw . Solid and dashed lines correspond to the perpendicular and parallel to the plane of the converter polarizations of an incoming LP 01 mode. Upper two curves correspond to the modal weights of the higher order LP 11 group after conversion, while lower two curves correspond to the remaining weights of the original LP 01 mode.

References and links

1.

Y. Danziger and D. Askegard, “Full Fiber Capacity Realized with High Order Mode Technology,” in IEC Annual Review, (2000).

2.

Kerbage C, Windeler RS, Eggleton BJ, Mach P, Dolinski M, and Rogers JA, “Tunable devices based on dynamic positioning of micro-fluids in micro-structured optical fiber,” Opt. Commun. 204, 179 (2002). [CrossRef]

3.

Steven G. Johnson, Mihai Ibanescu, M. Skorobogatiy, Ori Weisberg, Torkel D. Engeness, Marin Soljačić, Steven A. Jacobs, J. D. Joannopoulos, and Yoel Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9, 748 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748 [CrossRef] [PubMed]

4.

R.C. Youngquist, J.L. Brooks, and H.J. Shaw, “Two-mode fiber modal coupler,” Opt. Lett. 9, 177 (1984). [CrossRef] [PubMed]

5.

J. L. Brooks, R. C. Youngquist, and G. S. Kino “Active polarization coupler for birefringent fiber,” Opt. Lett. 9, 249 (1984). [CrossRef] [PubMed]

6.

W P. Risk, R. C. Youngquist, and G. S. Kino “Acousto-optic frequency shifting in birefringent fiber,” Opt. Lett. 9, 309 (1984). [CrossRef] [PubMed]

7.

J.N. Blake, B.Y. Kim, and H.J Shaw, “Fiber-optic modal coupler using periodic microbending,” Opt. Lett. 11, 177 (1986). [CrossRef] [PubMed]

8.

K.O. Hill, B. Malo, K.A. Vineberg, F. Bilodeau, D.C. Johnson, and L. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electronics Lett. 26, 1270 (1990). [CrossRef]

9.

C.D. Poole, C.D. Townsend, and K.T. Nelson, “Helical-grating two-mode fiber spatial-mode coupler,” J. Lightwave Techn. 9, 598 (1991). [CrossRef]

10.

K.S. Lee and T. Erdogan, “Fiber mode conversion with tilted gratings in an optical fiber,” J. Opt. Soc. Am. A 18, 1176 (2001). [CrossRef]

11.

K.S. Lee, “Coupling analysis of spiral fiber gratings,” Opt. Commun. 198, 317 (2001). [CrossRef]

12.

A.A. Ishaaya, G. Machavariani, N. Davidson, A.A. Friesem, and E. Hasman “Conversion of a high-order mode beam into a nearly Gaussian beam by use of a single interferometric element,” Opt. Lett. 28, 504 (2003). [CrossRef] [PubMed]

13.

P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. A 68, 1196–1201 (1978). [CrossRef]

14.

A.J. Fielding, K. Edinger, and C.C. Davis, “Experimental Observation of Mode Evolution in Single-Mode Tapered Optical Fibers,” J. Lightwave Techn. 17, 1649 (1999). [CrossRef]

15.

M. Skorobogatiy, Steven A. Jacobs, Steven G. Johnson, and Yoel Fink, “Geometric variations in high index-contrast waveguides, coupled mode theory in curvilinear coordinates,” Opt. Express 10, 1227 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-21-1227 [CrossRef] [PubMed]

16.

B. Z. Katsenelenbaum, L. Mercader del Río, M. Pereyaslavets, M. Sorolla Ayza, and M. Thumm, Theory of Nonuniform Waveguides: The Cross-Section Method (Inst. of Electrical Engineers, London, 1998) [CrossRef]

OCIS Codes
(060.2300) Fiber optics and optical communications : Fiber measurements
(060.2340) Fiber optics and optical communications : Fiber optics components

ToC Category:
Research Papers

History
Original Manuscript: September 11, 2003
Revised Manuscript: October 17, 2003
Published: November 3, 2003

Citation
M. Skorobogatiy, C. Anastassiou, Steven Johnson, O. Weisberg, Torkel Engeness, Steven Jacobs, Rokan Ahmad, and Yoel Fink, "Quantitative characterization of higher-order mode converters in weakly multimoded fibers," Opt. Express 11, 2838-2847 (2003)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-22-2838


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References

  1. Y. Danziger and D. Askegard, �??Full Fiber Capacity Realized with High Order Mode Technology,�?? in IEC Annual Review, (2000)
  2. Kerbage C, Windeler RS, Eggleton BJ, Mach P, Dolinski M and Rogers JA, �??Tunable devices based on dynamic positioning of micro-fluids in micro-structured optical fiber,�?? Opt. Commun. 204, 179 (2002). [CrossRef]
  3. Steven G. Johnson, Mihai Ibanescu, M. Skorobogatiy, Ori Weisberg, Torkel D. Engeness, Marin Solja¡�?i�?, Steven A. Jacobs, J. D. Joannopoulos and Yoel Fink, �??Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,�?? Opt. Express 9, 748 (2001), <a href=" http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13-748</a>. [CrossRef] [PubMed]
  4. R.C. Youngquist, J.L. Brooks and H.J. Shaw, �??Two-mode fiber modal coupler,�?? Opt. Lett. 9, 177 (1984) [CrossRef] [PubMed]
  5. J. L. Brooks, R. C. Youngquist and G. S. Kino �??Active polarization coupler for birefringent fiber,�?? Opt. Lett. 9, 249 (1984). [CrossRef] [PubMed]
  6. W P. Risk, R. C. Youngquist and G. S. Kino �??Acousto-optic frequency shifting in birefringent fiber,�?? Opt. Lett. 9, 309 (1984) [CrossRef] [PubMed]
  7. J.N. Blake, B.Y. Kim and H.J.Shaw, �??Fiber-optic modal coupler using periodic microbending,�?? Opt. Lett. 11, 177 (1986) [CrossRef] [PubMed]
  8. K.O. Hill, B. Malo, K.A. Vineberg, F. Bilodeau, D.C. Johnson and L. Skinner, �??Efficient mode conversion in telecommunication fibre using externally written gratings,�?? Electronics Lett. 26, 1270 (1990) [CrossRef]
  9. C.D. Poole, C.D. Townsend and K.T. Nelson, �??Helical-grating two-mode fiber spatial-mode coupler,�?? J. Lightwave Techn. 9, 598 (1991) [CrossRef]
  10. K.S. Lee and T. Erdogan, �??Fiber mode conversion with tilted gratings in an optical fiber,�?? J. Opt. Soc. Am. A 18, 1176 (2001). [CrossRef]
  11. K.S. Lee, �??Coupling analysis of spiral fiber gratings,�?? Opt. Commun. 198, 317 (2001). [CrossRef]
  12. A.A. Ishaaya, G. Machavariani, N. Davidson, A.A. Friesem and E. Hasman �??Conversion of a high-order mode beam into a nearly Gaussian beam by use of a single interferometric element,�?? Opt. Lett. 28 , 504 (2003) [CrossRef] [PubMed]
  13. P. Yeh, A. Yariv and E. Marom, �??Theory of Bragg fiber,�?? J. Opt. Soc. Am. A 68, 1196�??1201 (1978). [CrossRef]
  14. A.J. Fielding, K. Edinger and C.C. Davis, �??Experimental Observation of Mode Evolution in Single-Mode Tapered Optical Fibers,�?? J. Lightwave Techn. 17, 1649 (1999). [CrossRef]
  15. M. Skorobogatiy, Steven A. Jacobs, Steven G. Johnson and Yoel Fink, �??Geometric variations in high indexcontrast waveguides, coupled mode theory in curvilinear coordinates,�?? Opt. Express 10, 1227 (2002), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-21-1227">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-21-1227</a>. [CrossRef] [PubMed]
  16. B. Z. Katsenelenbaum, L. Mercader del Río, M. Pereyaslavets, M. Sorolla Ayza and M. Thumm, Theory of Nonuniform Waveguides: The Cross-Section Method (Inst. of Electrical Engineers, London, 1998) [CrossRef]

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