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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 27 — Dec. 19, 2011
  • pp: 26515–26528
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Angular-momentum coupled optical waves in chirally-coupled-core fibers

Xiuquan Ma, Chi-Hung Liu, Guoqing Chang, and Almantas Galvanauskas  »View Author Affiliations


Optics Express, Vol. 19, Issue 27, pp. 26515-26528 (2011)
http://dx.doi.org/10.1364/OE.19.026515


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Abstract

A new type of interaction between optical waves occurs in chirally-coupled-core (CCC) fibers. Instead of linear-translational symmetry of conventional cylindrical fibers, CCC fibers are helical-translation symmetric, and, consequently, interaction between CCC fiber modes involves both spin and orbital angular momentum of the waves. Experimentally this has been verified by observing a multitude of new phase-matching resonances in the transmitted super-continuum spectrum, and theoretically explained through modal theory developed in helical reference frame. This enables new degrees of freedom in controlling fiber modal properties.

© 2011 OSA

1. Introduction

2. CCC fiber geometry and quasi-phase-matched interactions between modes in the structure

3. Maxwell’s equations in curvilinear helical reference frame

For simplicity let’s assume that CCC fiber cladding is homogeneous and isotropic, and both the central core and side core are isotropic step-index profiles. Then, in the Cartesian reference frame {x, y, z}, the dielectric permittivity tensor ε^(x,y,z) of the CCC structure is expressed as:
ε^(x,y,z)=(εclad000εclad000εclad)+(Δε1(x,y)000Δε1(x,y)000Δε1(x,y))+(Δε2(x,y,z)000Δε2(x,y,z)000Δε2(x,y,z))
(2)
where εclad is the dielectric constant of the homogeneous cladding, Δε1(x,y) describes transversal profile of the central core which is cylindrically symmetric and, therefore, without z-dependence, and Δε2(x,y,z) describes three dimensional profile of the helical side core thus including its z-dependence.

In general, it can be shown [16

16. J. L. Wilson, C. Wang, A. E. Fathy, and Y. W. Kang, “Analysis of rapidly twisted hollow waveguides,” IEEE Trans. Microw. Theory Tech. 57(1), 130–139 (2009). [CrossRef]

,17

17. A. Nicolet, F. Zolla, and S. Guenneau, “Modelling of twisted optical waveguides with edge elements,” Eur. Phys. J. Appl. Phys. 28(2), 153–157 (2004). [CrossRef]

] that from the covariance of Maxwell’s equations it follows that in any generalized coordinate system these equations can be expressed in a mathematical form identical to their expression in Cartesian coordinates, reference-frame difference being solely captured by the change in the tensor form of permittivity and permeability in these equations:
{×E¯h=iωμ0μ^hH¯h×H¯h=iωε0ε^hE¯h
(3)
where subscripts ‘h’ indicate corresponding field vectors and tensors in helical coordinates. The dielectric permittivity tensor ε^h(X,Y) in the helical reference frame is expressed as:
ε^h(X,Y)=J˜ε^(x,y,z)J˜T=ε^straighth(X,Y)+ε^rotateh(X,Y)
(4)
ε^straighth(X,Y)=(εclad000εclad000εclad)+(Δε1(X,Y)000Δε1(X,Y)000Δε1(X,Y))+(Δε2(X,Y)000Δε2(X,Y)000Δε2(X,Y))
(5)
ε^rotateh(X,Y)=εclad(Y2K2XYK2YKXYK2X2K2XKYKXK0)
(6)
, where J˜ and J˜T are the Jacobian matrix [17

17. A. Nicolet, F. Zolla, and S. Guenneau, “Modelling of twisted optical waveguides with edge elements,” Eur. Phys. J. Appl. Phys. 28(2), 153–157 (2004). [CrossRef]

] and its transpose matrix of the coordinates transformation between helical and Cartesian reference frames. In Eq. (6) a simplification has been made by replacing the full transverse dielectric-permittivity distribution εclad+Δε1(X,Y)+Δε2(X,Y) by εclad, since in the weakly-guiding limit considered here the permittivity contributions from the cores are negligible compared to the magnitude of the glass permittivity. Note, that for the central core, due to the rotational invariance of the cylindrical symmetry, the distribution remains exactly the same as in the Cartesian coordinates Δε1(X,Y)=Δε1(x,y)|xX,yY. For the side core, since helical coordinates X and Y are rotating together with the side core, the longitudinal variation of the side core disappears in the helical coordinates and we can choose Δε2(X,Y)=Δε2(x,y,z=0)|xX,yY. Consequently, the component ε^straighth(X,Y) describes dielectric-permittivity distribution of straight center and side cores. The effects associated with rotation of the CCC structure now are solely captured by the component ε^rotateh(X,Y). Inspection of Eq. (6) reveals that the dielectric permittivity tensor ε^h(X,Y), consisting of both components, becomes z-coordinate invariant (although it becomes anisotropic due to component ε^rotateh), which verifies the intuitive expectation that in helical coordinates the CCC fiber geometry should be “unwound” and represented by a straight, Z-axis independent (but inhomogeneous and anisotropic) waveguide structure.

4. Helically symmetric eigenmodes in CCC structure

Enm±=(Enmo±jEnme)ej(βc,sn±nK)z
(10)

The two modes in this set are designated by the superscript “±”, where “+” and “-“ here refer to the same and the opposite senses of rotation respectively, with respect to the helical reference frame (or, equivalently, CCC structure) chirality. These two modes, formed by mixing two degenerate “static” straight-waveguide modes, are not degenerate in the sense that they each acquire different propagation-constant increments Δβ=±nK (for the n-th order modes).

Note, that n = 0 corresponds to TE0m modes and TM0m modes which are azimuthally invariant but with no optical angular momentum, and that modes with n = 1 (which include fundamental HE11 modes) carry only spin-angular momentum.

5. Quasi-phase matching in a CCC structure

For mathematically expressing QPM conditions it is convenient to group TE0mmodes,TM0m modes, andHEnm modes and EHnm modes into so-called LPlm modes (LPstands for “Linearly Polarized”) as shown in Fig. 2. The LPlm modes are customarily used in designating weakly-guiding cylindrically-symmetric fiber modes [19

19. E. Snitzer, “Cylindrical dielectric fiber modes,” J. Opt. Soc. Am. 51(5), 491 (1961). [CrossRef]

,20

20. J. A. Buck, Fundamentals of Optical Fibers, 2nd ed. (Wiley, New Jersey, 2004).

], which are groups of TE0m modes, TM0m modes, and HEnmand EHnm hybrid modes that become degenerate (their propagation constants β are equal) in the weakly guiding limit [21

21. D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10(10), 2252–2258 (1971). [CrossRef] [PubMed]

].

6. Effective single mode operation in CCC fibers

One remarkable and practically important effect that this QPM coupling in a CCC structure enables to achieve is single transverse mode propagation in a large diameter core fiber, at core sizes well beyond conventional single-mode fiber limit [20

20. J. A. Buck, Fundamentals of Optical Fibers, 2nd ed. (Wiley, New Jersey, 2004).

]. This has already been shown in Fig. 2d, where at wavelengths longer than ~950nm fundamental mode is transmitted with negligible loss (except within narrow-band resonances), while transmission of all higher order modes is suppressed by at least three orders of magnitude. In fact, this CCC fiber central core, which would by itself support five transverse spatial modes, becomes effectively single-mode in this domain, with performance indistinguishable from truly single-mode fiber. Qualitative difference between effectively-single-mode large core CCC fibers and conventional large mode area (LMA) fibers is clearly revealed in Fig. 4
Fig. 4 Qualitative difference between effectively-single-mode large core CCC fibers and conventional large mode area (LMA) fibers: (a) Broad-band spectra of 30um LMA fiber at different beam launching position, showing prominent spectral modulation. (b) Broad-band spectra of 35um CCC fiber at different beam launching position, showing no spectral modulation.
. Spectral measurements have been performed by sending a broad-band amplified-spontaneous-emission source operating in the 1010nm-1090nm spectral region into the fiber under test and obtaining the spectrum with an optical spectrum analyzer through a standard single mode fiber acting as a spatial filter [24

24. J. W. Nicholson, A. D. Yablon, S. Ramachandran, and S. Ghalmi, “Spatially and spectrally resolved imaging of modal content in large-mode-area fibers,” Opt. Express 16(10), 7233–7243 (2008). [CrossRef] [PubMed]

]. Here Fig. 4a shows the transmission spectra of a conventional multi-mode core (30µm diameter) fiber measured at different transversal beam-launching positions with respect to fiber axis, which shows prominent spectral modulation indicating the multimode nature of this LMA fiber. This spectral modulation (which can also be referred to as spectral “beating”) is a result of spatial interference between different transverse modes, which propagate in the test fiber with different group velocities [24

24. J. W. Nicholson, A. D. Yablon, S. Ramachandran, and S. Ghalmi, “Spatially and spectrally resolved imaging of modal content in large-mode-area fibers,” Opt. Express 16(10), 7233–7243 (2008). [CrossRef] [PubMed]

]. In other words, there will be no spectral “beating” at all if there is only one mode supported. Figure 4b shows similarly obtained transmission spectra of the same 1.5m-long 35µm central-core CCC fiber sample without a trace of any spectral modulation. This clearly shows that, despite large core size, CCC fiber performs as a truly single-mode fiber, producing fundamental mode output irrespective of fiber excitation conditions.

7. Conclusion

Acknowledgments

This work was supported by US Army Research Office grant W911NF051057.

References and links

1.

C. K. Kao, “Nobel Lecture: sand from centuries past: send future voices fast,” Rev. Mod. Phys. 82(3), 2299–2303 (2010). [CrossRef]

2.

J. L. Hall, “Nobel Lecture: defining and measuring optical frequencies,” Rev. Mod. Phys. 78(4), 1279–1295 (2006). [CrossRef]

3.

B. E. A. M. Saleh and C. Teich, Fundamentals of Photonics, 2nd ed.(Wiley, New Jersey, 2007).

4.

J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282(5393), 1476–1478 (1998). [CrossRef] [PubMed]

5.

L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (I.O.P. Publishing, London, 2003).

6.

X. Ma, I. N. Hu, and A. Galvanauskas, “Propagation Length Independent Nonlinearity Threshold in Stokes-Wave Suppressed SRS in Chirally-Coupled-Core Fibers,” Nonlinear Optics conference at Kauai, Hawaii, USA, July 17–22, 2011.

7.

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50(2), 115–125 (1936). [CrossRef]

8.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef] [PubMed]

9.

K. T. Gahagan and G. A. Swartzlander Jr., “Optical vortex trapping of particles,” Opt. Lett. 21(11), 827–829 (1996). [CrossRef] [PubMed]

10.

M. Dienerowitz, M. Mazilu, P. J. Reece, T. F. Krauss, and K. Dholakia, “Optical vortex trap for resonant confinement of metal nanoparticles,” Opt. Express 16(7), 4991–4999 (2008). [CrossRef] [PubMed]

11.

G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004). [CrossRef] [PubMed]

12.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007). [CrossRef]

13.

U. Peschel, T. Peschel, and F. Lederer, “A compact device for highly efficient dispersion compensation in fiber transmission,” Appl. Phys. Lett. 67(15), 2111–2113 (1995). [CrossRef]

14.

A. W. Snyder and J. D. Love, “Reflection at a curved dielectric interface—electromagnetic tunneling,” IEEE Trans. Microw. Theory Tech. 23(1), 134–141 (1975). [CrossRef]

15.

D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. 66(3), 216–220 (1976). [CrossRef]

16.

J. L. Wilson, C. Wang, A. E. Fathy, and Y. W. Kang, “Analysis of rapidly twisted hollow waveguides,” IEEE Trans. Microw. Theory Tech. 57(1), 130–139 (2009). [CrossRef]

17.

A. Nicolet, F. Zolla, and S. Guenneau, “Modelling of twisted optical waveguides with edge elements,” Eur. Phys. J. Appl. Phys. 28(2), 153–157 (2004). [CrossRef]

18.

K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed. (Academic Press, MA, 2006).

19.

E. Snitzer, “Cylindrical dielectric fiber modes,” J. Opt. Soc. Am. 51(5), 491 (1961). [CrossRef]

20.

J. A. Buck, Fundamentals of Optical Fibers, 2nd ed. (Wiley, New Jersey, 2004).

21.

D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10(10), 2252–2258 (1971). [CrossRef] [PubMed]

22.

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006). [CrossRef] [PubMed]

23.

R. Ulrich and A. Simon, “Polarization optics of twisted single-mode fibers,” Appl. Opt. 18(13), 2241–2251 (1979). [CrossRef] [PubMed]

24.

J. W. Nicholson, A. D. Yablon, S. Ramachandran, and S. Ghalmi, “Spatially and spectrally resolved imaging of modal content in large-mode-area fibers,” Opt. Express 16(10), 7233–7243 (2008). [CrossRef] [PubMed]

OCIS Codes
(060.2310) Fiber optics and optical communications : Fiber optics
(140.3510) Lasers and laser optics : Lasers, fiber
(270.0270) Quantum optics : Quantum optics

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: July 12, 2011
Revised Manuscript: August 25, 2011
Manuscript Accepted: November 29, 2011
Published: December 13, 2011

Citation
Xiuquan Ma, Chi-Hung Liu, Guoqing Chang, and Almantas Galvanauskas, "Angular-momentum coupled optical waves in chirally-coupled-core fibers," Opt. Express 19, 26515-26528 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-27-26515


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References

  1. C. K. Kao, “Nobel Lecture: sand from centuries past: send future voices fast,” Rev. Mod. Phys. 82(3), 2299–2303 (2010). [CrossRef]
  2. J. L. Hall, “Nobel Lecture: defining and measuring optical frequencies,” Rev. Mod. Phys. 78(4), 1279–1295 (2006). [CrossRef]
  3. B. E. A. M. Saleh and C. Teich, Fundamentals of Photonics, 2nd ed.(Wiley, New Jersey, 2007).
  4. J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282(5393), 1476–1478 (1998). [CrossRef] [PubMed]
  5. L. Allen, S. M. Barnett, and M. J. Padgett, Optical Angular Momentum (I.O.P. Publishing, London, 2003).
  6. X. Ma, I. N. Hu, and A. Galvanauskas, “Propagation Length Independent Nonlinearity Threshold in Stokes-Wave Suppressed SRS in Chirally-Coupled-Core Fibers,” Nonlinear Optics conference at Kauai, Hawaii, USA, July 17–22, 2011.
  7. R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50(2), 115–125 (1936). [CrossRef]
  8. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef] [PubMed]
  9. K. T. Gahagan and G. A. Swartzlander., “Optical vortex trapping of particles,” Opt. Lett. 21(11), 827–829 (1996). [CrossRef] [PubMed]
  10. M. Dienerowitz, M. Mazilu, P. J. Reece, T. F. Krauss, and K. Dholakia, “Optical vortex trap for resonant confinement of metal nanoparticles,” Opt. Express 16(7), 4991–4999 (2008). [CrossRef] [PubMed]
  11. G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12(22), 5448–5456 (2004). [CrossRef] [PubMed]
  12. G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007). [CrossRef]
  13. U. Peschel, T. Peschel, and F. Lederer, “A compact device for highly efficient dispersion compensation in fiber transmission,” Appl. Phys. Lett. 67(15), 2111–2113 (1995). [CrossRef]
  14. A. W. Snyder and J. D. Love, “Reflection at a curved dielectric interface—electromagnetic tunneling,” IEEE Trans. Microw. Theory Tech. 23(1), 134–141 (1975). [CrossRef]
  15. D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. 66(3), 216–220 (1976). [CrossRef]
  16. J. L. Wilson, C. Wang, A. E. Fathy, and Y. W. Kang, “Analysis of rapidly twisted hollow waveguides,” IEEE Trans. Microw. Theory Tech. 57(1), 130–139 (2009). [CrossRef]
  17. A. Nicolet, F. Zolla, and S. Guenneau, “Modelling of twisted optical waveguides with edge elements,” Eur. Phys. J. Appl. Phys. 28(2), 153–157 (2004). [CrossRef]
  18. K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed. (Academic Press, MA, 2006).
  19. E. Snitzer, “Cylindrical dielectric fiber modes,” J. Opt. Soc. Am. 51(5), 491 (1961). [CrossRef]
  20. J. A. Buck, Fundamentals of Optical Fibers, 2nd ed. (Wiley, New Jersey, 2004).
  21. D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10(10), 2252–2258 (1971). [CrossRef] [PubMed]
  22. L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006). [CrossRef] [PubMed]
  23. R. Ulrich and A. Simon, “Polarization optics of twisted single-mode fibers,” Appl. Opt. 18(13), 2241–2251 (1979). [CrossRef] [PubMed]
  24. J. W. Nicholson, A. D. Yablon, S. Ramachandran, and S. Ghalmi, “Spatially and spectrally resolved imaging of modal content in large-mode-area fibers,” Opt. Express 16(10), 7233–7243 (2008). [CrossRef] [PubMed]

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