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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 14 — Jul. 6, 2009
  • pp: 11565–11581
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A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part II: Stimulated Raman Scattering

Mark D. Turner, Tanya M. Monro, and Shahraam Afshar V.  »View Author Affiliations


Optics Express, Vol. 17, Issue 14, pp. 11565-11581 (2009)
http://dx.doi.org/10.1364/OE.17.011565


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Abstract

The significance of full vectorial pulse propagation through emerging waveguides has not been investigated. Here we report the development of a generalised vectorial model of nonlinear pulse propagation due to the effects of Stimulated Raman Scattering (SRS) in optical waveguides. Unlike standard models, this model does not use the weak guidance approximation, and thus accurately models the modal Raman gain of optical waveguides in the strong guidance regime. Here we develop a vectorial-based nonlinear Schrödinger Eq. (VNSE) to demonstrate how the standard model fails in certain regimes, with up to factors of 2.5 enhancement in Raman gain between the VNSE and the standard model. Using the VNSE we are able to explore opportunities for tailoring of the modal Raman gain spectrum to achieve effects such as gain flattening through design of the optical fiber.

© 2009 Optical Society of America

1. Introduction

Recently, the growing interest in optical waveguides with high nonlinearity [1

1. S. Afshar V and T. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part I: Kerr nonlinearity,” Opt. Express 17(4), 2298–2318 (2009). [CrossRef]

4

4. H. Ebendorff-Heidepriem, P. Petropoulos, S. Asimakis, V. Finazzi, R. Moore, K. Frampton, F. Koizumi, D. Richardson, and T. Monro, “Bismuth glass holey fibers with high nonlinearity,” Opt. Express 12(21), 5082–5087 (2004). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-12-21-5082. [CrossRef]

] and specific dispersion characteristics [5

5. G. Renversez, B. Kuhlmey, and R. McPhedran, “Dispersion management with microstructured optical fibers: ultraflattened chromatic dispersion with low losses,” Opt. Lett. 28(12), 989–991 (2003). URL http://ol.osa.org/abstract.cfm?URI=ol-28-12-989. [CrossRef]

8

8. S. Yiou, P. Delaye, A. Rouvie, J. Chinaud, R. Frey, G. Roosen, P. Viale, S. Février, P. Roy, J. Auguste, and J. Blondy, “Stimulated Raman scattering in an ethanol core microstructured optical fiber,” Opt. Express 13(12), 4786–4791 (2005). [CrossRef]

] has led to the development of emerging waveguides [1] that contain inhomogeneous and complex transverse structure [8

8. S. Yiou, P. Delaye, A. Rouvie, J. Chinaud, R. Frey, G. Roosen, P. Viale, S. Février, P. Roy, J. Auguste, and J. Blondy, “Stimulated Raman scattering in an ethanol core microstructured optical fiber,” Opt. Express 13(12), 4786–4791 (2005). [CrossRef]

, 9

9. S. Atakaramians, S. Afshar, V., B. Fischer, D. Abbott, and T. Monro, “Porous fibers: a novel approach to low loss THz waveguides,” Opt. Express 16(12), 8845–8854 (2008). [CrossRef]

], subwavelength inclusions (or holes) [9

9. S. Atakaramians, S. Afshar, V., B. Fischer, D. Abbott, and T. Monro, “Porous fibers: a novel approach to low loss THz waveguides,” Opt. Express 16(12), 8845–8854 (2008). [CrossRef]

11

11. Y. Lizé, E. Mägi, V. Ta’eed, J. Bolger, P. Steinvurzel, and B. Eggleton, “Microstructured optical fiber photonic wires with subwavelength core diameter,” Opt. Express 12(14), 3209–3217 (2004). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-12-14-3209. [CrossRef]

] and high refractive index contrast. These emerging waveguides often operate in regimes of strong guidance where light within the waveguide is tightly confined and thus weak guidance approximation is no longer valid.

Another feature of emerging waveguides is the use of inhomogeneous and complex structure within optical waveguides, which provides great flexibility for tailoring the optical characteristics of the waveguide. One example is a microstructured optical fiber (MOF) which can have any arbitrary fiber cross section giving the ability to tailor the nonlinearity, dispersion, absorption etc. One post processing technique availiable to MOFs containing holey structure is filling with gases and liquids to further engineer the optical characteristics of the fiber [8

8. S. Yiou, P. Delaye, A. Rouvie, J. Chinaud, R. Frey, G. Roosen, P. Viale, S. Février, P. Roy, J. Auguste, and J. Blondy, “Stimulated Raman scattering in an ethanol core microstructured optical fiber,” Opt. Express 13(12), 4786–4791 (2005). [CrossRef]

, 20

20. F. Benabid, J. C. Knight, G. Antonopoulos, and P. S. J. Russell, “Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” Science 298(5592), 399–402 (2002). USA. [CrossRef]

22

22. S. Konorov, D. Sidorov-Biryukov, A. Zheltikov, I. Bugar, D. Chorvat Jr, D. Chorvat, V. Beloglazov, N. Skibina, M. Bloemer, and M. Scalora, “Self-phase modulation of submicrojoule femtosecond pulses in a hollow-core photonic-crystal fiber,” Appl. Phys. Lett. 85, 3690 (2004). [CrossRef]

]. For example enhancement of stimulated Raman scattering by filling the core of a MOF with the nonlinear liquid ethanol [8

8. S. Yiou, P. Delaye, A. Rouvie, J. Chinaud, R. Frey, G. Roosen, P. Viale, S. Février, P. Roy, J. Auguste, and J. Blondy, “Stimulated Raman scattering in an ethanol core microstructured optical fiber,” Opt. Express 13(12), 4786–4791 (2005). [CrossRef]

], filling photonic crystal fibers with Hydrogen gas to produce Raman effects at low power thresholds [20

20. F. Benabid, J. C. Knight, G. Antonopoulos, and P. S. J. Russell, “Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” Science 298(5592), 399–402 (2002). USA. [CrossRef]

, 21

21. F. Benabid, G. Bouwmans, J. Knight, P. Russell, and F. Couny, “Ultrahigh Efficiency Laser Wavelength Conversion in a Gas-Filled Hollow Core Photonic Crystal Fiber by Pure Stimulated Rotational Raman Scattering in Molecular Hydrogen,” Phys. Rev. Lett. 93(12), 123,903 (2004).

] or with helium gas to decrease the effects of self-phase modulation [22

22. S. Konorov, D. Sidorov-Biryukov, A. Zheltikov, I. Bugar, D. Chorvat Jr, D. Chorvat, V. Beloglazov, N. Skibina, M. Bloemer, and M. Scalora, “Self-phase modulation of submicrojoule femtosecond pulses in a hollow-core photonic-crystal fiber,” Appl. Phys. Lett. 85, 3690 (2004). [CrossRef]

].

These features of emerging waveguides offer great potential for nonlinear optical devices. However, due to their assumptions of weak guidance, standard pulse propagation models of the effects of dispersion and nonlinearity are not valid in this regime of strong guidance. Hence there is a need for new formulations of nonlinear effects to accurately model nonlinear pulse propagation within these emerging waveguides.

The standard model for nonlinear effects in optical fibers such as Stimulated Raman Scattering (SRS) has been developed extensively [23

23. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).

, 24

24. C. Headley and G. P. Agrawal, “Unified description of ultrafast stimulated Raman scattering in optical fibers,” J. Opt. Soc. Am. B 13(10), 2170–2177 (1996). [CrossRef]

]. Experimental investigations within weakly guiding optical fibers agree with these standard models [23

23. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).

]. In this standard model (SM) of Raman effects in optical waveguides, the Raman gain is given by [23

23. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).

]:

g(m1W1)=gRAeff,
(1)

There have been attempts to create rigorous vectorial nonlinear models for silicon waveguides [27

27. X. G. Chen, N. C. Panoiu, and R. M. Osgood, “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. 42(1–2), 160–170 (2006). [CrossRef]

, 28

28. J. I. Dadap, N. C. Panoiu, X. G. Chen, I. W. Hsieh, X. P. Liu, C. Y. Chou, E. Dulkeith, S. J. McNab, F. N. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, and R. M. Osgood, “Nonlinear-optical phase modification in dispersion-engineered Si photonic wires,” Opt. Express 16(2), 1280–1299 (2008). [CrossRef]

] that include dispersion, Raman effects, two photon absorption and free carrier effects, which all have significance in silicon waveguides. Note that the models developed in [27

27. X. G. Chen, N. C. Panoiu, and R. M. Osgood, “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. 42(1–2), 160–170 (2006). [CrossRef]

, 28

28. J. I. Dadap, N. C. Panoiu, X. G. Chen, I. W. Hsieh, X. P. Liu, C. Y. Chou, E. Dulkeith, S. J. McNab, F. N. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, and R. M. Osgood, “Nonlinear-optical phase modification in dispersion-engineered Si photonic wires,” Opt. Express 16(2), 1280–1299 (2008). [CrossRef]

] only consider the linear and nonlinear interactions between two specified eigenmodes of the optical waveguide. However, a rigorous model must use a complete basis set of modes to fully describe the pulse propagation. Even in the case of a single mode optical fiber, there are two different polarisations to consider.

In [25

25. J. Driscoll, X. Liu, S. Yasseri, I. Hsieh, J. Dadap, and R. Osgood, “Large longitudinal electric fields (E_z) in silicon nanowire waveguides,” Opt. Express 17(4), 2797–2804 (2009). [CrossRef]

, 27

27. X. G. Chen, N. C. Panoiu, and R. M. Osgood, “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. 42(1–2), 160–170 (2006). [CrossRef]

] it is mentioned that the longitudinal components of the electric field could affect the nonlinearity of the waveguide. However, in the numerical analysis in [27

27. X. G. Chen, N. C. Panoiu, and R. M. Osgood, “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. 42(1–2), 160–170 (2006). [CrossRef]

], the Ez fields contributes less than 1% of the total vector field amplitude, such that the effect of Ez on nonlinearity can be ignored. Whilst [25

25. J. Driscoll, X. Liu, S. Yasseri, I. Hsieh, J. Dadap, and R. Osgood, “Large longitudinal electric fields (E_z) in silicon nanowire waveguides,” Opt. Express 17(4), 2797–2804 (2009). [CrossRef]

] investigates in which regimes the Ez fields become significant, the impact of Ez on nonlinear effects such as SRS is not discussed. Thus to the best of our knowledge, there has been no investigation into effects of the longitudinal component of the electric field to SRS.

This paper has two main purposes. Firstly, we develop a complete vectorial model for pulse propagation under the effects of SRS and dispersion within an arbitrary optical waveguide. Secondly, we investigate the effects of the longitudinal components of electromagnetic fields on Raman gain within emerging waveguides.

The VNSE simultaneously considers the set of eigenmodes of the optical waveguide, and their nonlinear interactions, which leads to new terms in the pulse propagation model not shown before in [23

23. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).

, 27

27. X. G. Chen, N. C. Panoiu, and R. M. Osgood, “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. 42(1–2), 160–170 (2006). [CrossRef]

, 28

28. J. I. Dadap, N. C. Panoiu, X. G. Chen, I. W. Hsieh, X. P. Liu, C. Y. Chou, E. Dulkeith, S. J. McNab, F. N. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, and R. M. Osgood, “Nonlinear-optical phase modification in dispersion-engineered Si photonic wires,” Opt. Express 16(2), 1280–1299 (2008). [CrossRef]

]. However, for simplicity here we only consider just SRS and dispersion. The models in [23

23. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).

, 27

27. X. G. Chen, N. C. Panoiu, and R. M. Osgood, “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. 42(1–2), 160–170 (2006). [CrossRef]

, 28

28. J. I. Dadap, N. C. Panoiu, X. G. Chen, I. W. Hsieh, X. P. Liu, C. Y. Chou, E. Dulkeith, S. J. McNab, F. N. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, and R. M. Osgood, “Nonlinear-optical phase modification in dispersion-engineered Si photonic wires,” Opt. Express 16(2), 1280–1299 (2008). [CrossRef]

] consider other important phenomena such as free carrier effects and two photon absorption. Our approach for deriving the VNSE for SRS here and Kerr nonlinearity in [1

1. S. Afshar V and T. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part I: Kerr nonlinearity,” Opt. Express 17(4), 2298–2318 (2009). [CrossRef]

] could be used to model these other effects as well. However, this is not within the scope of this paper.

2. Development of VNSE

To develop a vectorial based nonlinear Schrödinger Eq. we use perturbation theory. We define the unperturbed fields in terms of a particular eigenmode of the waveguide without dispersion and nonlinearity and then form the perturbed fields as a linear superposition of the basis set of unperturbed eigenmodes. The unperturbed system is defined by two independent sets of Maxwell’s Eqs. for the pump and Stokes fields, which in the absense of sources are:

×E(r,ω)=iμ0ωH(r,ω),
(2)
×H(r,ω)=iω[ε0E(r,ω)+PκL(r,ω)],
(3)

where κ=p, s denoting the pump and Stokes and the unperturbed fields are at two different frequencies ωp and ωs. The tilde above the fields represents performed done in the Fourier domain. The Fourier transform is defined as:

F(r,t)=12πF(r,ω)eiωtdω.
(4)

The linear polarisation of the optical waveguide is defined as:

P˜κL(r,ω)=ε0χ(1)(ω;ω)·E˜oκ,
(5)

n2(r,ω)=1+χ(1)(ω;ω).
(6)

The two differential Eqs. (2) and (3) can be solved to obtain a set of full vectorial eigenmodes of the unperturbed system. This set of both bound and leaky modes, in both the core and cladding of the optical waveguide, forms a complete set, such that any other electromagnetic field can be written as a superposition of these modes.

In practice these eigenmodes can be obtained numerically using the Finite Element method [29

29. K. Okamoto, Fundamentals of Optical Waveguides (Academic Press, 2006).

], the Multipole Method [30

30. B. Kuhlmey, T. White, G. Renversez, D. Maystre, L. Botten, C. de Sterke, and R. McPhedran, “Multipole method for microstructured optical fibers. II. Implementation and results,” J. Opt. Soc. Am. B 19(10), 2331–2340 (2002). [CrossRef]

, 31

31. T. White, B. Kuhlmey, R. McPhedran, D. Maystre, G. Renversez, C. de Sterke, and L. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B 19(10), 2322–2330 (2002). [CrossRef]

], etc. These numerical calculations are based entirely on the linear refractive index geometry of the waveguide.

To derive a full vectorial nonlinear pulse propagation Eq. that describes the evolution of a particular mode under the effects of dispersion and nonlinear SRS, we define the unperturbed fields as a single eigenmode of the optical waveguide i.e. the µth eigenmode with the form [32

32. A. Snyder and J. Love, Optical waveguide theory (Kluwer Academic Pub, 1983).

]:

E˜oκ=12Eκμδ(ωωκ),
(7)
H˜oκ=12Hκμδ(ωωκ),
(8)
Eκμ=eκμ(x,y)eiβκμzNκμ+c.c.,Hκμ=hκμ(x,y)eiβκμzNκμ+c.c.,
(9)

where β and β are the propagation constants for the µth mode of the unperturbed eigenmodes of the pump and Stokes fields and c.c. is the complex conjugate which we shall leave out from here for simplicity. The eigenmodes of the optical waveguide form an orthonormal basis set:

12(eκμ*×hκη)×ẑdA=Nκμηδμη=Nκμ,Nκμ=12(eκμ*×hκμ)×ẑdA.
(10)

E˜κ=12Σηa˜κη(z,ωωκ)Eκη,
(11)
H˜κ=12ηα˜κη(z,ωωκ)Hκη.
(12)

By using a complete basis set of modes including the discrete set of bound and leaky both forward and backward propagating, we have a more complete description of the pulse propagation than using a single mode. Even in the case of a single mode optical waveguide there may be situations where interactions between bound and leaky modes are significant. However, here we do not include the continuous set of radiation modes, as these modes typically contain information of fields far from the core of the optical waveguide [32

32. A. Snyder and J. Love, Optical waveguide theory (Kluwer Academic Pub, 1983).

]. Also, implementing this extra information into the basis set increases the complexity of the required numerical calculations. It should also be pointed out that we have not used the vector coupled-mode formalism developed in [33

33. A. Kireev and T. Graf, “Vector coupled-mode theory of dielectric waveguides,” IEEE J. Quantum Electron. 39(7), 866–873 (2003). [CrossRef]

], as here we consider the modes to be the solutions of the entire system, where in [33

33. A. Kireev and T. Graf, “Vector coupled-mode theory of dielectric waveguides,” IEEE J. Quantum Electron. 39(7), 866–873 (2003). [CrossRef]

] an array of individual waveguides each with their own set of modes is considered.

We shall assume here that since each of Maxwell’s Eqs. contain electric, magnetic and polarisation fields oscillating at either ωp or ωs, the perturbed Maxwell’s Eqs. can still be separated into two sets, one each for the pump and Stokes fields:

×E˜κ=iμ0ω2ηα˜κηHκm,
(13)
×H˜κ=iωε0n2(r,ω)2ηα˜κηEκηiωP˜κNL,
(14)

This is valid as long as the two quasi-monochromatic pump and Stokes pulses do not spectrally overlap. Having now explicitly defined Maxwell’s Eqs. and the fields for the unperturbed and perturbed systems, we now derive a pulse propagation Eq. of a particular mode propagating under the effects of dispersion and nonlinearity. The SM is derived using the wave Eq., which makes the assumption that ∇· E=0, by assuming that the permitivity of the optical waveguide has negligible spatial dependence. We do not use this technique since we are interested in waveguides that in general have permittivities with strong spatial dependence such as a MOF containing an array of subwavelength holes. Instead here we shall use the Reciprocal theorem [32

32. A. Snyder and J. Love, Optical waveguide theory (Kluwer Academic Pub, 1983).

], as used in [1

1. S. Afshar V and T. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part I: Kerr nonlinearity,” Opt. Express 17(4), 2298–2318 (2009). [CrossRef]

, 27

27. X. G. Chen, N. C. Panoiu, and R. M. Osgood, “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. 42(1–2), 160–170 (2006). [CrossRef]

], using the unperturbed and perturbed Stokes fields:

zFcs·ẑdA=·FcsdA,
(15)

where the vector field F cs is given by:

Fcs=E˜os*×H˜s+E˜s+H˜os*.
(16)

Eq. (15), will allow us to relate the unperturbed fields to the perturbed fields and create a propagation Eq. for Stokes pulses under dispersion and nonlinear effects such as SRS. The right hand side of Eq. (15), can be expanded using Eqs. (2), (3), (13), (14), Maxwell’s Eqs. for the unperturbed and perturbed system. A Taylor series expansion on ωn2(r,ω)=ωsn2(r,ωs)+Δωs[ω(ωn2(r,ω))]ω=ωs+O(Δωs2) is used to separate dispersion into the order of Δωs=ω-ωs. We then apply Eq. (10), the orthonormality condition to the left hand side of Eq. (15) and arrive with Eq. (17), which is the general nonlinear pulse propagation Eq.

a˜sμ(z,ωωs)z=iΔωsβsμ1a˜sμ(z,ωωs)+iΔωsnμβsη1a˜sη(z,ωωs)+O(Δωs2)
+i2Esμ*·ωP˜sNL(r,ω)dA
(17)

where

βsμ1=14Nsμ[μ0hsμ2+ε0[ω(ωn2(r,ω))]ω=ωs|esμ2]dA
(18)
βsη1=ei(βsηβsμ)z4NsηNsμ[μ0hsη.hsμ*+ε0[ω(ωn2(r,ω))]|ω=ωsesη.esμ*]dA
(19)

Eq. (17) is general and is of the same form as in [27

27. X. G. Chen, N. C. Panoiu, and R. M. Osgood, “Theory of Raman-mediated pulsed amplification in silicon-wire waveguides,” IEEE J. Quantum Electron. 42(1–2), 160–170 (2006). [CrossRef]

]. However, here we consider a basis set of modes for the perturbed fields, which leads to a series of interactions in the pulse propagation model in both dispersion and nonlinearity.

In a single mode step index optical fiber, there are two bound modes corresponding to the two polarisations of the fundamental mode. The rest of the modes of the optical waveguide contribute to the set of Radiation modes that attenuate rapidly along the waveguide. These two bound modes can potentially interact with each other through dispersion and nonlinearity and thus it is important to consider the sum of interactions in Eq. (17), even within single mode step index fibers.

αsμ(z,t)z=D̂asu(z,t)eiωstteβsμz2Nsμesμ*PsNL(r,t)dA
D̂=iβsμ1t+iημβsη1t+O(2t2),
(20)

To derive the nonlinear Raman polarisation, we use the formalism in [34

34. P. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University Press, 1990).

] for describing the nonlinear polarisation due to stimulated Raman scattering (SRS). The nonlinear Stokes polarisation can be split into its fast and slow time varying components:

PsNL(r,t)=12Pωs(r,t)eiωst+c.c.,,

where P ωs (r, t) is given by the convolution of the Electric fields with the 3rd order nonlinear response function Φωs;ωp,ωp,ωs(3) [34

34. P. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University Press, 1990).

].

Pωs(3)(t)=3ε02R(3)(r,tt1,tt2,tt3)Eωp(r,t1)Eωp*(r,t2)Eωs(r,t3)
exp(ir=13ωr(ttr))dt1dt2dt3,
(21)

where E-ω′=E*ω′,ω 1,2,3=ωp,-p,s and R (3) is the rank-4 Raman response tensor that is defined as [23

23. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).

, 35

35. Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express 15(25), 16,604–16,644 (2007).

]:

R(3)(tt1,tt2,tt3)=R(x,y,tt2)δ(tt2)δ(t2t3).
(22)

It is important to note that the spatial dependence of the Raman response tensor for waveguides must not be ignored in the case of optical waveguides containing complex cross-sectional geometry. Using Eq. (22) in Eq. (21) and expanding the electric field vectors using the time domain form of Eqs. (11), (12) the nonlinear Stokes polarisation becomes:

Pωs(3)(t)=3ε02η,σ,ξei(βpηβpσ+βsξ)NpηNpσNsξαpη(z,t)
×R(x,y,tt2)epηepσ*esξapσ*(z,t2)asξ(z,t2)
×exp(iΔω(tt2))dt2,
(23)

where Δω=ωp-ωs. Using Eq. (23) in Eq. (20) we arrive at the SRS pulse propagation Eq.:

asμ(z,t)z=D̂asμ(z,t)+iωs(1+iωst)eiβsμz4Nsμesμ*·Pωs(3)(t)dA
=D̂asμ(z,t)+i3ε0ωs8(1+iωst)n,σ,ξei(βpηβpσ+βsξiβsμz)NpηNpσNsξNsμapη(z,t)
esμ*·R(x,y,tt2)epηepη*esξapσ*(z,t2)asξ(z,t2)
exp(iΔω(tt2))dt2dA.
(24)

Eq. (24) is general and can be used for both short and long pulses, and for an arbitrary optical waveguide structure. Before we continue this analysis of SRS, we discuss another use for Eq. (24). In [1

1. S. Afshar V and T. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part I: Kerr nonlinearity,” Opt. Express 17(4), 2298–2318 (2009). [CrossRef]

], Kerr nonlinearity was assumed to be instantaneous, i.e. the nonlinear response function had the form R(x,y, t)=χ (3) ijkl(x,y)δ(t). This implies that there is no dispersion in the nonlinear coefficient n2.

However, for ultrashort pulses with large bandwidth, a finite response function k(x,y, t) may need to be considered for Kerr nonlinearity. If one lets p=s in Eq. (24) and R(x,y, t)=χ (3) ijkl(x,y)k(x,y, t), one will have a VNSE for Kerr nonlinerity that now considers the dispersion of Kerr nonlinearity. This is beyond the scope of this paper, and would be a subject for future study.

For silica glass, the Raman response function as defined by Eq. (25):

R(x,y,τ)=χxxxx(3)2(faha(τ)δijδkl+12fbhb(τ)(δikδjl+δilδjk)),
(25)

where i, j,k, l=x,y, z. Eq. (25) has two components corresponding to the isotropic and anisotropic contributions to SRS [23

23. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).

, 36

36. R. Hellwarth, “Third-order optical susceptibilities of liquids and solids,” Prog. Quantum Electron. 5(1), 2–68 (1977).

] which are denoted by a and b respectively. However due to the amorphous nature of glasses such as silica the isotropic contribution to SRS dominates throughout most of the Raman spectrum as discussed in [23

23. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).

, 36

36. R. Hellwarth, “Third-order optical susceptibilities of liquids and solids,” Prog. Quantum Electron. 5(1), 2–68 (1977).

]. We include the anisotropic component for completeness.

We now specialise the pulse propagation Eq. to MOFs that contain amorphous materials such as silica glass, or liquids and gases. We assume that all materials within the optical fiber have the same form of the Raman reponse as shown in Eq. (25). If this is not valid, the more general Eq. (24) must be used. The 3rd order susceptibility tensor component,χ (3) xxxx is given by [23

23. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).

]:

Xxxxx(3)(x,y)=4ε0cn2(x,y)n2(x,y)3
(26)

Again noting that here we consider the spatial dependence of n 2(x,y), n 2(x,y), fa,b(x,y) and ha,b(x,y,τ), which from here on we shall assume implicitly. Using Eqs. (25), (26) in (24) we find the general nonlinear pulse propagation due to dispersion and SRS for silica glass becomes:

asμ(z,t)z=D̂asμ(z,t)+iε02cωs4(1+iωst)η,σ,ξei(βpηβpσ+βsξiβsμz)NpηNpσNsξNsμapη(z,t)
×fan2n2(esμ*·epη)(epσ*·esξ)
×ha(tt2)apσ*(z,t2)asξ(z,t2)exp(iΔω(tt2))dt2
+12fbn2n2{(epσ*·epη)(esμ*·esξ)+(esμ*·epσ*)(epη·esξ*)}
×hb(tt2)apσ*(z,t2)asξ(z,t2)exp(iΔω(tt2))dt2dA.
(27)

Eq. (27) is a general pulse propagation Eq. for dispersion and SRS within optical fibers provided that the materials have the Raman response form given by Eq. (22). Eq. (27) does not use the adiabatic approximation and thus can be used for ultrashort pulses. Due to the full vectorial formulation as well as the use of a basis set of modes, the form of Eq. (27) is now different to that in [23

23. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).

].

If we now assume that the pump and Stokes pulses are much longer than the Raman response time, we can make the adiabatic approximation and bring the pulse amplitudes outside the convolution integral:

ha,b(t-t 2)a*(t 2)a(t 2)exp(-iΔω(t-t 2))dt 2=a*aa,b(-Δω),

From here on we shall only consider the imaginary part of a,b(Δω) which leads to Raman gain, whereas the real component leads to Raman-induced refractive index changes [23

23. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).

]. The effect of such index changes on the pulse propagation is ignored here for simplicity. Also, note that Im[a,bω)] is an odd function i.e. Im(a,b(-Δω))=-Im(a,bω)) [23

23. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).

]. We now use the definition of the bulk Raman coefficient measured in mW -1 [23

23. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).

]:

ha,b(tt2)apσ*(t2)asξ(t2)exp(iΔω(tt2))dt2=apσ*asξh˜a,b(Δω),.

where Δω=ωp-ωs. This leads to the following adiabatic pulse propagation Eq. for SRS:

asμ(Z,t)t=D̂asμ+ε02c28(1+iωst)η,σ,ξei(βpηβpσ+βsξiβsμz)NpηNpσNsξNsμapηapσ*asξ
gan2(esμ*·epη)(epσ*·esξ)
+12gbn2[(epσ*·epη)(esμ*·esξ)+(esμ*·epσ*)(epη·esξ)]dA
(28)

The VNSE for SRS within MOFs as defined by Eq. (28) is rather different to that in [23

23. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).

]. Here we have considered the spatial dependence of the refractive index profile and bulk Raman gain coefficient, the vectorial form of the fields, the difference in the pump and Stokes fields and the possibility of interactions between the basis set of modes. It is important to note that in a standard optical fibre in the weak guidance regime, the spatial dependence of the n 2(x,y;ωs) and ga,b(x,yω) can be ignored since material interfaces are usually located at positions where the fields have small intensity value, and thus these terms can be taken outside the integral in Eq. (28). Also in the weak guidance regime the modes are purely transverse thus the x-polarised and y-polarised modes are completely orthogonal (i.e. e*sx·e py=0). Hence any interference terms between x and y polarised modes will have zero isotropic Raman gain. However contributions coming from the anisotropic component of the Raman gain can occur as we shall discuss later.

From here on we shall only explicitly show the contributions of the terms that do not require any phase matching i.e. β-β+β-β=0 for any arbitrary optical waveguide. This contracts the summation of modes to σ=η and ξ=µ. For ση and/or ξµ, the phase term β-β+β-β in general will be non-zero and typically very large. Thus the beat length of these interactions will be of the order of µm, thus will not contribute to any physically observed Raman effects within typical waveguides. It is however, possible to phase match these modes such that β-β+β-β=0 even for ση and/or ξµ, by correct tailoring of the modes of the MOF such that these phase terms cancel [37

37. A. Efimov, A. Taylor, F. Omenetto, J. Knight, W. Wadsworth, and P. Russell “Phase-matched third harmonic generation in microstructured fibers,” Opt. Express 11(20), 2567–2576 (2003). [CrossRef]

, 38

38. D. Dimitropoulos, V. Raghunathan, R. Claps, and B. Jalali, “Phase-matching and Nonlinear Optical Processes in Silicon Waveguides,” Opt. Express 12(1), 149–160 (2004). [CrossRef]

].

asμ(z,t)z=D̂asμ+gμμ2[1+iωst]apμ2asμ
+ημgμη2[1+iωst]apη2asμ
+phaseterms.
(29)

gμη=ε02c24NpηNsμgan2epη·esμ*2+gbn2(epη·esμ2+epη2esμ2)dA.
(30)

g=g̅RAeff,
(31)
A̅eff=(epη*×hpη.ẑdA)(esμ*×hsμ.ẑdA)(epη*×hpη.ẑ)(esμ*×hsμ.ẑ)dA,
(32)
g̅R=ε02cgan2epη·esμ*2+gbn2(epη·esμ2+epη2esη2)dA(epη*×hpη·ẑ)(esμ*×hsμ·ẑ)dA.
(33)

The definition of the effective area in Eq. (32) is rather different to that in the SM which defines the effective area purely by the scalar transverse electric field distribution which is assumed to be equal at both the pump and Stokes wavelengths. However, it can be shown that Eq. (32) reduces to the standard definition of the effective mode area [23

23. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).

] in the limit of weak guidance, as discussed in further detail in Section 3.

To understand the physical meaning of the effective area defined by Eq. (32) we shall discuss the simpler case where there is only one frequency (i.e. p=s) and one mode under consideration (i.e. η=µ). Under these simplifications Eq. (32) becomes:

A̅eff=(eμ×hμ·ẑ)dA2(eμ×hμ·ẑ)2dA.
(34)

Eq. (34) is the definition of the effective area used in [1

1. S. Afshar V and T. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part I: Kerr nonlinearity,” Opt. Express 17(4), 2298–2318 (2009). [CrossRef]

]. The physical interpretation of this form of the effective area is easier to identify as: the area of power flow of the µth eigenmode along the z-direction.

Here, we consider SRS at two different frequencies (p, s) and possibly between different modes. Thus the definition of the effective area given by Eq. (32), is an “average” of the areas of power flow along the z-direction of the µth pump mode and ηth Stokes mode.

g=ε02c24NpηNsμgRn2epη·esμ*2dA,
g̅R=ε02c2gRn2epn·esμ*2dA(epη*×hpη·ẑ)(esμ*×hsμ·ẑ)dA.
(35)

The full form of the nonlinear pulse propagation Eq. as given by Eq. (28), is quite complicated and one must analyse the combination of propagation constants to see if the phase can be matched. For example in a single mode non-birefringent fiber then β-β=0 where η and σ are the x and y polarised modes respectively. In a standard optical fiber in the weak guidance regime, these modes have negligible overlap since the z-component of the electric field vector is negligible, but in the strong guidance regime where the z-component of the electric field vector becomes significant, the overlap can become significant. One may also need to consider the possibility of phase matching of higher order modes with lower order modes, forward and backward propagating modes for both the pump and Stokes fields.

3. Significance of VNSE

It was shown in [1

1. S. Afshar V and T. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part I: Kerr nonlinearity,” Opt. Express 17(4), 2298–2318 (2009). [CrossRef]

] for a high refractive index nanowire, as the size of the core decreases to subwavelength dimensions, the Ez fields have large magnitude and are confined more tightly to the nanowire. These features of the Ez fields makes these highly subwavelength core sized nanowires more desirable than previously considered with the SM that only considers the transverse fields.

To show the significance of this effect on Raman gain, we numerically model the effective area, g̅R and g(m -1 W -1) of a chalcogenide nanowire for a range of core diameters to observe in which regimes differences between the VNSE and SM can be significant. Here we take the refractive index of chalcogenide to be 2.4 [39

39. V. Ta’eed, N. Baker, L. Fu, K. Finsterbusch, M. Lamont, D. Moss, H. Nguyen, B. Eggleton, D. Choi, S. Madden, and B. Luther-Davies, “Ultrafast all-optical chalcogenide glass photonic circuits,” Opt. Express 15(15), 9205–9221 (2007). [CrossRef]

]. We assume that due to the amorphous nature of chalcogenide glass, the bulk Raman gain coefficient is isotropic at the peak of the gain spectrum as found for silica glass [23

23. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).

, 36

36. R. Hellwarth, “Third-order optical susceptibilities of liquids and solids,” Prog. Quantum Electron. 5(1), 2–68 (1977).

]. Also the pump and Stokes fields are co-polarised in all cases, except for VNSE ORTH in Fig. 2 where the pump and Stokes fields are taken to be orthogonally polarised. The pump wavelength is set to 1550 nm and here we consider only one value of the Raman gain spectrum, i.e. the peak which is located at 230 cm -1 [40

40. R. Slusher, G. Lenz, J. Hodelin, J. Sanghera, L. Shaw, and I. Aggarwal, “Large Raman gain and nonlinear phase shifts in high-purity As 2 Se 3 chalcogenide fibers,” J. Opt. Soc. Am. B 21(6), 1146–1155 (2004). [CrossRef]

] and corresponds to a Stokes wavelength of 1608 nm. At this Raman shift gR=5.1×10-11 mW -1 [40

40. R. Slusher, G. Lenz, J. Hodelin, J. Sanghera, L. Shaw, and I. Aggarwal, “Large Raman gain and nonlinear phase shifts in high-purity As 2 Se 3 chalcogenide fibers,” J. Opt. Soc. Am. B 21(6), 1146–1155 (2004). [CrossRef]

].

Thus we note that the energy within the mode can be confined more tightly than the transverse electric field distribution at nanowire dimensions located near the minimum effective area. This can be attributed to the z-component of the electric field vector as shown in [1

1. S. Afshar V and T. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part I: Kerr nonlinearity,” Opt. Express 17(4), 2298–2318 (2009). [CrossRef]

], that the z-component of the Poynting vector is dependant on the ez fields by:

Szβ|e t|2-i e t·∇t e*z.

Thus when ez becomes significant it can contribute to the effective area. As discussed in [1

1. S. Afshar V and T. Monro, “A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part I: Kerr nonlinearity,” Opt. Express 17(4), 2298–2318 (2009). [CrossRef]

], the ez fields maintain tighter confinement than the transverse electric fields for subwavelength core dimensions. Thus the ez fields help to keep the mode tightly confined to the nanowire, further reducing the effective area.

Fig. 1. Left: Effective area of a chalcogenide nanowire versus core diameter, using the VNLSE (Red) and SM (Blue). Right: Effective Raman gain coefficient of a chalcogenide nanowire versus core diameter using VNLSE (Red) and the bulk Raman gain coefficient is shown in blue.

On the right in Fig. 1 we plot the effective Raman gain coefficient of a chalcogenide nanowire for a range of core diameters with the VNSE in red and bulk Raman gain coefficient in blue which is gR=5.1×10-11 mW -1. For large core diameters the effective Raman gain coefficient is approximately equal to the bulk Raman gain coefficient. As the size of the core decreases, the effective Raman gain coefficient rises above the bulk Raman gain coefficient, showing that the contributions from the ez fields can create significant enhancement to the effective Raman gain coefficient. For even smaller core diameters the effective Raman gain coefficient decreases sharply from the peak. This occurs because the fields start to spread out into the air, decreasing the overlap of the fields with the Raman-active core.

Fig. 2. Modal Raman gain of a Chalcogenide nanowire for varying core diameter. SM in blue, VNSE in red, ASM in green and VNSE ORTH in Pink. Pump wavelength 1550 nm, Stokes wavelength at 1608 nm.

4. Tailoring the Raman Gain

Changing of the shape of the Raman gain spectrum can be very useful for applications such as telecommunications, where gain flattening techniques are applied to obtain an ultra-flat gain spectrum. Typical approaches to gain flattening of a Raman amplifier include using multiple pump sources [41

41. V. E. Perlin and H. G. Winful, “Optimal design of flat-gain wide-band fiber Raman amplifiers,” J. Lightwave Technol. 20(2), 250–254 (2002). [CrossRef]

, 42

42. S. Cui, J. S. Liu, and X. M. Ma, “A novel efficient optimal design method for gain-flattened multiwavelength pumped fiber Raman amplifier,” IEEE Photon. Technol. Lett. 16(11), 2451–2453 (2004). [CrossRef]

], designing the losses of the optical fiber [43

43. C. Kakkar and K. Thyagarajan, “High gain Raman amplifier with inherent gain flattening and dispersion compensation,” Opt. Commun. 250(1–3), 77–83 (2005). [CrossRef]

], tailoring of the composition of the glass [44

44. R. Jose and Y. Ohishi, “Higher nonlinear indices, Raman gain coefficients, and bandwidths in the TeO/sub 2/-ZnO-Nb/sub 2/O/sub 5/-MoO/sub 3/quaternary glass system,” Appl. Phys. Lett.90(21), 211,104-1-211,104-211,104-3 (2007). USA.

], etc. However, to the best of our knowledge, no reported techniques attempt to flatten the modal Raman gain spectrum by exploiting the wavelength dependence of the confinement of the fields to the optical fiber. The modal field distributions and the optical characteristics of a MOF can be tailored via modifying the complex fiber geometry, choice of the host glass material, as well as post fabrication processes such as filling the MOF with gases, liquids and atomic vapour. Thus MOFs offer flexibility for engineering the modal Raman gain to achieve gain enhancement, gain suppression or even change in the shape of the modal Raman gain spectrum.

Fig. 3. a) Approximation of the bulk Raman gain coefficient spectrum of tellurite glass. b) Modal Raman gain spectrum of a tellurite nanowire for 1.5µm core diameter (cyan) and 0.5µm core diameter (red) calculated with the VNSE. c) Ratio (R) of the two peaks for varying core diameter. d) Plot of the amount of decrease in modal Raman gain due to finite overlap of the Stokes fields with the pump fields and the Raman active core.

5. Conclusion and Discussion

The VNSE in its most general form can model pulse propagation due to dispersion and SRS, for pulses longer and shorter than the Raman response time. However, we have assumed that the pump and Stokes fields have no spectral overlap, and that any dispersion in the Raman susceptibility tensor is contained within the Raman response function.

References and links

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2.

M. Foster, A. Turner, M. Lipson, and A. Gaeta, “Nonlinear optics in photonic nanowires,” Opt. Express 16(2), 1300–1320 (2008). [CrossRef]

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G. Renversez, B. Kuhlmey, and R. McPhedran, “Dispersion management with microstructured optical fibers: ultraflattened chromatic dispersion with low losses,” Opt. Lett. 28(12), 989–991 (2003). URL http://ol.osa.org/abstract.cfm?URI=ol-28-12-989. [CrossRef]

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S. Atakaramians, S. Afshar, V., B. Fischer, D. Abbott, and T. Monro, “Porous fibers: a novel approach to low loss THz waveguides,” Opt. Express 16(12), 8845–8854 (2008). [CrossRef]

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M. Foster and A. Gaeta, “Ultra-low threshold supercontinuum generation in sub-wavelength waveguides,” Opt. Express 12(14), 3137–3143 (2004). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-12-14-3137. [CrossRef]

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Y. Lizé, E. Mägi, V. Ta’eed, J. Bolger, P. Steinvurzel, and B. Eggleton, “Microstructured optical fiber photonic wires with subwavelength core diameter,” Opt. Express 12(14), 3209–3217 (2004). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-12-14-3209. [CrossRef]

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S. Afshar, V., S. Warren-Smith, and T. Monro, “Enhancement of fluorescence-based sensing using microstructured optical fibres,” Opt. Express 15(26), 17,891–17,901 (2007).

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M. Foster, J. Dudley, B. Kibler, Q. Cao, D. Lee, R. Trebino, and A. Gaeta, “Nonlinear pulse propagation and supercontinuum generation in photonic nanowires: experiment and simulation,” Appl. Phys. B: Lasers and Optics 81(2), 363–367 (2005). [CrossRef]

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Q. Xu, V. Almeida, R. Panepucci, and M. Lipson, “Experimental demonstration of guiding and confining light in nanometer-size low-refractive-index material,” Opt. Lett. 29(14), 1626–1628 (2004). [CrossRef]

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M. Foster, A. Turner, R. Salem, M. Lipson, and A. Gaeta, “Broad-band continuous-wave parametric wavelength conversion in silicon nanowaveguides,” Opt. Express 15(20), 12,949–12,958 (2007).

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V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. 29(11), 1209–1211 (2004). URL http://ol.osa.org/abstract.cfm?URI=ol-29-11-1209. [CrossRef]

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G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).

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C. Headley and G. P. Agrawal, “Unified description of ultrafast stimulated Raman scattering in optical fibers,” J. Opt. Soc. Am. B 13(10), 2170–2177 (1996). [CrossRef]

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J. Driscoll, X. Liu, S. Yasseri, I. Hsieh, J. Dadap, and R. Osgood, “Large longitudinal electric fields (E_z) in silicon nanowire waveguides,” Opt. Express 17(4), 2797–2804 (2009). [CrossRef]

26.

K. Thyagarajan and C. Kakkar, “Novel fiber design for flat gain Raman amplification using single pump and dispersion compensation in S band,” J. Lightwave Technol. 22(10), 2279–2286 (2004). [CrossRef]

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35.

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36.

R. Hellwarth, “Third-order optical susceptibilities of liquids and solids,” Prog. Quantum Electron. 5(1), 2–68 (1977).

37.

A. Efimov, A. Taylor, F. Omenetto, J. Knight, W. Wadsworth, and P. Russell “Phase-matched third harmonic generation in microstructured fibers,” Opt. Express 11(20), 2567–2576 (2003). [CrossRef]

38.

D. Dimitropoulos, V. Raghunathan, R. Claps, and B. Jalali, “Phase-matching and Nonlinear Optical Processes in Silicon Waveguides,” Opt. Express 12(1), 149–160 (2004). [CrossRef]

39.

V. Ta’eed, N. Baker, L. Fu, K. Finsterbusch, M. Lamont, D. Moss, H. Nguyen, B. Eggleton, D. Choi, S. Madden, and B. Luther-Davies, “Ultrafast all-optical chalcogenide glass photonic circuits,” Opt. Express 15(15), 9205–9221 (2007). [CrossRef]

40.

R. Slusher, G. Lenz, J. Hodelin, J. Sanghera, L. Shaw, and I. Aggarwal, “Large Raman gain and nonlinear phase shifts in high-purity As 2 Se 3 chalcogenide fibers,” J. Opt. Soc. Am. B 21(6), 1146–1155 (2004). [CrossRef]

41.

V. E. Perlin and H. G. Winful, “Optimal design of flat-gain wide-band fiber Raman amplifiers,” J. Lightwave Technol. 20(2), 250–254 (2002). [CrossRef]

42.

S. Cui, J. S. Liu, and X. M. Ma, “A novel efficient optimal design method for gain-flattened multiwavelength pumped fiber Raman amplifier,” IEEE Photon. Technol. Lett. 16(11), 2451–2453 (2004). [CrossRef]

43.

C. Kakkar and K. Thyagarajan, “High gain Raman amplifier with inherent gain flattening and dispersion compensation,” Opt. Commun. 250(1–3), 77–83 (2005). [CrossRef]

44.

R. Jose and Y. Ohishi, “Higher nonlinear indices, Raman gain coefficients, and bandwidths in the TeO/sub 2/-ZnO-Nb/sub 2/O/sub 5/-MoO/sub 3/quaternary glass system,” Appl. Phys. Lett.90(21), 211,104-1-211,104-211,104-3 (2007). USA.

45.

R. Stegeman, L. Jankovic, K. Hongki, C. Rivero, G. Stegeman, K. Richardson, P. Delfyett, G. Yu, A. Schulte, and T. Cardinal, “Tellurite glasses with peak absolute Raman gain coefficients up to 30 times that of fused silica,” Opt. Lett. 28(13), 1126–1128 (2003). USA. [CrossRef]

46.

Q. Guanshi, R. Jose, and Y. Ohishi, “Design of ultimate gain-flattened O+ E and S+ C+ L ultrabroadband fiber amplifiers using a new fiber Raman gain medium,” J. Lightwave Technol. 25(9), 2727–2738 (2007). USA.

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A. Mori, H. Masuda, K. Shikano, and M. Shimizu, “Ultra-wide-band tellurite-based fiber Raman amplifier,” J. Lightwave Technol. 21(5), 1300–1306 (2003). USA. [CrossRef]

OCIS Codes
(190.4360) Nonlinear optics : Nonlinear optics, devices
(190.4370) Nonlinear optics : Nonlinear optics, fibers
(190.5650) Nonlinear optics : Raman effect
(060.4005) Fiber optics and optical communications : Microstructured fibers

ToC Category:
Nonlinear Optics

History
Original Manuscript: May 4, 2009
Revised Manuscript: June 19, 2009
Manuscript Accepted: June 22, 2009
Published: June 25, 2009

Citation
Mark D. Turner, Tanya M. Monro, and Shahraam Afshar V., "A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part II: Stimulated Raman Scattering," Opt. Express 17, 11565-11581 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-14-11565


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References

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