## A full vectorial model for pulse propagation in emerging waveguides with subwavelength structures part II: Stimulated Raman Scattering

Optics Express, Vol. 17, Issue 14, pp. 11565-11581 (2009)

http://dx.doi.org/10.1364/OE.17.011565

Acrobat PDF (188 KB)

### 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

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. 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]

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. 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]

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. 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]

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. 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]

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]

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. 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. 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]

**13(12)**, 4786–4791 (2005).
[CrossRef]

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. 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]

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]

*g*is the bulk Raman gain coefficient of the host material measured in metres per watt, and

_{R}*A*is the effective area measured in

_{e f f}*m*

^{2}. The standard model (SM) is based upon the weak guidance approximation which assumes the fields are purely transverse and it neglects the spatial dependence of both the refractive index and bulk Raman gain coefficient. The SM also assumes no differences in the modal field distributions of the pump and Stokes fields. These approximations are only valid for optical fibers in the weak guidance regime, where the core size is large and the refractive index contrast is small. It has been shown that in the regime of subwavelength features with high refractive index contrast, the fields can have significant longitudinal components of the electric field vector and thus the mode can no longer can be considered as purely transverse, which can have significant impact on Kerr 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]

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]

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]

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. 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]

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. 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]

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. 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]

**42(1–2)**, 160–170 (2006).
[CrossRef]

*E*fields contributes less than 1% of the total vector field amplitude, such that the effect of

_{z}*E*on nonlinearity can be ignored. Whilst [25

_{z}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]

*E*fields become significant, the impact of

_{z}*E*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.

_{z}**42(1–2)**, 160–170 (2006).
[CrossRef]

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]

**42(1–2)**, 160–170 (2006).
[CrossRef]

**16(2)**, 1280–1299 (2008).
[CrossRef]

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]

*E*on SRS, and show that it can effect the calculations for the modal Raman gain, especially in the regime of high index and subwavelength structured waveguides.

_{z}**17(4)**, 2298–2318 (2009).
[CrossRef]

## 2. Development of VNSE

*κ*=

*p, s*denoting the pump and Stokes and the unperturbed fields are at two different frequencies

*ω*and

_{p}*ω*. The tilde above the fields represents performed done in the Fourier domain. The Fourier transform is defined as:

_{s}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. 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]

*µ*eigenmode with the form [32]:

^{th}*β*and

_{pµ}*β*are the propagation constants for the

_{sµ}*µ*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:

^{th}**E**

_{κ}and

**H**

_{κ}. Unlike previous models that consider just the interaction between two modes of the optical waveguide [23, 27

**42(1–2)**, 160–170 (2006).
[CrossRef]

**16(2)**, 1280–1299 (2008).
[CrossRef]

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

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

*a*(

_{κη}*z, t*) are the modal amplitudes and |

*a*(

_{κη}*z, t*)|

^{2}2 is the pulse power envelope for the

*η*(pump/Stokes) mode. Physically these modal amplitudes describe how the modes of the optical fiber evolve as they propagate along the waveguide with the effects of both dispersion and nonlinearity where |

^{th}*a*(0,

_{κη}*t*)|

^{2}is the initial pulse of the

*η*mode at the start of the waveguide.

^{th}*ω*or

_{p}*ω*, the perturbed Maxwell’s Eqs. can still be separated into two sets, one each for the pump and Stokes fields:

_{s}**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], as used in [1

**17(4)**, 2298–2318 (2009).
[CrossRef]

**42(1–2)**, 160–170 (2006).
[CrossRef]

**F**

*is given by:*

_{cs}*ω*=

_{s}*ω*-

*ω*. 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.

_{s}**42(1–2)**, 160–170 (2006).
[CrossRef]

**P**

*ω*(

_{s}**r**,

*t*) is given by the convolution of the Electric fields with the 3rd order nonlinear response function

**E**-

*′=*

_{ω}**E***

*′,*

_{ω}*ω*

_{1,2,3}=

*ω*and

_{p,-p,s}**R**

^{(3)}is the rank-4 Raman response tensor that is defined as [23, 35]:

**17(4)**, 2298–2318 (2009).
[CrossRef]

*R*(

*x,y, t*)=

*χ*

^{(3)}

*(*

_{ijkl}*x,y*)

*δ(t)*. This implies that there is no dispersion in the nonlinear coefficient

*n*.

_{2}*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.

*i, j,k, l*=

*x,y, z*. Eq. (25) has two components corresponding to the isotropic and anisotropic contributions to SRS [23, 36] 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, 36]. We include the anisotropic component for completeness.

*χ*

^{(3)}

*is given by [23]:*

_{xxxx}*n*

^{2}(

*x,y*),

*n*

^{2}(

*x,y*),

*fa,b*(

*x,y*) and

*h*(

_{a,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:

*h*(

_{a,b}*t*-

*t*

_{2})

*a**

*pσ*(

*t*

_{2})

*a*(

_{sξ}*t*

_{2})exp(-

*i*Δ

*ω*(

*t*-

*t*

_{2}))

*dt*

_{2}=

*a**

*(-Δω),*

_{pσ}a_{sξ}h̃_{a,b}*h̃*(Δω) which leads to Raman gain, whereas the real component leads to Raman-induced refractive index changes [23]. The effect of such index changes on the pulse propagation is ignored here for simplicity. Also, note that

_{a,b}*Im*[

*h̃*(Δ

_{a,b}*ω*)] is an odd function i.e.

*Im*(

*h̃*(-Δ

_{a,b}*ω*))=-

*Im*(

*h̃*(Δ

_{a,b}*ω*)) [23]. We now use the definition of the bulk Raman coefficient measured in

*mW*

^{-1}[23]:

*ω*=

*ω*-

_{p}*ω*. This leads to the following adiabatic pulse propagation Eq. for SRS:

_{s}*n*

^{2}(

*x,y;ω*) and

_{s}*g*(

_{a,b}*x,y*,Δ

*ω*) 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**

*=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.*

_{py}**17(4)**, 2298–2318 (2009).
[CrossRef]

*β*-

_{pη}*β*+

_{pσ}*β*-

_{sξ}*β*=0 for any arbitrary optical waveguide. This contracts the summation of modes to

_{sµ}*σ*=

*η*and

*ξ*=

*µ*. For

*σ*≠

*η*and/or

*ξ*≠

*µ*, the phase term

*β*-

_{pη}*β*+

_{pσ}*β*-

_{sξ}*β*in general will be non-zero and typically very large. Thus the beat length of these interactions will be of the order of

_{sµ}*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*-

_{pη}*β*+

_{pσ}*β*-

_{sξ}*β*=0 even for

_{sµ}*σ*≠

*η*and/or

*ξ*≠

*µ*, 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. 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]

*g*(

_{µη}*m*

^{-1}

*W*

^{-1}) is the modal Raman gain experienced by the

*µ*Stokes mode due to the presence of the

^{th}*η*pump mode, which is now defined as:

^{th}*g̅R*as the effective Raman gain coefficient with respect to the intensity of fields inside the fiber and redefine the effective area to be

*A̅*where:

_{eff}*p*=

*s*) and one mode under consideration (i.e.

*η*=

*µ*). Under these simplifications Eq. (32) becomes:

**17(4)**, 2298–2318 (2009).
[CrossRef]

*µ*eigenmode along the z-direction.

^{th}*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

*µ*pump mode and

^{th}*η*Stokes mode.

^{th}*A̅*and the nonlinear properties that are determined by

_{eff}*g*(

_{a,b}*x,y*). If one assumes that the anisotropic component of the Raman susceptibility is negligible within all materials of the optical waveguide, then

*f*=0 and

_{b}*f*=

_{R}*f*, and the analysis further simplifies to:

_{a}*β*-

_{pη}*β*=0 where

_{pσ}*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

**17(4)**, 2298–2318 (2009).
[CrossRef]

**17(4)**, 2797–2804 (2009).
[CrossRef]

*E*) can become significant in the presence of subwavelength features with high refractive index contrast. In [25

_{z}**17(4)**, 2797–2804 (2009).
[CrossRef]

**17(4)**, 2298–2318 (2009).
[CrossRef]

*E*fields have large magnitude and are confined more tightly to the nanowire. These features of the

_{z}*E*fields makes these highly subwavelength core sized nanowires more desirable than previously considered with the SM that only considers the transverse fields.

_{z}*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]

*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]

*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]

*E*fields, and thus the scalar approximation of the fields used by the SM is valid. However, at subwavelength core diameters, the VNSE predicts smaller effective areas with a minumum of 0.233

_{z}*µm*

^{2}than the SM with a minimum of 0.36

*µm*

^{2}.

**17(4)**, 2298–2318 (2009).
[CrossRef]

*e*fields by:

_{z}*S*∝

_{z}*β*|

**e**

*|*

_{t}^{2}-

*i*

**e**

*·∇*

_{t}

_{t}*e**

*.*

_{z}*e*becomes significant it can contribute to the effective area. As discussed in [1

_{z}**17(4)**, 2298–2318 (2009).
[CrossRef]

*e*fields maintain tighter confinement than the transverse electric fields for subwavelength core dimensions. Thus the

_{z}*e*fields help to keep the mode tightly confined to the nanowire, further reducing the effective area.

_{z}*g*=5.1×10

_{R}^{-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.

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]

*E*components. However, for subwavelength core sizes, significant differences can be seen between the 3 models with a factor of over 2.5 observed between the VNSE and SM for a core diameter of 0.5

_{z}*µm*. Both the VNSE and ASM show a sharp decrease in modal Raman gain for core sizes smaller than 0.5

*µm*, which is due to the increase in effective area as well as decrease in overlap with the Raman active core as seen above in Fig. 1.

**e**

*·*

_{px}**e**

*=0. However, at the subwavelength core sizes around 0.5*

_{sy}*µm*, the z-component of the fields can no longer be considered zero and hence

**e**

*·*

_{px}**e**

*≠0 as discussed in Section 2. Thus the orthogonally polarised pump and Stokes fields will make a non-zero contributions to Eq. (35), to the modal Raman gain. This contribution to the modal Raman gain can be as large as 10% of the modal Raman gain due to the co-polarised Stokes and pump fields for the same core diameter.*

_{sy}## 4. Tailoring the Raman Gain

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]

**e**

*will create a wavelength-dependent overlap of the Stokes fields with both the pump fields and the Raman active materials. Therefore, the shape of the modal Raman gain spectrum will not only depend on the bulk Raman gain coefficient spectrum,*

_{sµ}*gR*(Δ

*ω*), but also the wavelength dependence of the waveguide’s modal field distributions over the entire Raman spectrum. In contrast, the SM assumes that the effective area to be constant over the Raman spectrum and hence the shape of the modal Raman gain to be governed entirely by the bulk Raman gain coefficient. In [1

**17(4)**, 2298–2318 (2009).
[CrossRef]

47. 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]

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]

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]

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]

*µm*is shown. At this core size, both pump and Stokes fields are confined tightly to the Raman active core over all the Stokes wavelengths within the spectrum and the shape of the modal Raman gain looks identical to that of the bulk Raman gain coefficient with

*R*≈2.0.

*µm*as shown in Fig. 3(b) in red, a significant change in the shape of the modal Raman gain spectrum is observed with a decrease of the ratio of the peaks to

*R*=1.35. At this highly subwavelength core diameter, as we increase the Stokes wavelength, the Stokes modal field distributions further spread out from the core into the air, increasing the effective area as well as decreasing the overlap of the Stokes fields with both the Raman active core and pump modal field distribution. The resultant effect at this core diameter is a decrease in the confinement of the Stokes modal field distributions for increasing Raman shift (longer wavelength). The resultant modal Raman gain spectrum is a significantly flattened gain spectrum making this optical fiber more suitable in a Raman amplifier system.

*R*, the amount of decrease in the modal Raman gain relative to the bulk Raman gain spectrum for a number of core diameters. As expected, the larger Raman shifts are affected more by this effect and these effects become more significant as we decrease the core size to subwavelength dimensions. Decreases in the modal Raman gain of up to nearly 60% are observed demonstrating the importance of considering these effects in the regime of strong guidance.

## 5. Conclusion and Discussion

**17(4)**, 2298–2318 (2009).
[CrossRef]

*E*fields, that in this regime of strong guidance, have large magnitude and tight confinement to the Raman active core. This increase in the modal Raman gain at subwavelength dimensions is expected have a great impact on creating highly efficient Raman devices.

_{z}**42(1–2)**, 160–170 (2006).
[CrossRef]

## References and links

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 |

2. | M. Foster, A. Turner, M. Lipson, and A. Gaeta, “Nonlinear optics in photonic nanowires,” Opt. Express |

3. | M. Lamont, C. de Sterke, and B. Eggleton, “Dispersion engineering of highly nonlinear As_2S_3 waveguides for parametric gain and wavelength conversion,” Opt. Express |

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 |

5. | G. Renversez, B. Kuhlmey, and R. McPhedran, “Dispersion management with microstructured optical fibers: ultraflattened chromatic dispersion with low losses,” Opt. Lett. |

6. | A. Mussot, M. Beaugeois, M. Bouazaoui, and T. Sylvestre, “Tailoring CW supercontinuum generation in microstructured fibers with two-zero dispersion wavelengths,” Opt. Express |

7. | J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Rev. Mod. Phys. |

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 |

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 |

10. | M. Foster and A. Gaeta, “Ultra-low threshold supercontinuum generation in sub-wavelength waveguides,” Opt. Express |

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. | S. Afshar, V., S. Warren-Smith, and T. Monro, “Enhancement of fluorescence-based sensing using microstructured optical fibres,” Opt. Express |

13. | 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 |

14. | 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. |

15. | M. Foster, A. Turner, R. Salem, M. Lipson, and A. Gaeta, “Broad-band continuous-wave parametric wavelength conversion in silicon nanowaveguides,” Opt. Express |

16. | V. R. Almeida, Q. Xu, C. A. Barrios, and M. Lipson, “Guiding and confining light in void nanostructure,” Opt. Lett. |

17. | M. Nagel, A. Marchewka, and H. Kurz, “Low-index discontinuity terahertz waveguides,” Opt. Express |

18. | H. Fukuda, K. Yamada, T. Shoji, M. Takahashi, T. Tsuchizawa, T. Watanabe, J. Takahashi, and S. Itabashi, “Four-wave mixing in silicon wire waveguides,” Opt. Express |

19. | G. Wiederhecker, C. Cordeiro, F. Couny, F. Benabid, S. Maier, J. Knight, C. Cruz, and H. Fragnito, “Field enhancement within an optical fibre with a subwavelength air core,” Nature |

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 |

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. |

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. |

23. | G. P. Agrawal, |

24. | C. Headley and G. P. Agrawal, “Unified description of ultrafast stimulated Raman scattering in optical fibers,” J. Opt. Soc. Am. B |

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 |

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. |

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. |

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 |

29. | K. Okamoto, |

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 |

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 |

32. | A. Snyder and J. Love, |

33. | A. Kireev and T. Graf, “Vector coupled-mode theory of dielectric waveguides,” IEEE J. Quantum Electron. |

34. | P. Butcher and D. Cotter, |

35. | Q. Lin, O. J. Painter, and G. P. Agrawal, “Nonlinear optical phenomena in silicon waveguides: Modeling and applications,” Opt. Express |

36. | R. Hellwarth, “Third-order optical susceptibilities of liquids and solids,” Prog. Quantum Electron. |

37. | A. Efimov, A. Taylor, F. Omenetto, J. Knight, W. Wadsworth, and P. Russell
“Phase-matched third harmonic generation in microstructured fibers,” Opt. Express |

38. | D. Dimitropoulos, V. Raghunathan, R. Claps, and B. Jalali, “Phase-matching and Nonlinear Optical Processes in Silicon Waveguides,” Opt. Express |

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 |

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 |

41. | V. E. Perlin and H. G. Winful, “Optimal design of flat-gain wide-band fiber Raman amplifiers,” J. Lightwave Technol. |

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. |

43. | C. Kakkar and K. Thyagarajan, “High gain Raman amplifier with inherent gain flattening and dispersion compensation,” Opt. Commun. |

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. |

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. |

47. | A. Mori, H. Masuda, K. Shikano, and M. Shimizu, “Ultra-wide-band tellurite-based fiber Raman amplifier,” J. Lightwave Technol. |

**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

- 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]
- M. Foster, A. Turner, M. Lipson, and A. Gaeta, "Nonlinear optics in photonic nanowires," Opt. Express 16(2), 1300-1320 (2008). [CrossRef]
- M. Lamont, C. de Sterke, and B. Eggleton, "Dispersion engineering of highly nonlinear As_2S_3 waveguides for parametric gain and wavelength conversion," Opt. Express 15(15), 9458-9463 (2007). [CrossRef]
- 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]
- 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]
- A. Mussot, M. Beaugeois, M. Bouazaoui, and T. Sylvestre, "Tailoring CW supercontinuum generation in microstructured fibers with two-zero dispersion wavelengths," Opt. Express 15(18), 11,553-11,563 (2007). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-15-18-11553.
- J. M. Dudley, G. Genty, and S. Coen, "Supercontinuum generation in photonic crystal fiber," Rev. Mod. Phys. 78(4), 1135-1184 (2006). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- 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).
- 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]
- 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]
- 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).
- 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]
- M. Nagel, A. Marchewka, and H. Kurz, "Low-index discontinuity terahertz waveguides," Opt. Express 14(21), 9944-9954 (2006). [CrossRef]
- H. Fukuda, K. Yamada, T. Shoji, M. Takahashi, T. Tsuchizawa, T. Watanabe, J. Takahashi, and S. Itabashi, "Four-wave mixing in silicon wire waveguides," Opt. Express 13(12), 4629-4637 (2005). [CrossRef]
- G. Wiederhecker, C. Cordeiro, F. Couny, F. Benabid, S. Maier, J. Knight, C. Cruz, and H. Fragnito, "Field enhancement within an optical fibre with a subwavelength air core," Nature 1(2), 115-118 (2007).
- 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]
- 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).
- S. Konorov, D. Sidorov-Biryukov, A. Zheltikov, I. Bugar, D. ChorvatJr, 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]
- G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).
- 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]
- 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]
- 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]
- 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]
- 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]
- K. Okamoto, Fundamentals of Optical Waveguides (Academic Press, 2006).
- 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]
- 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]
- A. Snyder and J. Love, Optical waveguide theory (Kluwer Academic Pub, 1983).
- A. Kireev and T. Graf, "Vector coupled-mode theory of dielectric waveguides," IEEE J. Quantum Electron. 39(7), 866-873 (2003). [CrossRef]
- P. Butcher and D. Cotter, The Elements of Nonlinear Optics (Cambridge University Press, 1990).
- 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. Hellwarth, "Third-order optical susceptibilities of liquids and solids," Prog. Quantum Electron. 5(1), 2-68 (1977).
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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), 211,104-1-211,104-3 (2007). USA.
- 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]
- 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.
- 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]

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