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Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 14 — Jul. 10, 2006
  • pp: 6572–6582
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Approximate empirical relations for nonlinear photonic crystal fibers

Kunimasa Saitoh, Takeshi Fujisawa, Takahito Kirihara, and Masanori Koshiba  »View Author Affiliations


Optics Express, Vol. 14, Issue 14, pp. 6572-6582 (2006)
http://dx.doi.org/10.1364/OE.14.006572


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Abstract

Approximate empirical relations for nonlinear photonic crystal fibers (PCFs) are newly proposed. Replacing a PCF with a conventional step-index fiber, closed form expressions for the effective refractive index and the effective core area of nonlinear PCFs are derived. To define the equivalent cladding index, the effective index of the so-called fundamental space-filling mode, which is calculated using empirical relations for the effective normalized frequency, is introduced, and thus, nonlinear guided waves propagating in PCFs can be easily characterized without the need for numerical computations. The validity of the method proposed here is ensured by comparing the calculated results with those obtained by a full-vector finite-element method.

© 2006 Optical Society of America

1. Introduction

In this paper, approximate empirical relations for nonlinear PCFs are proposed. Replacing a PCF with a conventional step-index fiber (SIF) [5

5. M. Koshiba and K. Saitoh, “Applicability of classical optical fiber theories to holey fibers,” Opt. Lett. 29, 1739–1741 (2004). [CrossRef] [PubMed]

], closed form expressions for the effective refractive index and effective core area [6

6. R.A. Sammut and C. Pask, “Gaussian and equivalent-step-index approximations for nonlinear waveguides,” J. Opt. Soc. Am. B 8, 395–402 (1991). [CrossRef]

, 7

7. Y. Chen, “Nonlinear fibers with arbitrary nonlinearity,” J. Opt. Soc. Am. B 8, 2338–2341 (1991). [CrossRef]

] of nonlinear PCFs can be derived. To define the equivalent cladding index, the effective index of the so-called fundamental space-filling mode, which is calculated using empirical relations [8

8. K. Saitoh and M. Koshiba, “Empirical relations for simple design of photonic crystal fibers,” Opt. Express 13, 267–274 (2005). [CrossRef] [PubMed]

], for the effective normalized frequency, is introduced, and thus, nonlinear guided waves propagating in PCFs can be easily characterized without any need for numerical computations. Numerical results for nonlinear PCFs with nonsaturable and saturable nonlinearities are presented. The validity of the method proposed here is ensured by comparing the calculated results with those obtained by a full-vector FEM [2

2. T. Fujisawa and M. Koshiba, “Finite element characterization of chromatic dispersion in nonlinear holey fibers,” Opt. Express 11, 1481–1489 (2003). [CrossRef] [PubMed]

].

Fig. 1. (a) A geometry of PCF and (b) its equivalent classical optical fiber model.

2. Approximate empirical relations for nonlinear PCFs

2.1 Replacing PCFs with SIFs

We consider a PCF as shown in Fig. 1(a), where d is the hole diameter, Λ is the hole pitch of triangular lattice structure, and the refractive index of the background material is given as n. To utilize the approximate analytical solutions, originally derived for axially symmetric nonlinear optical fibers, a complex PCF geometry as shown in Fig. 1(a) is replaced with the classical SIF equivalent model, as shown in Fig. 1(b). Here, nco is the equivalent refractive index of the core and is given as n. aeff is the effective core radius and is given as Λ/3 [5

5. M. Koshiba and K. Saitoh, “Applicability of classical optical fiber theories to holey fibers,” Opt. Lett. 29, 1739–1741 (2004). [CrossRef] [PubMed]

]. ncl is the effective refractive index of the cladding, and is given as the effective index of the so-called fundamental space filling mode nFSM. Usually, the use of numerical methods is mandatory to obtain nFSM. However, it takes much time to calculate nFSM if a pure numerical method is used. Therefore, recently proposed empirical relations for PCF designs [8

8. K. Saitoh and M. Koshiba, “Empirical relations for simple design of photonic crystal fibers,” Opt. Express 13, 267–274 (2005). [CrossRef] [PubMed]

] are utilized to obtain nFSM. According to Ref. [8

8. K. Saitoh and M. Koshiba, “Empirical relations for simple design of photonic crystal fibers,” Opt. Express 13, 267–274 (2005). [CrossRef] [PubMed]

], the normalized frequency V of a linear PCF is given by

V(λΛ,dΛ)=2πλaeffnco2ncl2
=A1(dΛ)+A2(dΛ)1+A3(dΛ)exp{A4(dΛ)λΛ}
(1)

with λ being the free-space wavelength, while the values of the fitting coefficients, Ai (i=1 to 4) are summarized in Ref. [8

8. K. Saitoh and M. Koshiba, “Empirical relations for simple design of photonic crystal fibers,” Opt. Express 13, 267–274 (2005). [CrossRef] [PubMed]

]. As a result, nco, aeff, ncl=nFSM can be obtained without time-consuming numerical calculations.

2.2 Approximate empirical relations for nonlinear PCFs with nonsaturable nonlinearity

We consider the case that the background material of PCF has instantaneous Kerr-type nonlinearity whose nonlinear refractive index n is given by

n2=nL2+nL2n2Z0E2
(2)

where nL stands for the linear part of the refractive index of the material, n 2 [m2/W] is the nonlinear coefficient, Z 0 is the free-space impedance, and E is the electric field. If the guided mode field distribution of nonlinear SIF shown in Fig. 1(b) is approximated as a Gaussian field ϕ given by

ϕ=Aexp(r2w2)
(3)

where A is the amplitude of field, and w is the spot size where the amplitude of the field drops as 1/e, the intensity-dependent effective refractive index neff is given as [6

6. R.A. Sammut and C. Pask, “Gaussian and equivalent-step-index approximations for nonlinear waveguides,” J. Opt. Soc. Am. B 8, 395–402 (1991). [CrossRef]

]:

neff2=nco21k02(1aeff2+2w2)+nco2n2w2A28Z0(1aeff2+4w2).
(4)

Here, k 0 is the free-space wavenumber, while the spot size w is given by

w=aefflnVNL
(5)

and VNL is defined as

VNL=V1k02nco2n2w2A28Z0.
(6)

By taking into account the fact that the optical power P and the critical optical power Pc at which VNL→∞ are given by

P=02π0ncoϕ22Z0rdrdθ=πncow2A24Z0
(7)

and

Pc=λ22πncon2,
(8)

respectively, the effective index of nonlinear PCFs is given as

neff2=nco2(λ2πaeff)2[1+2lnVNLPPc(1+4lnVNL)].
(9)

In addition, the effective core area of nonlinear PCFs Aeff is given as

Aeff=πw2=πaeff2lnVNL.
(10)

In this case, VNL defined in Eq. (6) can be rewritten as

VNL=V1PPc.
(11)

By assigning the refractive index of the core nco, the effective core radius aeff, the normalized frequency V, and operating wavelength λ, empirical relation for the effective index of nonlinear PCF neff can be obtained as a function of optical power P with moderate accuracies by using Eq. (9). This approach was originally developed for nonlinear SIFs based on a scalar approximation, however, since the accurate analysis of the cladding effective index in PCF has to be based on a full-vectorial formalism, we use the approach proposed in Ref. [8

8. K. Saitoh and M. Koshiba, “Empirical relations for simple design of photonic crystal fibers,” Opt. Express 13, 267–274 (2005). [CrossRef] [PubMed]

] to calculate the cladding effective index.

2.3 Approximate empirical relations for nonlinear PCF with saturable nonlinearity

We consider the case that the background material of PCF has saturable nonlinearity whose nonlinear refractive index n is given as:

n2=(nsat2ncl2){1exp[ncl2n2Z0(nsat2ncl2)E2]}
(12)

where nsat stands for the saturation coefficient. By assuming Gaussian field given in Eq. (3) for the electric field distributions of SIFs as shown in Fig. 1(b), the intensity-dependent effective refractive index neff is given as [7

7. Y. Chen, “Nonlinear fibers with arbitrary nonlinearity,” J. Opt. Soc. Am. B 8, 2338–2341 (1991). [CrossRef]

]:

neff2=nco2(λ2πaeff)2[1R02+V2exp(1R02)]
+(λ2πaeff)2Vsat2{11Q[1exp(Q)]}
(13)

where the normalized spot size R0, Vsat, and Q are defined as

R0=w2aeff,
(14)
Vsat=k0aeffnsat2ncl2,
(15)

and

Q=4PVsat2R02Pc,
(17)

respectively, and nFSM is calculated by using the empirical relations [8

8. K. Saitoh and M. Koshiba, “Empirical relations for simple design of photonic crystal fibers,” Opt. Express 13, 267–274 (2005). [CrossRef] [PubMed]

]. The normalized spot size R 0 is calculated by the following equation:

1V2[1PPcm=0(Q)m(1+0.5m)2m!]=exp(1R02)
(18)

Here, m is an integer and usually, a value of m=500 is enough to obtain convergence. Because R 0 is not explicitly calculated from Eq. (18), numerical methods such as bisectional methods have to be used. However, the calculation time is almost negligible for obtaining R 0. After calculating R 0, neff can be easily obtained by using Eq. (13). In addition, the effective core area Aeff is given by

Aeff=2πR02aeff2=2πR023Λ2.
(19)

3. Guiding properties of nonlinear PCFs

3.1 PCF with nonsaturable nonlinearity

We consider a PCF as shown in Fig. 1(a), and we assume that the refractive index of the background material is given as in Eq. (2). The linear part of the refractive index of silica is taken as nL=1.45. Solid curves in Figs. 2(a), (b), (c), and (d) show the effective refractive index of nonlinear PCFs as a function of P/Pc for λ/Λ=0.1, 0.2, 0.3, and 0.4, respectively. We can see that the value of the effective refractive index is increased for higher optical power. In these figures, the results obtained by full-vector FEM [2

2. T. Fujisawa and M. Koshiba, “Finite element characterization of chromatic dispersion in nonlinear holey fibers,” Opt. Express 11, 1481–1489 (2003). [CrossRef] [PubMed]

] are also depicted (dots) and are found to be in good agreement with those obtained by the approximate empirical relations proposed here. From these figures, we can observe that for smaller values of λ/Λ (relatively shorter wavelength), some discrepancies in the results can be seen for higher values of P/Pc (around 0.9 to 1.0). For smaller values of λ/Λ, the field is strongly confined in the core region, and the nonlinearity is enhanced. Therefore, if the optical power is increased to the value near Pc, the guided mode behaves like a Townes soliton [1

1. A. Ferrando, M. Zacares, P.F. de Cordoba, D. Binosi, and J.A. Monsoriu, “Spatial soliton formation in photonic crystal fibers,” Opt. Express 11, 452–459 (2003). [CrossRef] [PubMed]

, 9

9. R.Y. Chiao, E. Garmire, and C.H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964). [CrossRef]

] which is a self-localized guided mode in the bulk nonlinear medium. The critical power for Townes soliton is given by PTownes≈0.93Pc [10

10. G. Fibich and A.L. Gaeta, “Critical power for self-focusing in bulk media and in hollow waveguides,” Opt. Lett. 25, 335–337 (2000). [CrossRef]

]. Dashed curves in Figs. 2(a), (b), (c), and (d) show the effective refractive index obtained by Eq. (9) when Pc is replaced with PTownes. For λ/Λ=0.1 (Fig. 2 (a)), the results obtained by FEM agree well with dashed curves. On the other hand, for larger values of λ/Λ, the results obtained by FEM agree well with solid curves. These results indicate that for relatively stronger nonlinearity (smaller values of λ/Λ), the light becomes insensitive to the presence of the cladding air holes around the critical power of Townes soliton (PTownes), while for relatively weak nonlinearity (larger values of λ/Λ), the light is confined by the nonlinearity effect and by the index-guiding effect. This is confirmed by the field distributions as shown in Fig. 3. Figs. 3(a) and (b) show the field distributions obtained by FEM with d/Λ=0.4, P/Pc=0.9, for λ/Λ=0.1 and 0.4, respectively. We can see that for λ/Λ=0.1, the field confinement in the core region is stronger than that for λ/Λ=0.4 and there are almost no overlaps with cladding air holes. This observation was also confirmed by our full-vector FEM in the regime of P=PTownes, where a stable solution could not be obtained. Therefore the argument that the field in the regime around P=PTownes is best described as a Townes soliton is physically correct.

Fig. 2. Effective refractive index with nonsaturable nonlinearity as a function of P/Pc for λ/Λ=(a) 0.1, (b) 0.2, (c) 0.3, and (d) 0.4.
Fig. 3. Field distributions of nonlinear PCFs with d/Λ=0.4 and P/Pc=0.9, for λ/Λ=(a) 0.1 and (b) 0.4.
Fig. 4. Errors of approximate empirical relations as a function of the normalized power for λ/Λ=(a) 0.1, (b) 0.4, and (c) 1.5.

Figures 4(a), (b), and (c) show the errors of the approximate solutions Δneff as a function of the normalized power for λ/Λ=0.1, 0.4 and 1.5, respectively. Here, Δneff is defined as

Δneff=neffneff,FEMneff,FEM
(20)

where neff,FEM is the effective refractive index of nonlinear PCFs obtained by using FEM [2

2. T. Fujisawa and M. Koshiba, “Finite element characterization of chromatic dispersion in nonlinear holey fibers,” Opt. Express 11, 1481–1489 (2003). [CrossRef] [PubMed]

]. Solid and dashed curves in the figures are the results obtained for Pc2/(2πncon 2) and Pc=PTownes, respectively. We can see that the errors become smaller for smaller values of λ/Λ. For λ/Λ=0.4, |Δneff| lies within 0.5%. For larger values of λ/Λ, the accuracy of the solution obtained by the present method becomes worse. For λ/Λ=1.5, |Δneff| is 5 to 15% around the critical power and 2 to 3% around the half of critical power. From Eq. (9), it is clear that neff is independent to d/Λ when P=0.5Pc (P=0.465Pc for Townes soliton) and given by

neff2=nco212(3λ2πΛ)2.
(21)

It is interesting to note that the results obtained by FEM also show this feature.

Fig. 5. Normalized effective core area with nonsaturable nonlinearity as a function of P/Pc for λ/Λ=(a) 0.1, (b) 0.2, (c) 0.3, and (d) 0.4.

To obtain the converged solutions by using FEM around the critical power, it takes some tens or hundreds of iterations. On the other hand, by using the present method, solutions can be obtained almost instantaneously with moderate accuracy. Therefore, it is very useful to use the present method for initial design predictions or grasping general tendencies of the characteristics of nonlinear PCF.

Solid curves in Figs. 5(a), (b), (c), and (d) show the normalized effective core area as a function of P/Pc for λ/Λ=0.1, 0.2, 0.3, and 0.4, respectively. Dashed curves in Fig. 4 represent the normalized effective core area obtained by using the Eq. (10), by replacing Pc with PTownes. The results obtained by the present method agree well with those obtained by FEM (dots) for larger values of d/Λ and smaller values of λ/Λ. This is because for smaller values of d/Λ, field is leaked into the cladding region because of the weak confinement and the Gaussian field assumption used to derive Eq. (9) is not satisfied. For larger values of λ/Λ, the situation is the same. Because the wavelength of light is relatively long compared with Λ, the field is leaked into the cladding region, and therefore the Gaussian field assumption is not valid any more.

Fig. 6. Effective refractive index with saturable nonlinearity as a function of P/Pc for d/Λ (a)=0.4 and (b)=0.8, where the normalized wavelength is λ/Λ=0.1.
Fig. 7. Effective refractive index with saturable nonlinearity as a function of P/Pc for d/Λ (a)=0.4 and (b)=0.8, where the normalized wavelength is λ/Λ=0.4

3.2 PCF with saturable nonlinearity

We consider a PCF as shown in Fig. 1(a), and we assume that the refractive index of silica is given as in Eq. (12). Solid curves in Figs. 6(a) and (b) show the effective refractive index of the nonlinear PCF as a function of P/Pc for λ/Λ=0.1 with d/Λ=0.4 and 0.8, respectively. Solid curves in Figs. 7(a) and (b) show the same thing as in Fig. 6 except for λ/Λ=0.4. By decreasing the value of Vsat/V (smaller values of nsat), we can see strong saturation of the effective refractive index at high optical power. The results obtained by the present method agree well with those obtained by FEM and the general tendencies are well described. For smaller values of d/Λ, some discrepancies in the results can be seen. This is due to the fact that in the linear regime (P/Pc=0), the results are not so accurate because of the inappropriateness of the Gaussian field assumption. Solid curves in Figs. 8(a) and (b) represent the normalized effective core area of the nonlinear PCF as a function of P/Pc for λ/Λ=0.1 with d/Λ=0.4 and 0.8, respectively. Solid curves in Figs. 9(a) and (b) show the same thing as in Fig. 8 except for λ/Λ=0.4. Again, the general behavior of effective core area is well described by the approximate empirical relations.

Fig. 8. Normalized effective core area with saturable nonlinearity as a function of P/Pc for d/Λ (a)=0.4 and (b)=0.8, where the normalized wavelength is λ/Λ=0.1.
Fig. 9. Normalized effective core area with saturable nonlinearity as a function of P/Pc for d/Λ(a)=0.4 and (b)=0.8, where the normalized wavelength is λ/Λ=0.4.

4. Conclusion

We have proposed approximate empirical relations for nonlinear PCFs. Replacing a PCF with a conventional step-index fiber, closed form expressions for the effective refractive index and effective core area of nonlinear PCFs have been derived. The validity of the proposed method was confirmed by comparing the results with those obtained by FEM. We have confirmed that, for the intervals of λ/Λ<0.4 and 0.3<d/Λ<0.8, a standard error for Eqs. (9) and (13) is expected to be less than 1%. Although the accuracy of solutions obtained by the present approach is inferior to those obtained by general numerical methods due to its approximation, solutions are almost instantaneously obtained. Therefore, the proposed approach is very useful for the initial design or grasping general characteristics of nonlinear PCFs. We believe that the present methodology can applied to any PCF configurations by carefully define all the critical parameters like the effective core radius of the equivalent model and the effective cladding index.

References and links

1.

A. Ferrando, M. Zacares, P.F. de Cordoba, D. Binosi, and J.A. Monsoriu, “Spatial soliton formation in photonic crystal fibers,” Opt. Express 11, 452–459 (2003). [CrossRef] [PubMed]

2.

T. Fujisawa and M. Koshiba, “Finite element characterization of chromatic dispersion in nonlinear holey fibers,” Opt. Express 11, 1481–1489 (2003). [CrossRef] [PubMed]

3.

A. Ferrando, M. Zacares, P.F. de Cordoba, D. Binosi, and J.A. Monsoriu, “Vortex solitons in photonic crystal fibers,” Opt. Express 12, 817–822 (2004). [CrossRef] [PubMed]

4.

A. Ferrando, M. Zacares, P. Andrees, P.F. de Cordoba, and J.A. Monsoriu, “Nodal solitons and the nonlinear breaking of discrete symmetry,” Opt. Express 13, 1072–1078 (2005). [CrossRef] [PubMed]

5.

M. Koshiba and K. Saitoh, “Applicability of classical optical fiber theories to holey fibers,” Opt. Lett. 29, 1739–1741 (2004). [CrossRef] [PubMed]

6.

R.A. Sammut and C. Pask, “Gaussian and equivalent-step-index approximations for nonlinear waveguides,” J. Opt. Soc. Am. B 8, 395–402 (1991). [CrossRef]

7.

Y. Chen, “Nonlinear fibers with arbitrary nonlinearity,” J. Opt. Soc. Am. B 8, 2338–2341 (1991). [CrossRef]

8.

K. Saitoh and M. Koshiba, “Empirical relations for simple design of photonic crystal fibers,” Opt. Express 13, 267–274 (2005). [CrossRef] [PubMed]

9.

R.Y. Chiao, E. Garmire, and C.H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479–482 (1964). [CrossRef]

10.

G. Fibich and A.L. Gaeta, “Critical power for self-focusing in bulk media and in hollow waveguides,” Opt. Lett. 25, 335–337 (2000). [CrossRef]

OCIS Codes
(190.3270) Nonlinear optics : Kerr effect
(190.4370) Nonlinear optics : Nonlinear optics, fibers
(230.3990) Optical devices : Micro-optical devices
(260.5950) Physical optics : Self-focusing

ToC Category:
Photonic Crystal Fibers

History
Original Manuscript: May 22, 2006
Revised Manuscript: July 4, 2006
Manuscript Accepted: July 5, 2006
Published: July 10, 2006

Citation
Kunimasa Saitoh, Takeshi Fujisawa, Takahito Kirihara, and Masanori Koshiba, "Approximate empirical relations for nonlinear photonic crystal fibers," Opt. Express 14, 6572-6582 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-14-6572


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References

  1. A. Ferrando, M. Zacares, P.F. de Cordoba, D. Binosi, and J.A. Monsoriu, "Spatial soliton formation in photonic crystal fibers," Opt. Express 11, 452-459 (2003). [CrossRef] [PubMed]
  2. T. Fujisawa and M. Koshiba, "Finite element characterization of chromatic dispersion in nonlinear holey fibers," Opt. Express 11, 1481-1489 (2003). [CrossRef] [PubMed]
  3. A. Ferrando, M. Zacares, P.F. de Cordoba, D. Binosi, and J.A. Monsoriu, "Vortex solitons in photonic crystal fibers," Opt. Express 12, 817-822 (2004). [CrossRef] [PubMed]
  4. A. Ferrando, M. Zacares, P. Andrees, P.F. de Cordoba, and J.A. Monsoriu, "Nodal solitons and the nonlinear breaking of discrete symmetry," Opt. Express 13, 1072-1078 (2005). [CrossRef] [PubMed]
  5. M. Koshiba and K. Saitoh, "Applicability of classical optical fiber theories to holey fibers," Opt. Lett. 29, 1739-1741 (2004). [CrossRef] [PubMed]
  6. R.A. Sammut and C. Pask, "Gaussian and equivalent-step-index approximations for nonlinear waveguides," J. Opt. Soc. Am. B 8, 395-402 (1991). [CrossRef]
  7. Y. Chen, "Nonlinear fibers with arbitrary nonlinearity," J. Opt. Soc. Am. B 8, 2338-2341 (1991). [CrossRef]
  8. K. Saitoh and M. Koshiba, "Empirical relations for simple design of photonic crystal fibers," Opt. Express 13, 267-274 (2005). [CrossRef] [PubMed]
  9. R.Y. Chiao, E. Garmire, and C.H. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964). [CrossRef]
  10. G. Fibich and A.L. Gaeta, "Critical power for self-focusing in bulk media and in hollow waveguides," Opt. Lett. 25, 335-337 (2000). [CrossRef]

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