## Approximate empirical relations for nonlinear photonic crystal fibers

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

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]

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]

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

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]

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

## 2. Approximate empirical relations for nonlinear PCFs

### 2.1 Replacing PCFs with SIFs

*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,

*n*is the equivalent refractive index of the core and is given as

_{co}*n. a*is the effective core radius and is given as Λ/3 [5

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

*n*is the effective refractive index of the cladding, and is given as the effective index of the so-called fundamental space filling mode

_{cl}*n*. Usually, the use of numerical methods is mandatory to obtain

_{FSM}*n*. However, it takes much time to calculate

_{FSM}*n*if a pure numerical method is used. Therefore, recently proposed empirical relations for PCF designs [8

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

*n*. According to Ref. [8

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

*V*of a linear PCF is given by

*A*(

_{i}*i*=1 to 4) are summarized in Ref. [8

**13**, 267–274 (2005). [CrossRef] [PubMed]

*n*can be obtained without time-consuming numerical calculations.

_{co}, a_{eff}, n_{cl}=n_{FSM}### 2.2 Approximate empirical relations for nonlinear PCFs with nonsaturable nonlinearity

*n*is given by

*n*stands for the linear part of the refractive index of the material,

_{L}*n*

_{2}[m

^{2}/W] is the nonlinear coefficient,

*Z*

_{0}is the free-space impedance, and

*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*

**E***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

*n*is given as [6

_{eff}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]

*k*

_{0}is the free-space wavenumber, while the spot size

*w*is given by

*V*is defined as

_{NL}*P*and the critical optical power

*P*at which

_{c}*V*→∞ are given by

_{NL}*A*is given as

_{eff}*V*defined in Eq. (6) can be rewritten as

_{NL}*n*, the effective core radius

_{co}*a*, the normalized frequency

_{eff}*V*, and operating wavelength λ, empirical relation for the effective index of nonlinear PCF

*n*can be obtained as a function of optical power

_{eff}*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

**13**, 267–274 (2005). [CrossRef] [PubMed]

### 2.3 Approximate empirical relations for nonlinear PCF with saturable nonlinearity

*n*is given as:

*n*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

_{sat}*n*is given as [7

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

*R*, and

_{0}, V_{sat}*Q*are defined as

*n*is calculated by using the empirical relations [8

_{FSM}**13**, 267–274 (2005). [CrossRef] [PubMed]

*R*

_{0}is calculated by the following equation:

*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},

*n*can be easily obtained by using Eq. (13). In addition, the effective core area

_{eff}*A*is given by

_{eff}## 3. Guiding properties of nonlinear PCFs

### 3.1 PCF with nonsaturable nonlinearity

*n*=1.45. Solid curves in Figs. 2(a), (b), (c), and (d) show the effective refractive index of nonlinear PCFs as a function of

_{L}*P/P*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

_{c}**11**, 1481–1489 (2003). [CrossRef] [PubMed]

*P/P*(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

_{c}*P*, the guided mode behaves like a Townes soliton [1

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

*P*≈0.93

_{Townes}*P*[10

_{c}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]

*P*is replaced with

_{c}*P*. 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 (

_{Townes}*P*), 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

_{Townes}*d*/Λ=0.4,

*P/P*=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

_{c}*P=P*, where a stable solution could not be obtained. Therefore the argument that the field in the regime around

_{Townes}*P=P*is best described as a Townes soliton is physically correct.

_{Townes}*n*as a function of the normalized power for λ/Λ=0.1, 0.4 and 1.5, respectively. Here, Δ

_{eff}*n*is defined as

_{eff}*n*is the effective refractive index of nonlinear PCFs obtained by using FEM [2

_{eff,FEM}**11**, 1481–1489 (2003). [CrossRef] [PubMed]

*P*=λ

_{c}^{2}/(2π

*n*

_{co}n_{2}) and

*P*=

_{c}*P*, respectively. We can see that the errors become smaller for smaller values of λ/Λ. For λ/Λ=0.4, |Δ

_{Townes}*n*| 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, |Δ

_{eff}*n*| 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

_{eff}*n*is independent to

_{eff}*d*/Λ when

*P*=0.5

*P*(

_{c}*P*=0.465

*P*for Townes soliton) and given by

_{c}*P/P*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

_{c}*P*with

_{c}*P*. The results obtained by the present method agree well with those obtained by FEM (dots) for larger values of

_{Townes}*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.

### 3.2 PCF with saturable nonlinearity

*P/P*for λ/Λ=0.1 with

_{c}*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

*V*(smaller values of

_{sat}/V*n*), 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

_{sat}*d*/Λ, some discrepancies in the results can be seen. This is due to the fact that in the linear regime (

*P/P*=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

_{c}*P/P*for λ/Λ=0.1 with

_{c}*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.

## 4. Conclusion

*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 |

2. | T. Fujisawa and M. Koshiba, “Finite element characterization of chromatic dispersion in nonlinear holey fibers,” Opt. Express |

3. | A. Ferrando, M. Zacares, P.F. de Cordoba, D. Binosi, and J.A. Monsoriu, “Vortex solitons in photonic crystal fibers,” Opt. Express |

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 |

5. | M. Koshiba and K. Saitoh, “Applicability of classical optical fiber theories to holey fibers,” Opt. Lett. |

6. | R.A. Sammut and C. Pask, “Gaussian and equivalent-step-index approximations for nonlinear waveguides,” J. Opt. Soc. Am. B |

7. | Y. Chen, “Nonlinear fibers with arbitrary nonlinearity,” J. Opt. Soc. Am. B |

8. | K. Saitoh and M. Koshiba, “Empirical relations for simple design of photonic crystal fibers,” Opt. Express |

9. | R.Y. Chiao, E. Garmire, and C.H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. |

10. | G. Fibich and A.L. Gaeta, “Critical power for self-focusing in bulk media and in hollow waveguides,” Opt. Lett. |

**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

- 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]
- T. Fujisawa and M. Koshiba, "Finite element characterization of chromatic dispersion in nonlinear holey fibers," Opt. Express 11, 1481-1489 (2003). [CrossRef] [PubMed]
- 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]
- 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]
- M. Koshiba and K. Saitoh, "Applicability of classical optical fiber theories to holey fibers," Opt. Lett. 29, 1739-1741 (2004). [CrossRef] [PubMed]
- 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]
- Y. Chen, "Nonlinear fibers with arbitrary nonlinearity," J. Opt. Soc. Am. B 8, 2338-2341 (1991). [CrossRef]
- K. Saitoh and M. Koshiba, "Empirical relations for simple design of photonic crystal fibers," Opt. Express 13, 267-274 (2005). [CrossRef] [PubMed]
- R.Y. Chiao, E. Garmire, and C.H. Townes, "Self-trapping of optical beams," Phys. Rev. Lett. 13, 479-482 (1964). [CrossRef]
- 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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