## Efficient calculation of higher-order optical waveguide dispersion |

Optics Express, Vol. 18, Issue 19, pp. 19522-19531 (2010)

http://dx.doi.org/10.1364/OE.18.019522

Acrobat PDF (2164 KB)

### Abstract

An efficient numerical strategy to compute the higher-order dispersion parameters of optical waveguides is presented. For the first time to our knowledge, a systematic study of the errors involved in the higher-order dispersions’ numerical calculation process is made, showing that the present strategy can accurately model those parameters. Such strategy combines a full-vectorial finite element modal solver and a proper finite difference differentiation algorithm. Its performance has been carefully assessed through the analysis of several key geometries. In addition, the optimization of those higher-order dispersion parameters can also be carried out by coupling to the present scheme a genetic algorithm, as shown here through the design of a photonic crystal fiber suitable for parametric amplification applications.

© 2010 OSA

## 1. Introduction

*β*. As this is frequency dependent, mathematically the effects of medium dispersion can be taken into account by expanding

*β*of the considered mode in a Taylor series around a central frequency

*β*in the neighborhood of

## 2. Numerical strategy for HODP calculation

2. L. Valor and J. Zapata, “Efficient finite element analysis of waveguides with lossy inhomogeneous anisotropic materials characterized by arbitrary permittivity and permeability tensors,” IEEE Trans. Microw. Theory Tech. **43**(10), 2452–2459 (1995). [CrossRef]

3. M. Koshiba and Y. Tsuji, “Curvilinear Hybrid Edge/Nodal Elements with Triangular Shape for Guided-Wave Problems,” J. Lightwave Technol. **18**(5), 737–743 (2000). [CrossRef]

4. F. L. Teixeira and W. C. Chew, “General Closed-Form PML Constitutive Tensors to Match Arbitrary Bianisotropic and Dispersive Linear Media,” IEEE Microwave Guided Wave Lett. **8**(6), 223–230 (1998). [CrossRef]

3. M. Koshiba and Y. Tsuji, “Curvilinear Hybrid Edge/Nodal Elements with Triangular Shape for Guided-Wave Problems,” J. Lightwave Technol. **18**(5), 737–743 (2000). [CrossRef]

## 3. HODP calculation: numerical considerations

### 3.1 Finite element choice and numerical integration

### 3.2 Differentiation method choice

5. Finite Difference Schemes of One Variable - Wolfram Demonstrations, http://demonstrations.wolfram.com/FiniteDifferenceSchemesOfOneVariable/

*h*. The choice of

*h*is a compromise. Unfortunately, there is no way to estimate in advance what is the best value to be chosen. Here

*h*’s choice was made by several numerical simulations which allowed finding a more appropriate

*h*. Thus, for all simulations was taken

### 3.3 Considerations about chain rule use

*ω*and obtain the other parameters by derivatives with respect to

*ω*(referenced here as

*ω*-approach), or apply the chain rule in order to write the derivatives with respect to

*ω*as derivatives with respect to

*λ*and use these derivatives’ values to calculate HODP (referenced here as

*λ*-approach). From a mathematical view point, both approaches are identical and produce the same result. However, from a numerical view point, sometimes using the second one can produce less accurate results. This is because, using the

*ω*-approach, we may write,while using the

*λ*-approach,, where

*f*represents a linear combination of the derivatives in its argument, whose coefficients are nonlinear functions of

*λ*. As can be seen from Eq. (3), in the

*ω*-approach there is an error accumulation only from one derivative term (derivative of order

*i*) and, from Eq. (4), in the

*λ*-approach there is an error accumulation from several different derivative terms (order 1 to

*i*). In other words, the error accumulation from Eq. (3) tends to be smaller than the one from Eq. (4).

*ω*.

### 3.4 Error calculation

*RE*(initials from Relative Error) and defined by the equation, where

*i*indicates the derivative order.

## 4. Assessment

### 4.1 Rectangular waveguide analysis

*ω*and, because of that, this case has been chosen as reference for the assessment process.

*a*= 1

*μm*e

*b*=

*a*/2 were assumed [see Fig. 2(a)]. Due to its symmetry only half waveguide was simulated utilizing rectilinear FE.

*μm*band (optical communications’ band). Two points close to its extremes were picked up: 1299.41

*nm*and 1702.33

*nm*. For these two wavelengths the

*RE*, given in Eq. (5), was evaluated and plotted versus the number of Degrees Of Freedom (DOF) in the mesh. The corresponding results are shown in Figs. 2(b) and 2(c), for 1299.41

*nm*and 1702.33

*nm*, respectively. In Fig. 2(d) the

*RE*versus wavelength is presented for 75191 DOF.

6. L. H. Gabrielli, H. E. Hernández-Figueroa, and H. L. Fragnito, “Robustness Optimization of Fiber Index Profiles for Optical Parametric Amplifiers,” J. Lightwave Technol. **27**(24), 5571–5579 (2009). [CrossRef]

*nm*and 1702.33

*nm*, the

*RE*for 75191 DOF. The largest error is obtained for

*nm*and about 2.5% for 1702.33

*nm*. These precision values are considered very good even when considering up to sixth-order dispersion. If only up to fourth-order dispersion is considered, the errors are 0.00% and 0.01% for 1299.41

*nm*and 1702.33

*nm*, respectively. Finally, from Figs. 2(b) and 2(c) may be noted that for 10000 DOF a good approximation has already been reached.

### 4.2 Step-index profile fiber analysis

6. L. H. Gabrielli, H. E. Hernández-Figueroa, and H. L. Fragnito, “Robustness Optimization of Fiber Index Profiles for Optical Parametric Amplifiers,” J. Lightwave Technol. **27**(24), 5571–5579 (2009). [CrossRef]

6. L. H. Gabrielli, H. E. Hernández-Figueroa, and H. L. Fragnito, “Robustness Optimization of Fiber Index Profiles for Optical Parametric Amplifiers,” J. Lightwave Technol. **27**(24), 5571–5579 (2009). [CrossRef]

*ε*), allowing for results to be used as reference in error calculation. Because numerical data is in tabular form in this case, they must undergo a numerical differentiation process in order to obtain the remaining HODP. This implies that the reference has inherent differentiation process’ errors.

*RE*versus DOF for 1299.41

*nm*and 1702.33

*nm*is shown in Figs. 3(b) and 3(c), respectively. In Fig. 3(d) the

*RE*versus wavelength for 75099 DOF is displayed. Here the lowest accuracy is also obtained for the highest-order derivatives.

*nm*and 1702.33

*nm*, all numerical values of interest. The maximum error occurs for

*nm*and approximately 0.89% for 1702.33

*nm*. These precision values are considered excellent for the case of sixth-order dispersion. If only up to fourth-order dispersion is considered, the maximum errors are 0.00% and 0.01%, for 1299.41

*nm*and 1702.33

*nm*, respectively. Furthermore, analyzing Figs. 3(b) and 3(c), an excellent approximation could have already been obtained for a little more than 20000 DOF. In a global sense, an impressively good approximation was obtained.

## 5. An inverse problem in PCF

7. S. Arismar Cerqueira Jr., “Recent progress and novel applications of photonic crystal fibers,” Rep. Prog. Phys. **73**(2), 024401 (2010). [CrossRef]

8. E. Kerrinckx, L. Bigot, M. Douay, and Y. Quiquempois, “Photonic crystal fiber design by means of a genetic algorithm,” Opt. Express **12**(9), 1990–1995 (2004). [CrossRef] [PubMed]

*D*e

*D*is given byand

*c*is the light speed in vacuum. The parameters’ desired values were taken as being

*λ*= 1.55

*μm*. These specific objectives come from some criteria to obtain a plane gain in Fiber Optical Parametric Amplifiers (FOPA) [9

9. J. M. Chavéz Boggio, J. D. Marconi, S. R. Bickham, and H. L. Fragnito, “Spectrally flat and broadband double-pumped fiber optical parametric amplifiers,” Opt. Express **15**(9), 5288–5309 (2007). [CrossRef] [PubMed]

*F*) adopted,, where

*σ*represents a weight, which was taken as being equal to the order of magnitude of

*F*, a PCF with SF6 glass from Schott Glass [10

10. Schott Corporation, North America, http://www.us.schott.com.

*n*

_{2}) higher than pure silica (the

*n*

_{2}used was obtained via the formula reported in [11

11. X. Feng, A. K. Mairaj, D. W. Hewak, and T. M. Monro, “Nonsilica Glasses for Holey Fibers,” J. Lightwave Technol. **23**(6), 2046–2054 (2005). [CrossRef]

*γ*attainment. Additionally, the non-inclusion of

*γ*in

*F*allows reducing the problem’s optimization complexity.

*μm*and hole diameters d

_{1}= 0.36

*μm*, d

_{2}= 0.64

*μm,*d

_{3}= 1.44

*μm*and d

_{4}= 1.44

*μm*. Here it is important to point out that all geometric dimensions were swept with a two decimal places’ discretization (a minimum variation of 0.01

*μm*in all the dimensions was allowed). The obtained nonlinear coefficient was

*μm*and the HODP versus wavelength plots can be seen in Figs. 5(b), 5(c) and 5(d). The results are in good accordance with the established goals, as can be seen from those figures.

## 6. Conclusions

## Acknowledgements

## References and links

1. | G. P. Agrawal, |

2. | L. Valor and J. Zapata, “Efficient finite element analysis of waveguides with lossy inhomogeneous anisotropic materials characterized by arbitrary permittivity and permeability tensors,” IEEE Trans. Microw. Theory Tech. |

3. | M. Koshiba and Y. Tsuji, “Curvilinear Hybrid Edge/Nodal Elements with Triangular Shape for Guided-Wave Problems,” J. Lightwave Technol. |

4. | F. L. Teixeira and W. C. Chew, “General Closed-Form PML Constitutive Tensors to Match Arbitrary Bianisotropic and Dispersive Linear Media,” IEEE Microwave Guided Wave Lett. |

5. | Finite Difference Schemes of One Variable - Wolfram Demonstrations, http://demonstrations.wolfram.com/FiniteDifferenceSchemesOfOneVariable/ |

6. | L. H. Gabrielli, H. E. Hernández-Figueroa, and H. L. Fragnito, “Robustness Optimization of Fiber Index Profiles for Optical Parametric Amplifiers,” J. Lightwave Technol. |

7. | S. Arismar Cerqueira Jr., “Recent progress and novel applications of photonic crystal fibers,” Rep. Prog. Phys. |

8. | E. Kerrinckx, L. Bigot, M. Douay, and Y. Quiquempois, “Photonic crystal fiber design by means of a genetic algorithm,” Opt. Express |

9. | J. M. Chavéz Boggio, J. D. Marconi, S. R. Bickham, and H. L. Fragnito, “Spectrally flat and broadband double-pumped fiber optical parametric amplifiers,” Opt. Express |

10. | Schott Corporation, North America, http://www.us.schott.com. |

11. | X. Feng, A. K. Mairaj, D. W. Hewak, and T. M. Monro, “Nonsilica Glasses for Holey Fibers,” J. Lightwave Technol. |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(190.4970) Nonlinear optics : Parametric oscillators and amplifiers

(230.7370) Optical devices : Waveguides

(260.2030) Physical optics : Dispersion

(060.5295) Fiber optics and optical communications : Photonic crystal fibers

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: July 12, 2010

Revised Manuscript: August 12, 2010

Manuscript Accepted: August 15, 2010

Published: August 30, 2010

**Citation**

J. A. Mores Jr., G. N. Malheiros-Silveira, H. L. Fragnito, and H. E. Hernández-Figueroa, "Efficient calculation of higher-order optical waveguide dispersion," Opt. Express **18**, 19522-19531 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-19-19522

Sort: Year | Journal | Reset

### References

- G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed (Academic Press, 1995).
- L. Valor and J. Zapata, “Efficient finite element analysis of waveguides with lossy inhomogeneous anisotropic materials characterized by arbitrary permittivity and permeability tensors,” IEEE Trans. Microw. Theory Tech. 43(10), 2452–2459 (1995). [CrossRef]
- M. Koshiba and Y. Tsuji, “Curvilinear Hybrid Edge/Nodal Elements with Triangular Shape for Guided-Wave Problems,” J. Lightwave Technol. 18(5), 737–743 (2000). [CrossRef]
- F. L. Teixeira and W. C. Chew, “General Closed-Form PML Constitutive Tensors to Match Arbitrary Bianisotropic and Dispersive Linear Media,” IEEE Microwave Guided Wave Lett. 8(6), 223–230 (1998). [CrossRef]
- Finite Difference Schemes of One Variable - Wolfram Demonstrations, http://demonstrations.wolfram.com/FiniteDifferenceSchemesOfOneVariable/
- L. H. Gabrielli, H. E. Hernández-Figueroa, and H. L. Fragnito, “Robustness Optimization of Fiber Index Profiles for Optical Parametric Amplifiers,” J. Lightwave Technol. 27(24), 5571–5579 (2009). [CrossRef]
- S. Arismar Cerqueira., “Recent progress and novel applications of photonic crystal fibers,” Rep. Prog. Phys. 73(2), 024401 (2010). [CrossRef]
- E. Kerrinckx, L. Bigot, M. Douay, and Y. Quiquempois, “Photonic crystal fiber design by means of a genetic algorithm,” Opt. Express 12(9), 1990–1995 (2004). [CrossRef] [PubMed]
- J. M. Chavéz Boggio, J. D. Marconi, S. R. Bickham, and H. L. Fragnito, “Spectrally flat and broadband double-pumped fiber optical parametric amplifiers,” Opt. Express 15(9), 5288–5309 (2007). [CrossRef] [PubMed]
- Schott Corporation, North America, http://www.us.schott.com .
- X. Feng, A. K. Mairaj, D. W. Hewak, and T. M. Monro, “Nonsilica Glasses for Holey Fibers,” J. Lightwave Technol. 23(6), 2046–2054 (2005). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.