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Optics Express

Optics Express

  • Editor: J. H. Eberly
  • Vol. 9, Iss. 13 — Dec. 17, 2001
  • pp: 687–697
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Designing the properties of dispersion-flattened photonic crystal fibers

Albert Ferrando, Enrique Silvestre, Pedro Andrés, Juan J. Miret, and Miguel V. Andrés  »View Author Affiliations


Optics Express, Vol. 9, Issue 13, pp. 687-697 (2001)
http://dx.doi.org/10.1364/OE.9.000687


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Abstract

We present a systematic study of group-velocity-dispersion properties in photonic crystal fibers (PCF’s). This analysis includes a thorough description of the dependence of the fiber geometrical dispersion on the structural parameters of a PCF. The interplay between material dispersion and geometrical dispersion allows us to established a well-defined procedure to design specific predetermined dispersion profiles. We focus on flattened, or even ultraflattened, dispersion behaviors both in the telecommunication window (around 1.55 µm) and in the Ti-Za laser wavelength range (around 0.8 µm). We show the different possibilities of obtaining normal, anomalous, and zero dispersion curves in the above frequency domains and discuss the limits for the existence of the above dispersion profiles.

© Optical Society of America

1 Introduction

One of the most appealing features of photonic crystal fibers (PCF’s) is their high flexibility based on the particular geometry of their refractive index distribution. The transverse section of a PCF is a two-dimensional (2D) silica-air photonic crystal in which an irregularity of the refractive index, or defect, is generated. In PCF’s guidance occurs in the region where the defect is located, which determines an effective PCF core. Analogously, one can define an effective PCF cladding constituted by the region surrounding the core, or defect area, that has the form of a perfectly periodic 2D photonic crystal. As compared to conventional fibers, it is apparent that PCF’s enjoy a more complex geometrical structure because of their 2D photonic crystal cladding. This fact allows us to manipulate the geometrical parameters of the fiber (e.g., the air-hole radius a and the lattice period, or pitch, Λ of a 2D triangular photonic crystal cladding) to generate an enormous variety of different configurations.

The peculiarities of the guidance in the core depend on the nature of the defect, which can generate donor or acceptor guided modes by an analogous mechanism leading to impurity states in electronic crystals [1

1. A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, and M. V. Andrés, “Donor and acceptor guided modes in photonic crystal fibers,” Opt. Lett. 25, 1238–1330 (2000). [CrossRef]

]. On the other hand, the functional form of the dispersion relation of guided modes is very sensitive to the 2D photonic crystal cladding. For this reason, one expects to be able to control, at least to some extent, the dispersion properties of guided modes by manipulating the geometry of the photonic crystal cladding. It was soon realized that PCF’s exhibited dispersion properties very different than those corresponding to ordinary fibers. As an example, some PCF configurations presenting a point of zero dispersion well below the characteristic zero dispersion point of silica at 1.3 µm where found [2

2. D. Mogilevtsev, T. A. Birks, and P. S. J. Russell, “Dispersion of photonic crystal fibers,” Opt. Lett. 23, 1662–1664 (1998). [CrossRef]

, 3

3. M. J. Gander, R. McBride, J. D. C. Jones, D. Mogilevtsev, T. A. Birks, J. C. Knight, and P. S. J. Russell, “Experimental measurement of group velocity dispersion in photonic crystal fibers,” Electron. Lett. 35, 63–64 (1999). [CrossRef]

, 4

4. P. J. Bennet, T. M. Monro, and D. J. Richardson, “Toward practical holey fiber technology: fabrication, splicing, modeling, and fabrication,” Opt. Lett. 24, 1203–1205 (1999). [CrossRef]

], as well as some other showing flattened dispersion profiles [5

5. A. Ferrando, E. Silvestre, J. J. Miret, J. A. Monsoriu, M. V. Andrés, and P. S. J. Russell, “Designing a photonic crystal fibre with flattened chromatic dispersion,” Electron. Lett. 24, 325–327 (1999). [CrossRef]

, 6

6. J. Broeng, D. Mogilevtsev, S. E. Barkou, and A. Bjarklev, “Photonic crys tal fibers : a new clas s of optical waveguides,” Opt. Fib. Tech. 5, 305–330 (1999). [CrossRef]

, 7

7. A. Ferrando, E. Silvestre, J. J. Miret, and P. Andrés, “Nearly zero ultraflattened dispersion in photonic crystal fibers,” Opt. Lett. 25, 790–792 (2000). [CrossRef]

]. Since the number of different photonic crystal configurations is significant, one can deduce that it must be possible to elaborate a procedure to tailor the dispersion of PCF modes in an efficient way. The success in the achievement of such a procedure, that has to be necessarily smart and cannot be based on pure guesses, will ideally provide a useful design tool to determine the PCF geometrical parameters necessary to obtain a desired dispersion profile with specific characteristics. A first approach to design the dispersion properties of PCF’s using a systematic procedure has been already suggested in Ref.[7

7. A. Ferrando, E. Silvestre, J. J. Miret, and P. Andrés, “Nearly zero ultraflattened dispersion in photonic crystal fibers,” Opt. Lett. 25, 790–792 (2000). [CrossRef]

].

Fig. 1. Transformations of the lattice structure with the dimensionless parameters f and M: (a) two structures with different filling fraction f and same magnification M (a/Λ≠a′/Λ); (b) two structures with different magnification M and same filling fraction f (a/Λ=a′/Λ).

2 Designing Procedure

In this paper, we will focus on the dispersion properties of triangular silica-air PCF’s with circular holes, although our procedure can be easily adapted to other geometries and materials. Geometrically, a triangular lattice is characterized by the air-hole radius a and the lattice period, or pitch, Λ. However, in the design procedure we have recognized that it is more convenient to consider two alternate dimensionless parameters instead. First, we consider the so-called filling fraction f, defined as f=(4π/3)(a/Λ)2, that involves the dimensionless ratio a/Λ and provides the proportion of air with respect to silica in the photonic crystal structure. A change in f produces a variation of the amount of air in the structure, as shown in Fig. 1(a). The second parameter we take into account is the magnification M, which simply consists in a simultaneous scale transformation of both a and Λ in the same amount, as shown in Fig. 1(b). In order to define M operatively, it is convenient to select a reference value of the pitch (in our case, w e choose Λ=2.3 µm). The magnification M has also an appealing practical interest. In the pulling process during the fabrication of the fiber, M is the parameter that can be controlled in a natural way. This is so because, under optimal conditions, the pulling process should preserve the proportions of the original structure.

The definition of the dispersion coefficient of a PCF is

Dλcd2neffdλ2,
(1)

where the effective refractive index of the mode is given by n eff=β[λ, n m(λ)]/k 0, β is the propagation constant, k 0=2π/λ is the free-space w ave number, and n m=n m(λ) is the chromatic dispersion of the material, silica in this case. According to the above equation, there are two different sources of obtaining non-zero dispersion due to the existence of two different types of dependence of β on λ. One of them is originated by the explicit dependence of the propagation constant of the mode on λ and it occurs even if the material is, or it can be considered, non-dispersive (n m(λ)=const). Since the dispersion generated in this way is not produced by the chromatic dispersion of the material but by the geometry of the PCF refractive index distribution that determines the dispersion relation of the guided mode, β=β[λ,n m(λ)=const], we call it geometrical dispersion. Its definition is, accordingly, the same as in Eq. (1) but supplemented with the condition that the material is non-dispersive;DgDnm()=const. The second source of dispersion is certainly given by the implicit dependence of β on λ through the chromatic dispersion of the material, n m=n m(λ). Consequently, this type of dispersion is called material dispersion, D m, and we calculate it as in Eq. (1) by substituting n eff(λ) by n m(λ).

Our design procedure is based on the possibility to approximate the real dispersion D by a sum of the geometrical and material dispersion [11

11. D. Davidson, Optical-Fiber Transmission (E. E. Bert Basch, ed., Howard W. Sams & Co, 1987).

];

D(λ)Dg(λ)+Dm(λ).
(2)

The problem of designing the dispersion of a PCF becomes clearer when D is written in this way. The virtue of Eq. (2) is that permits to split both sources of dispersion into two different terms explicitly.

Since we consider air-silica PCF’s, the chromatic dispersion of silica n m(λ) is an input of the problem and consequently, so is D m. All the design power is stored in the geometrical dispersion, In this sense it is very important to recognize the following fact. The effective refractive index of a guided mode n eff, for the calculation of which we assume no material dispersion, explicitly depends on the photonic crystal cladding parameters, a and Λ, and the wavelength λ. Inasmuch as n eff is a dimensionless function, this dependence can only occur through dimensionless ratios of these three parameters. For our discussion, it is convenient to take as independent parameters a/λ and Λ/λ, so that n eff=n eff(a/λ,Λ/λ). This property determines the dependence of D g on M completely. According to the definition of the geometrical dispersion, it is clear that under a scale transformation of λ, we obtain

Dg(λ;M,f)=1MDg(λM;f).
(3)

Consequently, it is enough to calculate the dispersion curve for one reference configuration (fixing the filling fraction f and setting M=1, or equivalently, fixing a and Λ=2.3µm) to analytically obtain all the family of dispersion curves parametrized by M, as shown in Fig. 2(a). The linear part of these curves modifies its slope and it is simulteanously shitfed when M is changed.

On the contrary, there is no simple analytical approach to predict the behavior of D g with the filling fraction f. In practice, the only way to determine this dependence is by calculating D g numerically. We thus start with a reference configuration, e.g. M=1 (i.e., with Λ=2.3 µm) and evaluate the geometrical dispersion curves for different filling fractions f simply by changing a. The result is represented in Fig. 2(b). The remarkable feature of these curves is that, besides they are shifted, the slope of their linear part is approximately preserved when the filling fraction is changed. This property will show to be very helpful in the design process.

The design procedure is better visualized by means of a graphical representation of the geometrical, material and total dispersion. For convenience, the total dispersion is calculated using Eq. (2), but written in a slightly different form,

D(λ)Dg(λ)(Dm(λ)).
(4)
Fig. 2. Dependence of the geometrical dispersion curves on: (a) the magnification M; and (b) the filling fraction f.
Fig. 3. The total dispersion D (red curve) is, in a first-order approximation, the result of substracting the sign-changed material dispersion -D m (black curve) from the geometrical dispersion D g (blue curve). A typical case exhibiting positive ultraflattened dispersion in the 1.55 µm window is obtained.

In Fig. 3, the curves corresponding to the geometrical dispersion D g, the sign-changed material dispersion -D m, and the total dispersion D, are represented in blue, black and red, respectively. According to Eq. (4), the red curve corresponding to total dispersion is obtained by subtracting the values of the black curve from the blue one.

The key factor to achieve this particularly interesting dispersion property is the control of the slope of the linear part of D g. The sign-changed material dispersion -D m is a smooth curve in most of the infrared region, so that it can be well approximated by a linear function around different λ’s belonging to this region over pretty wide intervals. It is clear, in view of Fig. 3, that in the region of λ’s in which the linear part of both the black and blue curves can be set parallel, the total dispersion will achieve an ideally perfect flattened behavior.

The strategy to obtain such a behavior is then straightforward. We start by determining the slope of the black curve at some specific wavelength. In the region where the material dispersion curve is smooth, this slope is approximately the same for a reasonably wide neighborhood around the specified wavelength. Once the slope of the D m curve is fixed, we perform a scale transformation of D g parametrized by the magnification M in such a way that provides an scaled D g curve having a linear region with the same given slope. If the wavelength region (centered at the specified λ) where D m behaves linearly overlaps the wavelength region of linear behavior of D g, we will obtain an ultraflattened total dispersion curve in the overlapping wavelength range.

This process fixes the value of M. However, it remains still one degree of freedom to play with, the filling fraction f. As shown in Fig. 2(b), note that a change in f does not alter the value of the slope of the linear part of D g. Therefore, if we proceed to change the value of f preserving the value of M obtained above, the difference between both curves will change, and also the overlapping range, but the parallelism condition will remain unaltered. This means that one simultaneously modifies the value of the total dispersion D and the width of the wavelength window where ultraflattened behavior occurs by acting on the filling fraction f. Since these two properties are not independent, one has the choice to select f either to obtain a desired value of D or to maximize the range of ultraflattened dispersion operation. In both cases, all possible configurations will provide ultraflattened dispersion profiles.

3 Some Specific Designs

Fig. 4. Ultraflattened dispersion behavior for three different PCF configurations near the communication window with: (a) positive dispersion (a=0.4 µm and Λ=3.12 µm); (b) nearly-zero dispersion (a=0.316 µm and Λ=2.62 µm); and (c) negative dispersion (a=0.27 µm and Λ=2.19 µm). The ultraflattened behavior bandwidth, that corresponds to an allowed dispersion variation of 2 psnm-1 km-1, is668 nm, 523nm, and 411nm, respectively.

The strategy to pursue in such a situation has to be necessarily different altough based on similar ideas. This strategy is based upon the two following observations. The first one is that the value of -D m for silica at 0.8 µm is approximately 120 psnm-1 km-1, a high value to compensate with D g if one is looking for positive or nearly zero total dispersion. The second one is that, in this wavelength range, the curvature of the geometrical and material curves, unlike in the 1.55 µm region, has always opposite signs. This fact is clearly appreciated in Fig. 2, where the curvature of Dg is negative around 0.8 µm in all cases, whereas, according to Fig. 3, the curvature of -D m remains positive even for values of λ beyond the zero material dispersion point at 1.3 µm. The issue now is not to play with the slope of D g, as before, but to be able to achieve values of the geometrical dispersion large enough to compensate for the high value of the sign-changed material dispersion at this wavelength. Our strategy will consist in finding configurations whose geometrical dispersion around 0.8 µm exceeds the value of -D m in the same wavelength region, as shown in Fig. 5. Because of the opposite sign of the curvature of these two curves, it is granted that the profile of the total dispersion will have the form shown by the red curve in Fig. 5. It will include one point of zero third-order dispersion, located at the wavelength for which the difference between the D g and -D m curves reaches a maximum, or, equivalently, at the point for which the negative slopes of both curves are equal. This type of behavior has been already proven to exist in PCF’s, although in a different wavelength window and only for nearly-zero dispersion [5

5. A. Ferrando, E. Silvestre, J. J. Miret, J. A. Monsoriu, M. V. Andrés, and P. S. J. Russell, “Designing a photonic crystal fibre with flattened chromatic dispersion,” Electron. Lett. 24, 325–327 (1999). [CrossRef]

].

Fig. 5. As in Fig. 3 but for a typical case exhibiting positive flattened dispersion in the 0.8 µm window.

We start with our reference configuration curve (that with M=1, or Λ=2.3 µm, in Fig. 2 (a)), whose maximum occurs close to 0.8 µm. At this wavelength, the geometrical dispersion has a small value (D g ≈ 25 psnm-1 km-1) as compared to that of -D m at the same wavelength. The properties depicted in Fig. 2 for D g will guide us in increasing the values of the geometrical dispersion. According to Fig. 5, we will focus on the region of these curves that have negative slope and near maximum. It is clear from Fig. 2(b) that, as we increase f, the maximum of the D g curve moves upwards and simultaneously is shifted to the right. Despite that we are able to increase the value of the maximum of the D g curve, this maximum moves away from the 0.8 µm window. We can relocate the D g curve in such a way the region near maximum moves back to the desired window by acting nowon the magnification M. By reducing M, we simultaneously displace the maximum to shorter wavelengths and increase its value, as depicted in Fig. 2(a). The global effect on the value of the dispersion of this twofold operation is additive, so that we can considerably increase the value of the geometrical dispersion in the 0.8 µm window by a suitable selection first of f (increase) and then of M (decrease). The high value of -D m at 0.8 µm can be in fact overcome, as shown by the positive dispersion curve in Fig. 6.

Fig. 6. Flattened dispersion behavior for three different PCF configurations centered near the Ti-Za window at 0.8 µm: (a) with positive dispersion (a=0.28µm and Λ=0.88µm); (b) with nearly-zero dispersion (a=0.27µm and Λ=0.90µm); and (c) with negative dispersion (a=0.255µm and Λ=0.91µm). The allowed variation of the flattened dispersion profiles is 2 ps nm-1 km-1 and their corresponding flattened dispersion bandwidths are 58 nm, 57nm, and 59nm, respectively.

4 Conclusions

Fig. 7. Four flattened dispersion curves corresponding to different values of the dispersion centered near 1.55 µm. With positive dispersion: (a) D≈+45 psnm -1 km-1 with a=0.49 µm and Λ=2.32 µm, and (b) D ≈ +22 psnm -1 km-1 with a=0.40 µm and Λ=2.71 µm. With negative dispersion: (c) D≈-23 psnm -1 km-1 with a=0.28 µm and Λ=2.16 µm, and (d) D≈-43 psnm -1 km-1 with a=0.27 µm and Λ=1.93 µm. The allowed variation of the flattened dispersion profiles is 2 ps nm-1 km-1 and their corresponding flattened dispersion bandwidths are 270nm, 294 nm, 259 nm, and 195nm, respectively.

This research was supported by the Plan Nacional I+D+I (grant TIC2001-2895-C02-02), Ministerio de Ciencia y Tecnología, Spain.

References and links

1.

A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, and M. V. Andrés, “Donor and acceptor guided modes in photonic crystal fibers,” Opt. Lett. 25, 1238–1330 (2000). [CrossRef]

2.

D. Mogilevtsev, T. A. Birks, and P. S. J. Russell, “Dispersion of photonic crystal fibers,” Opt. Lett. 23, 1662–1664 (1998). [CrossRef]

3.

M. J. Gander, R. McBride, J. D. C. Jones, D. Mogilevtsev, T. A. Birks, J. C. Knight, and P. S. J. Russell, “Experimental measurement of group velocity dispersion in photonic crystal fibers,” Electron. Lett. 35, 63–64 (1999). [CrossRef]

4.

P. J. Bennet, T. M. Monro, and D. J. Richardson, “Toward practical holey fiber technology: fabrication, splicing, modeling, and fabrication,” Opt. Lett. 24, 1203–1205 (1999). [CrossRef]

5.

A. Ferrando, E. Silvestre, J. J. Miret, J. A. Monsoriu, M. V. Andrés, and P. S. J. Russell, “Designing a photonic crystal fibre with flattened chromatic dispersion,” Electron. Lett. 24, 325–327 (1999). [CrossRef]

6.

J. Broeng, D. Mogilevtsev, S. E. Barkou, and A. Bjarklev, “Photonic crys tal fibers : a new clas s of optical waveguides,” Opt. Fib. Tech. 5, 305–330 (1999). [CrossRef]

7.

A. Ferrando, E. Silvestre, J. J. Miret, and P. Andrés, “Nearly zero ultraflattened dispersion in photonic crystal fibers,” Opt. Lett. 25, 790–792 (2000). [CrossRef]

8.

E. Silvestre, M. V. Andrés, and P. Andrés, “Biorthonormal-basis method for the vector description of optical-fiber modes,” J. Lightwave Technol. 16, 923–928 (1998). [CrossRef]

9.

A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, and M. V. Andrés, “Full-vector analysis of a realistic photonic crystal fiber,” Opt. Lett. 24, 276–278 (1999). [CrossRef]

10.

A. Ferrando, E. Silvestre, J. J. Miret, P. Andrés, and M. V. Andrés, “Vector description of higher-order modes in photonic crystal fibers,” J. Opt. Soc. Am. A 17, 1333–1340 (2000). [CrossRef]

11.

D. Davidson, Optical-Fiber Transmission (E. E. Bert Basch, ed., Howard W. Sams & Co, 1987).

OCIS Codes
(060.2270) Fiber optics and optical communications : Fiber characterization
(060.2280) Fiber optics and optical communications : Fiber design and fabrication
(060.2430) Fiber optics and optical communications : Fibers, single-mode

ToC Category:
Focus Issue: Photonic crystal fiber

History
Original Manuscript: November 5, 2001
Published: December 17, 2001

Citation
Albert Ferrando, Enrique Silvestre, Pedro Andres, Juan Miret, and Miguel Andres, "Designing the properties of dispersion-flattened photonic crystal fibers," Opt. Express 9, 687-697 (2001)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-13-687


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References

  1. A. Ferrando, E. Silvestre, J. J. Miret, P. Andres, and M. V. Andres, "Donor and acceptor guided modes in photonic crystal fibers," Opt. Lett. 25, 1238-1330 (2000). [CrossRef]
  2. D. Mogilevtsev, T. A. Birks, and P. S. J. Russell, "Dispersion of photonic crystal fibers," Opt. Lett. 23, 1662-1664 (1998). [CrossRef]
  3. M. J. Gander, R. McBride, J. D. C. Jones, D. Mogilevtsev, T. A. Birks, J. C. Knight, and P. S. J. Russell, "Experimental measurement of group velocity dispersion in photonic crystal fibers," Electron. Lett. 35, 63-64 (1999). [CrossRef]
  4. P. J. Bennet, T. M. Monro, and D. J. Richardson, "Toward practical holey fiber technology: fabrication, splicing, modeling, and fabrication," Opt. Lett. 24, 1203-1205 (1999). [CrossRef]
  5. A. Ferrando, E. Silvestre, J. J. Miret, J. A. Monsoriu, M. V. Andres, and P. S. J. Russell, "Designing a photonic crystal fibre with flattened chromatic dispersion," Electron. Lett. 24, 325-327 (1999). [CrossRef]
  6. J. Broeng, D. Mogilevtsev, S. E. Barkou, and A. Bjarklev, "Photonic crystal fibers: a new class of optical waveguides," Opt. Fib. Tech. 5, 305-330 (1999). [CrossRef]
  7. A. Ferrando, E. Silvestre, J. J. Miret, and P. Andres, "Nearly zero ultraflattened dispersion in photonic crystal fibers," Opt. Lett. 25, 790-792 (2000). [CrossRef]
  8. E. Silvestre, M. V. Andres, and P. Andres, "Biorthonormal-basis method for the vector description of optical-fiber modes," J. Lightwave Technol. 16, 923-928 (1998). [CrossRef]
  9. A. Ferrando, E. Silvestre, J. J. Miret, P. Andres, and M. V. Andres, "Full-vector analysis of a realistic photonic crystal fiber," Opt. Lett. 24, 276-278 (1999). [CrossRef]
  10. A. Ferrando, E. Silvestre, J. J. Miret, P. Andres, and M. V. Andres, "Vector description of higher-order modes in photonic crystal fibers," J. Opt. Soc. Am. A 17, 1333-1340 (2000). [CrossRef]
  11. D. Davidson, Optical-Fiber Transmission (E. E. Bert Basch, ed., Howard W. Sams & Co, 1987).

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