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Optics Express

Optics Express

  • Editor: J. H. Eberly
  • Vol. 8, Iss. 10 — May. 7, 2001
  • pp: 547–554
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Analysis of the space filling modes of photonic crystal fibers

Zhaoming Zhu and Thomas G. Brown  »View Author Affiliations


Optics Express, Vol. 8, Issue 10, pp. 547-554 (2001)
http://dx.doi.org/10.1364/OE.8.000547


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Abstract

We study the cladding modes of photonic crystal fibers (PCFs) using a fully vectorial method. This approach enables us to analyze the modes and incorporate material dispersion in a straightforward fashion. We find the field flow lines, intensity distribution and polarization properties of these modes. The effective cladding indices of different PCFs are investigated in detail.

© Optical Society of America

1. Introduction

Photonic crystal fibers (PCFs) are optical fibers which guide light in a defect surrounded by a periodic array of air holes running along the entire length of the fiber. These fibers have been shown to possess numerous unusual properties, including highly tunable dispersion and nonlinearity [1

1. D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, “Group-velocity dispersion in photonic crystal fibers,” Opt. Lett. 23, 1662–1664 (1998). [CrossRef]

,2

2. T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Holey optical fibers: an efficient modal model,” J. Lightwave Technol. 17, 1093–1102 (1999). [CrossRef]

] and single-mode operation at all wavelengths [3

3. T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997). [CrossRef] [PubMed]

]. Such properties are of fundamental interest in optical physics (soliton formation and propagation, for example) and are also of practical importance in the design of increasingly sophisticated broadband optical telecommunications networks.

Two different guiding mechanisms for PCFs have been identified. The first mechanism uses a defect mode in a two-dimensional photonic band gap; the second is analogous to conventional guiding, and relies on a form of total internal reflection. The former utilizes structure which stops propagation in any transverse direction, is typically narrowband, but, in principle, allows light to propagate in the air core [4

4. J. C. Knight, J. Boreng, T. A. Birks, and P. St. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282, 1476–1478 (1998). [CrossRef] [PubMed]

,5

5. R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allen, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999). [CrossRef] [PubMed]

]. The latter achieves a total internal reflection condition because the effective index of the cladding is lower that the dielectric core. This type of PCF, which we consider in this paper, does not need the strict periodicity of air holes or the high air filling ratio required for the existence of a photonic band gap.

2. The Method

H(x,y,z;t)=[Ht(x,y)+Hz(x,y)ẑ]exp(iβziωt)
(1)

2H+k2εH=(lnε)×(×H),
(2)

we get the equation for H t :

LHt=(t2+k2ε)Ht+t(lnε)×(t×Ht)=β2Ht,
(3)

where k=ω/c=2π/λ is the wavevector and ε=ε(x, y) is the transverse dielectric constant profile. We write H t as a column vector

Ht=(HxHy)

then Eq. (3) is the coupled equation for Hx and Hy .

To solve the coupled equations for Hx and Hy , we expand ε(x,y), lnε (x,y) and Ht (x,y) as

ε(x,y)=Gε̂(G)exp(iG·xt),
lnε(x,y)=Gκ̂(G)exp(iG·xt),
Hj(x,y)=GĤj(G)exp(iG·xt),j=x,y.
(4)

where x t = x+ŷ y, and G(l)=lx b x +ly b y is a vector in the reciprocal space. Here lx and ly are any two integers that we denote collectively by l, b x and b y are the primitive vector of the reciprocal lattice. When these expansions are substituted into Eq. (3), it becomes the algebraic eigenvalue problem

GLGGĤ(G)=β2Ĥ(G),
(5)

where LGG are the matrix coefficients of the operator L in the plane-wave basis.

The elements of the matrix L can be written as

[LGG]u,v=[G2δG,G+k2Gε̂(G)]δu,v+[QGG]u,v,(u,v=x,y)
(6)

with

[QGG]x,x=κ̂(GG)(GyGy)Gy,
[QGG]x,y=κ̂(GG)(GyGy)Gx,
[QGG]y,x=κ̂(GG)(GxGx)Gy,
[QGG]y,y=κ̂(GG)(GxGx)Gx.
(7)

When considering the air holes are circular, we obtain the Fourier coefficients ε^(G)and κ^(G)in Eq. (4) as

ε̂(G)={εb+(εaεb)f,G=0(εaεb)f2J1(GR)GR,G0
(8)

and

κ̂(G)={lnεb+(lnεalnεb)f,G=0(lnεalnεb)f2J1(GR)GR,G0,
(9)

where f=πR 2/Ac is the air filling ratio, R is the radius of the air holes, Ac is the area of the primitive cell. εa and εb are the dielectric constants of air and silica, respectively.

3. Cladding modes

We consider PCFs with claddings of triangular lattice of circular air holes, as illustrated in Fig. 1. Λ is the lattice pitch, R is the radius of the air-holes. The two vectors of the primitive cell are a1=(12Λ,32Λ) and a2=(12Λ,32Λ). The area of the primitive cell is Ac=32Λ2. The filling ratio is thus f=2π3(RΛ)2. The primitive vector in the reciprocal space are b1=2πΛ(1,13) and b2=2πΛ(1,13). The dielectric constant of silica εb is calculated using the Sellmeier equation with the parameters for fused silica as given in Ref. 11. The dielectric constant of air εa is assumed to be 1.0 at all wavelengths considered.

Fig. 1. Schematic of the triangular lattice of air holes in the cladding. The lattice pitch is Λ, and R is the radius of the air holes. Dielectric constants of air and silica are εa =1.0 and εb (λ), respectively.

The fundamental cladding mode (space filling mode) is defined to be the mode that exhibits the highest effective index. We use the method described in the above section and calculated the dispersion curves for the several lower order cladding modes, as shown in Fig. 2. The chosen parameters correspond to a previously investigated sample [10

10. 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]

]: Λ=2.3 µm, R=0.3 µm. The fundamental cladding mode is a two-fold degenerate mode. Fig. 3 shows the transverse magnetic fields for these two degenerate modes at λ=1.5 µm. We can see that they are essentially linearly polarized, with minute deviation from linear polarization near silica-air interfaces. In Fig. 4 we show the transverse magnetic field intensity of the x-polarized fundamental mode. As expected, the field is strongly concentrated in the high dielectric region (silica).

Fig. 2. Dispersions of the some lower cladding modes. The fundamental mode (FCM) is a two-fold degenerate mode. The next lower mode (HCM) includes six nearly degenerate modes. The material dispersion of the silica core is also shown in the figure.
Fig. 3. Transverse magnetic fields of the two degenerate modes of the fundamental cladding mode.

The effective cladding index neff,cl is a function of wavelength λ, dielectric constant of silica εb , lattice pitch Λ, and air hole radius R. Due to fact that Maxwell’s equations have no fundamental length scale, neff,cl is in fact a function of εb and normalized parameters λ/Λ and R/Λ. In Fig. 6 we show a surface plot of neff,cl . The material dispersion of silica is considered in the calculations.

Fig. 4. Transverse magnetic field intensity of the fundamental cladding mode (x-polarized).
Fig. 5. Dispersion curves calculated using the present vectorial method (solid lines) and the scalar method (dashed lines) for claddings of different air-hole radius R.

Once the effective cladding indices are determined, one can use them to determine the width of the spectral range over which a PCF is single-mode [3

3. T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997). [CrossRef] [PubMed]

]. The V parameter of the PCF defined as V=kΛ(ncore2 -neff,cl2 )1/2 gives a good estimation of the number of guided modes inside the PCF. Fig. 7 shows how V varies with wavelength for PCFs of different lattice pitches and air-filling ratios. We can see that the V is well below the red dashed line (which corresponds to the single-mode cutoff value for traditional step-index optical fibers) for PCFs with small air holes. It is clearly that PCFs with relatively small air holes can be single mode over a very large spectral range. To appreciate the difference between the vectorial and scalar methods, we also show the results from the scalar approximation in Fig. 7(a). The relative difference between the two approaches reaches about 10% in the results shown. So, in order to accurately predict the single-mode behavior of PCFs (especially near the single-mode cutoff), it is advisable to use the vectorial method.

Fig. 6. Effective cladding index as a function of λ/Λ and R/Λ. Λ is chosen to be 2.3 µm and the material dispersion of silica is considered in the calculations.
Fig. 7. Calculated parameter V vs. wavelength for claddings of different pitches and air-hole radii. The red dashed line indicates the single-mode cutoff value of V (=2.405). (a) Lattice pitch Λ=2.3 µm. Results from the vectorial (scalar) method are shown by blue (black) solid lines. (b) Lattice pitch Λ=5.0 µm. Only the vectorial results are shown.

In all the numerical calculations, a total of 289 plane waves was used. We estimate that the relative error in the calculated effective cladding index is less than 0.06% at λ=2 µm and R=0.8 µm. More plane waves are needed to keep the accuracy for increasing wavelength λ and air-hole radius R.

4. Discussion and Conclusion

The method presented in this paper has broad applicability beyond the analysis of cladding modes. In fact, under the so-called supercell approximation, our approach is readily used in characterization of PCFs with various cladding and core structures, such as irregular air-hole cladding PCFs and multiple-core PCFs. Such structures can yield either accidental birefringence (due to disorder, perhaps) or may be engineered for very high birefringence (for polarization compensation). The presented method provides a full vectorial and efficient tool in modeling PCFs. It should be emphasized that the localized function method used in Refs. [2

2. T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Holey optical fibers: an efficient modal model,” J. Lightwave Technol. 17, 1093–1102 (1999). [CrossRef]

,12

12. D. Mogilevtsev, T. A. Birks, and P. St. J. Russell“Localized function method for modeling defect modes in 2-d photonic crystals,” J. Lightwave Technol.17, 2078–2081 (1999);T. M. Monro, D. J. Richardson, N.G. R. Broderick, and P. J. Bennett, “Modeling large air fraction holey optical fibers,” J. Lightwave Technol.18, 50–56 (2000). [CrossRef]

] incorporates material dispersion easily as we do. However, this method is incapable of calculating the effective cladding index of PCFs, because it makes use of the fact that the fields of guided modes are localized around the defect core.

In summary, we present a rigorous and efficient method to analyze the cladding modes in PCFs. It overcomes the limitation of general plane-wave-expansion based techniques that material dispersions are not easily taken into account. Using this approach, we analyze the cladding modes in PCFs with different air-filling ratios. Comparisons with scalar approximation are also given.

References and links

1.

D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, “Group-velocity dispersion in photonic crystal fibers,” Opt. Lett. 23, 1662–1664 (1998). [CrossRef]

2.

T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Holey optical fibers: an efficient modal model,” J. Lightwave Technol. 17, 1093–1102 (1999). [CrossRef]

3.

T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997). [CrossRef] [PubMed]

4.

J. C. Knight, J. Boreng, T. A. Birks, and P. St. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282, 1476–1478 (1998). [CrossRef] [PubMed]

5.

R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allen, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999). [CrossRef] [PubMed]

6.

M. Midrio, M. P. Singh, and C. G. Someda, “The space filling mode of holey fibers: an analytical vectorial solution,” J. Lightwave Technol. 18, 1031 (2000). [CrossRef]

7.

A. A. Maradudin and A. R. McGurn, “Out of plane propagation of electromagnetic waves in two-dimensional periodic dielectric medium,” J. Mod. Opt. 41, 275–284 (1994). [CrossRef]

8.

J. Broeng, T. Sondergaard, S. E. Barkou, P. M. Barbeito, and A. Bjarklev, “Waveguidance by the photonic bandgap effect in optical fiber,” J. Opt. A—Pure Appl. Opt. 1, 477–482 (1999). [CrossRef]

9.

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]

10.

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]

11.

G. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1995).

12.

D. Mogilevtsev, T. A. Birks, and P. St. J. Russell“Localized function method for modeling defect modes in 2-d photonic crystals,” J. Lightwave Technol.17, 2078–2081 (1999);T. M. Monro, D. J. Richardson, N.G. R. Broderick, and P. J. Bennett, “Modeling large air fraction holey optical fibers,” J. Lightwave Technol.18, 50–56 (2000). [CrossRef]

OCIS Codes
(060.2270) Fiber optics and optical communications : Fiber characterization
(060.2400) Fiber optics and optical communications : Fiber properties

ToC Category:
Research Papers

History
Original Manuscript: March 22, 2001
Published: May 7, 2001

Citation
Zhaoming Zhu and Thomas Brown, "Analysis of the space filling modes of photonic crystal fibers," Opt. Express 8, 547-554 (2001)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-10-547


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References

  1. D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, "Group-velocity dispersion in photonic crystal fibers," Opt. Lett. 23, 1662-1664 (1998). [CrossRef]
  2. T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, "Holey optical fibers: an efficient modal model," J. Lightwave Technol. 17, 1093-1102 (1999). [CrossRef]
  3. T. A. Birks, J. C. Knight, and P. St. J. Russell, "Endlessly single-mode photonic crystal fiber," Opt. Lett. 22, 961-963 (1997). [CrossRef] [PubMed]
  4. J. C. Knight, J. Boreng, T. A. Birks, and P. St. J. Russell, "Photonic band gap guidance in optical fibers," Science 282, 1476-1478 (1998). [CrossRef] [PubMed]
  5. R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks, P. St. J. Russell, P. J. Roberts, and D. C. Allen, "Single-mode photonic band gap guidance of light in air," Science 285, 1537-1539 (1999). [CrossRef] [PubMed]
  6. M. Midrio, M. P. Singh, and C. G. Someda, "The space filling mode of holey fibers: an analytical vectorial solution," J. Lightwave Technol. 18, 1031 (2000). [CrossRef]
  7. A. A. Maradudin and A. R. McGurn, "Out of plane propagation of electromagnetic waves in two-dimensional periodic dielectric medium," J. Mod. Opt. 41, 275-284 (1994). [CrossRef]
  8. J. Broeng, T. Sondergaard, S. E. Barkou, P. M. Barbeito, and A. Bjarklev, "Waveguidance by the photonic bandgap effect in optical fiber," J. Opt. A - Pure Appl. Opt. 1, 477-482 (1999). [CrossRef]
  9. 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]
  10. 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]
  11. G. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1995).
  12. D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, "Localized function method for modeling defect modes in 2-d photonic crystals," J. Lightwave Technol. 17, 2078-2081 (1999); T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, "Modeling large air fraction holey optical fibers," J. Lightwave Technol. 18, 50-56 (2000). [CrossRef]

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