By using a multipole moment approach, we analyze the loss of an air-core photonic crystal fiber and demonstrate that it is possible reduce the transmission loss that is due to photon radiation leakage through the photonic crystal cladding to a level below 0.01 dB/km, with eight rings of air holes. An analogy is drawn between air-core photonic crystal fiber modes and Bragg fiber modes. The influence of material absorption in the silica glass is discussed

Xu, Yong

Yariv, Amnon

English

Caltech Authors - Main

October 15, 2003 / Vol. 28, No. 20 / OPTICS LETTERS 1885Loss analysis of air-core photonic crystal fibers
Yong Xu and Amnon Yariv
Department of Applied Physics, MS 128-95, California Institute of Technology, Pasadena, California 91125Received April 7, 2003
By using a multipole moment approach, we analyze the loss of an air-core photonic crystal fiber and demon-
strate that it is possible reduce the transmission loss that is due to photon radiation leakage through the
photonic crystal cladding to a level below 0.01 dBkm, with eight rings of air holes. An analogy is drawn
between air-core photonic crystal fiber modes and Bragg fiber modes. The inf luence of material absorption
in the silica glass is discussed. © 2003 Optical Society of America
OCIS codes: 060.2280, 230.1480.Recently the use of Bragg ref lection to confine light
in an air-core fiber has attracted considerable atten-
tion.1 – 8 Some advantages of air-core fibers are their
potential to achieve propagation losses of less than
0.2 dBkm (the current value of telecommunication
fibers) and to greatly increase the power threshold
for the onset of nonlinear effects. The air-core f ibers
considered in the literature can be generally classified
into two types: photonic crystal fibers2 –5 (PCFs) and
Bragg fibers.1,6,7,8 Some important advantages of
PCFs are the excellent material properties of the silica
glass and the compatibility of PCFs with the current
optical f iber manufacturing technologies. Yet Bragg
ref lection in an air-core PCF with a finite number
of cladding air holes always leads to some radiation
leakage of light from the fiber core, which is diff icult
to calculate owing to the structural complexity of
the PCF. The method of plane-wave expansion in a
supercell,3 because of an artificially imposed periodic
boundary condition, cannot give the propagation loss.
Many other numerical approaches do not have the
numerical precision to resolve propagation loss of
the order of decibels per kilometer or less. In this
Letter we use newly developed multipole moment
approach4,9 to analyze the modal loss of an air-
core PCF.
As shown in Fig. 1(a), the air-core PCF is formed
by replacing the inner seven air cylinders in a trian-
gular lattice with one bigger air core of radius rco 
3.106 mm. The cladding air cylinders have a lattice
pitch L  2.875 mm, a radius r  1.394 mm, and a
finite number of rings Nring. The index of the silica
glass takes the value 1.45. According to group the-
ory, the symmetry operations of the PCF forms a C6v
point group, and, as a result, the guided modes can
be divided into eight classes.10 For each symmetry
class, the azimuthal dependence of the guided modes is
given by
P
m fmrexpimu, with m  6n 6 m0, where
n  0, 61, 62, . . . and m0 is an integer from 0 and 3
that depends on the symmetry class.10 In this Letter
we consider only the guided modes that are conf ined
within the air core and are dominated by the compo-
nents with small m. As a result, we limit our dis-
cussion to the guided modes that belong to symmetry
classes 1 and 2 (which are nondegenerate with m0  0
and to those of symmetry classes 3 and 4 (which are
degenerate with m0  1).100146-9592/03/201885-03$15.00/0In Fig. 1(b) we give the GM finger diagram for the
fiber shown in Fig. 1(a), and the guided modes with
symmetry classes 1–4. We calculated the GM finger
diagram by counting the number of photonic crystal
cladding modes with the Bloch vector along the GM
direction in reciprocal space.4 In Fig. 2 we plot the z
component of the Poynting vector Pz of the class 1
mode, which is clearly dominated by the m  0 compo-
nent. We can denote the class 1 mode as the TM-like
Fig. 1. (a) Cross section of an air-core photonic crystal
fiber. The dashed line denotes a ring of air cylinders.
(b) GM f inger diagram of the triangular lattice cladding
(with the color density proportional to the number of prop-
agating states) and dispersions of the guided modes with
different symmetry. The straight line is the light line,
which is given by b  vc.© 2003 Optical Society of America
1886 OPTICS LETTERS / Vol. 28, No. 20 / October 15, 2003Fig. 2. Distribution of the z component of the Poynting
vector.
mode, because the m  0 component consists entirely
of the Ez, Hu, and Er field components and resembles
the TM mode of a Bragg fiber.1,6 The class 2 mode,
as shown in Fig. 2, is also dominated by the m  0
component, with a small part being the m  6 compo-
nent. The m  0 component of the class 2 mode re-
sembles the TE mode of a Bragg fiber1,6 and can be
similarly denoted a TE-like mode. The power f lux of
symmetry class 3 and class 4 modes, however, clearly
resembles the fundamental HE11 mode of a step-index
fiber (therefore we denote it HE-like) and consists al-
most entirely of the m  1 component, as required by
group theory.
For an air-core PCF, as in the case of Bragg fibers,8
propagation loss as should decreases exponentially as
the number of air cylinder rings Nring increases: as ~
1DsNring , where s refers to the symmetry class of the
mode and Ds is a constant. In Figs. 3(a)–3(c) we show
the modal propagation loss as a function of Nring at
three wavelengths, together with the exponential fit
and the exponential constant Ds. In every case, the
exponential f it gives an excellent approximation of the
modal loss. As a result, we can extrapolate the modal
loss to a PCF with eight rings of air holes. We find
that at 1.53 mm the loss of the class 3 mode drops below
the level of 0.01 dBkm, a value 20 times below that of
current telecommunication fibers. This value is cal-
culated for an ideal PCF, i.e., without any air-hole de-
formation or fiber nonuniformity, and can be regarded
as a lower limit for the propagation loss in a particular
PCF. This lower limit is much smaller than the cur-
rent experimental record of 13 dBkm,5 which suggests
that the main limiting factor in lowering propagation
loss is imperfection in the PCF.
In previous calculations we ignored the modal
attenuation that is due to optical absorption in the
silica glass. However, as the loss that is due to
photon leakage approaches the level of loss in a con-
ventional silica fiber, it becomes necessary to include
the absorption in the silica glass, which can be taken
into account from the following simple considerations.
We introduce a small imaginary part Imnglass into
the refractive index of the silica glass and calculate
the effective index of the guided mode, neff , defined
as the complex propagation constant divided by the
vacuum wave vector. The imaginary part of the
effective index, Imneff , should have a linear depen-
dence on Imnglass, whose linear coeff icient tells us
the percentage of the optical f ield in the silica glass.
We apply this procedure to the fiber illustrated in
Figs. 3(a)–3(c). We choose three rings of air holes
and use a wavelength of 1.53 mm. The results areshown in Fig. 3(d) and demonstrate excellent linear
dependence between Imneff  and Imnglass. From
a linear f it of the results we find that 13% of the
optical field falls inside the silica glass. Therefore,
for the class 3 mode we can expect that the absorption
loss that is due to the silica glass will be of the order
of 0.03 dBkm, a value comparable with that of the
photon leakage loss.
In the case of Bragg fibers, the propagation loss of
the TE modes is well described by a formula derived
from the intuitive picture of a plane wave bouncing be-
tween two planar Bragg ref lectors.8 A similar result
can be derived for the air-core PCF. First, we notice
that the guided modes shown in Fig. 2 are all domi-
nated by components with angular quantum number
m of either 0 or 1. We can define an effective az-
imuthal wave vector ku  mrco at the interface be-
tween the air core and the photonic crystal cladding
to characterize the angular variation of the electro-
magnetic f ield. For the m  0 and the m  1 com-
ponents at l  1.55 mm, the values of azimuthal wave
vector ku normalized by vacuum wave vector vc are,
respectively, 0 and 0.08. Since ku is smaller than vc,
we can transform the problem of propagation of light
Fig. 3. (a)–(c) Propagation loss of the modes with sym-
metry classes 1–4. Dashed lines are fits assuming an ex-
ponential dependence. (d) Imaginary part of the modal
effective index of the class 3 mode as a function of the
imaginary part of the silica refractive index.
October 15, 2003 / Vol. 28, No. 20 / OPTICS LETTERS 1887Fig. 4. (a) Wave zigzagging between two photonic crystal
stacks. (b) Modal loss of the class 1 mode (TM-like) with
five rings of air holes: circles, numerical results from the
multipole expansion calculations; triangles were obtained
from Eq. (1).
within the PCF air core into that of a wave bounc-
ing between two parallel triangular lattice photonic
crystal stacks with an incident angle cos21cbv[see
Fig. 4(a)], where b is the propagation constant. In
this equivalent picture the incident photon wave vec-
tor is confined within the xz plane. As a result, the
corresponding photon Bloch vector within the triangu-
lar lattice cladding must be along the GM direction in
reciprocal space, which justifies the use of the GM f in-
ger diagram in Fig. 1(b). We emphasize that analysis
is different from most analyses of air-core PCFs in the
literature, in which the use of the complete finger dia-
gram that counts the PCF cladding modes along both
the GM and the GK directions may lead to an under-
estimate of the frequency range of bandgap guiding.
We point out that the existence of confined PCF modes
outside the complete bandgap was shown in Fig. 2 of
Ref. 4.
If we equate the propagation loss of a PCF with
that of its equivalent photonic crystal stack shown in
Fig. 4(a) and assume a transmission coeff icient T for
each bounce, we can approximate the PCF propagation
loss (in decibels per kilometer) as
a 
2.14 3 109
rco
T
∑
1 2
µ
bc
v
∂2∏12
dBkm , (1)
where rco is given in micrometers. Parameter T de-
scribes the transmission through the finite stack of
the photonic crystal cladding and is calculated with
a method described in Ref. 11, which is different from
the previously used multipole moment method. Inas-
much as in general T depends on the polarizationof the incident wave, we define the incident wave in
Fig. 4(a) as TM (TE) polarized if the magnetic (the
electric) field is along the êy direction. In Fig. 4(b) we
show the loss of the TM-like mode in an air-core PCF
calculated from the multipole moment method as well
as from the estimate given by Eq. (1) (assuming a TM
incident wave). The fiber has five rings of air holes;
the rest of the parameters are the same as those used in
the previous calculations. The two results in Fig. 4(b)
agree fairly well with each other, which not only vali-
dates the picture of a wave zigzagging within the air
core but also conf irms the use of the GM finger dia-
gram in the analysis of PCF, because the photon wave
vector shown in Fig. 4(a) samples the Bloch wave vec-
tor only along the GM direction. For the TE-like or
the HE-like modes Eq. (1) does not work so well, a fact
that can be attributed to the presence of both the TE
component and the TM component. For the HE-like
mode we use Eq. (1) to estimate the propagation loss
for both the TE-polarized wave and the TM-polarized
wave. Comparing these results with those obtained
from the multipole moment method, we find that the
loss of the HE-like mode largely follows the estimate
that assumes a TM incident wave. In contrast, the
loss of the TE-like modes varies from the estimates for
which a TE-polarized wave is assumed and those for
which a TM-polarized wave is assumed.
References
1. P. Yeh, A. Yariv, and E. Marom, J. Opt. Soc. Am. 68,
1196 (1978).
2. R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks,
P. St. J. Russell, P. J. Roberts, and D. C. Allan, Science
285, 1537 (1999).
3. J. Broeng, S. E. Barkou, T. Søndergaard, and A.
Bjarklev, Opt. Lett. 25, 96 (2000).
4. T. P. White, R. C. McPhedran, L. C. Botten, G. H.
Smith, and C. M. de Sterke, Opt. Express 9, 721 (2001),
http://www.opticsexpress.com.
5. N. Venkataraman, M. T. Gallagher, D. Müller, C. M.
Smith, J. A. West, K. W. Koch, and J. C. Fajardo, pre-
sented at the European Conference on Optical Commu-
nications, Copenhagen, Denmark, September 12–18,
2002.
6. Y. Xu, R. K. Lee, and A. Yariv, Opt. Lett. 25, 1756
(2000).
7. S. D. Hart, G. R. Maskaly, B. Temelkuran, P. H.
Prideaux, J. D. Joannopoulos, and Y. Fink, Science
296, 510 (2002).
8. Y. Xu, A. Yariv, J. G. Fleming, and S.-Y. Lin, Opt.
Express 11, 1039 (2003), http://www.opticsexpress.com.
9. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D.
Maystre, G. Renversez, C. M. de Sterke, and L. C.
Botten, J. Opt. Soc. Am. B 19, 2322 (2002).
10. P. R. McIssac, IEEE Trans. Microwave Theory Tech.
MTT-23, 421 (1975).
11. G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A.
Nicorovici, Phys. Rev. E 66, 056604 (2002).


Loss analysis of air-core photonic crystal fibers

October 15, 2003 / Vol. 28, No. 20 / OPTICS LETTERS 1885Loss analysis of air-core photonic crystal fibers
Yong Xu and Amnon Yariv
Department of Applied Physics, MS 128-95, California Institute of Technology, Pasadena, California 91125Received April 7, 2003
By using a multipole moment approach, we analyze the loss of an air-core photonic crystal fiber and demon-
strate that it is possible reduce the transmission loss that is due to photon radiation leakage through the
photonic crystal cladding to a level below 0.01 dBkm, with eight rings of air holes. An analogy is drawn
between air-core photonic crystal fiber modes and Bragg fiber modes. The inf luence of material absorption
in the silica glass is discussed. © 2003 Optical Society of America
OCIS codes: 060.2280, 230.1480.Recently the use of Bragg ref lection to confine light
in an air-core fiber has attracted considerable atten-
tion.1 – 8 Some advantages of air-core fibers are their
potential to achieve propagation losses of less than
0.2 dBkm (the current value of telecommunication
fibers) and to greatly increase the power threshold
for the onset of nonlinear effects. The air-core f ibers
considered in the literature can be generally classified
into two types: photonic crystal fibers2 –5 (PCFs) and
Bragg fibers.1,6,7,8 Some important advantages of
PCFs are the excellent material properties of the silica
glass and the compatibility of PCFs with the current
optical f iber manufacturing technologies. Yet Bragg
ref lection in an air-core PCF with a finite number
of cladding air holes always leads to some radiation
leakage of light from the fiber core, which is diff icult
to calculate owing to the structural complexity of
the PCF. The method of plane-wave expansion in a
supercell,3 because of an artificially imposed periodic
boundary condition, cannot give the propagation loss.
Many other numerical approaches do not have the
numerical precision to resolve propagation loss of
the order of decibels per kilometer or less. In this
Letter we use newly developed multipole moment
approach4,9 to analyze the modal loss of an air-
core PCF.
As shown in Fig. 1(a), the air-core PCF is formed
by replacing the inner seven air cylinders in a trian-
gular lattice with one bigger air core of radius rco 
3.106 mm. The cladding air cylinders have a lattice
pitch L  2.875 mm, a radius r  1.394 mm, and a
finite number of rings Nring. The index of the silica
glass takes the value 1.45. According to group the-
ory, the symmetry operations of the PCF forms a C6v
point group, and, as a result, the guided modes can
be divided into eight classes.10 For each symmetry
class, the azimuthal dependence of the guided modes is
given by
P
m fmrexpimu, with m  6n 6 m0, where
n  0, 61, 62, . . . and m0 is an integer from 0 and 3
that depends on the symmetry class.10 In this Letter
we consider only the guided modes that are conf ined
within the air core and are dominated by the compo-
nents with small m. As a result, we limit our dis-
cussion to the guided modes that belong to symmetry
classes 1 and 2 (which are nondegenerate with m0  0
and to those of symmetry classes 3 and 4 (which are
degenerate with m0  1).100146-9592/03/201885-03$15.00/0In Fig. 1(b) we give the GM finger diagram for the
fiber shown in Fig. 1(a), and the guided modes with
symmetry classes 1–4. We calculated the GM finger
diagram by counting the number of photonic crystal
cladding modes with the Bloch vector along the GM
direction in reciprocal space.4 In Fig. 2 we plot the z
component of the Poynting vector Pz of the class 1
mode, which is clearly dominated by the m  0 compo-
nent. We can denote the class 1 mode as the TM-like
Fig. 1. (a) Cross section of an air-core photonic crystal
fiber. The dashed line denotes a ring of air cylinders.
(b) GM f inger diagram of the triangular lattice cladding
(with the color density proportional to the number of prop-
agating states) and dispersions of the guided modes with
different symmetry. The straight line is the light line,
which is given by b  vc.© 2003 Optical Society of America
1886 OPTICS LETTERS / Vol. 28, No. 20 / October 15, 2003Fig. 2. Distribution of the z component of the Poynting
vector.
mode, because the m  0 component consists entirely
of the Ez, Hu, and Er field components and resembles
the TM mode of a Bragg fiber.1,6 The class 2 mode,
as shown in Fig. 2, is also dominated by the m  0
component, with a small part being the m  6 compo-
nent. The m  0 component of the class 2 mode re-
sembles the TE mode of a Bragg fiber1,6 and can be
similarly denoted a TE-like mode. The power f lux of
symmetry class 3 and class 4 modes, however, clearly
resembles the fundamental HE11 mode of a step-index
fiber (therefore we denote it HE-like) and consists al-
most entirely of the m  1 component, as required by
group theory.
For an air-core PCF, as in the case of Bragg fibers,8
propagation loss as should decreases exponentially as
the number of air cylinder rings Nring increases: as ~
1DsNring , where s refers to the symmetry class of the
mode and Ds is a constant. In Figs. 3(a)–3(c) we show
the modal propagation loss as a function of Nring at
three wavelengths, together with the exponential fit
and the exponential constant Ds. In every case, the
exponential f it gives an excellent approximation of the
modal loss. As a result, we can extrapolate the modal
loss to a PCF with eight rings of air holes. We find
that at 1.53mm the loss of the class 3 mode drops below
the level of 0.01 dBkm, a value 20 times below that of
current telecommunication fibers. This value is cal-
culated for an ideal PCF, i.e., without any air-hole de-
formation or fiber nonuniformity, and can be regarded
as a lower limit for the propagation loss in a particular
PCF. This lower limit is much smaller than the cur-
rent experimental record of 13 dBkm,5 which suggests
that the main limiting factor in lowering propagation
loss is imperfection in the PCF.
In previous calculations we ignored the modal
attenuation that is due to optical absorption in the
silica glass. However, as the loss that is due to
photon leakage approaches the level of loss in a con-
ventional silica fiber, it becomes necessary to include
the absorption in the silica glass, which can be taken
into account from the following simple considerations.
We introduce a small imaginary part Imnglass into
the refractive index of the silica glass and calculate
the effective index of the guided mode, neff , defined
as the complex propagation constant divided by the
vacuum wave vector. The imaginary part of the
effective index, Imneff , should have a linear depen-
dence on Imnglass, whose linear coeff icient tells us
the percentage of the optical f ield in the silica glass.
We apply this procedure to the fiber illustrated in
Figs. 3(a)–3(c). We choose three rings of air holes
and use a wavelength of 1.53 mm. The results areshown in Fig. 3(d) and demonstrate excellent linear
dependence between Imneff  and Imnglass. From
a linear f it of the results we find that 13% of the
optical field falls inside the silica glass. Therefore,
for the class 3 mode we can expect that the absorption
loss that is due to the silica glass will be of the order
of 0.03 dBkm, a value comparable with that of the
photon leakage loss.
In the case of Bragg fibers, the propagation loss of
the TE modes is well described by a formula derived
from the intuitive picture of a plane wave bouncing be-
tween two planar Bragg ref lectors.8 A similar result
can be derived for the air-core PCF. First, we notice
that the guided modes shown in Fig. 2 are all domi-
nated by components with angular quantum number
m of either 0 or 1. We can define an effective az-
imuthal wave vector ku  mrco at the interface be-
tween the air core and the photonic crystal cladding
to characterize the angular variation of the electro-
magnetic f ield. For the m  0 and the m  1 com-
ponents at l  1.55 mm, the values of azimuthal wave
vector ku normalized by vacuum wave vector vc are,
respectively, 0 and 0.08. Since ku is smaller thanvc,
we can transform the problem of propagation of light
Fig. 3. (a)–(c) Propagation loss of the modes with sym-
metry classes 1–4. Dashed lines are fits assuming an ex-
ponential dependence. (d) Imaginary part of the modal
effective index of the class 3 mode as a function of the
imaginary part of the silica refractive index.
October 15, 2003 / Vol. 28, No. 20 / OPTICS LETTERS 1887Fig. 4. (a) Wave zigzagging between two photonic crystal
stacks. (b) Modal loss of the class 1 mode (TM-like) with
five rings of air holes: circles, numerical results from the
multipole expansion calculations; triangles were obtained
from Eq. (1).
within the PCF air core into that of a wave bounc-
ing between two parallel triangular lattice photonic
crystal stacks with an incident angle cos21cbv[see
Fig. 4(a)], where b is the propagation constant. In
this equivalent picture the incident photon wave vec-
tor is confined within the xz plane. As a result, the
corresponding photon Bloch vector within the triangu-
lar lattice cladding must be along the GM direction in
reciprocal space, which justifies the use of the GM f in-
ger diagram in Fig. 1(b). We emphasize that analysis
is different from most analyses of air-core PCFs in the
literature, in which the use of the complete finger dia-
gram that counts the PCF cladding modes along both
the GM and the GK directions may lead to an under-
estimate of the frequency range of bandgap guiding.
We point out that the existence of confined PCF modes
outside the complete bandgap was shown in Fig. 2 of
Ref. 4.
If we equate the propagation loss of a PCF with
that of its equivalent photonic crystal stack shown in
Fig. 4(a) and assume a transmission coeff icient T for
each bounce, we can approximate the PCF propagation
loss (in decibels per kilometer) as
a 
2.14 3 109
rco
T
∑
1 2
µ
bc
v
∂2∏12
dBkm , (1)
where rco is given in micrometers. Parameter T de-
scribes the transmission through the finite stack of
the photonic crystal cladding and is calculated with
a method described in Ref. 11, which is different from
the previously used multipole moment method. Inas-
much as in general T depends on the polarizationof the incident wave, we define the incident wave in
Fig. 4(a) as TM (TE) polarized if the magnetic (the
electric) field is along the eˆy direction. In Fig. 4(b) we
show the loss of the TM-like mode in an air-core PCF
calculated from the multipole moment method as well
as from the estimate given by Eq. (1) (assuming a TM
incident wave). The fiber has five rings of air holes;
the rest of the parameters are the same as those used in
the previous calculations. The two results in Fig. 4(b)
agree fairly well with each other, which not only vali-
dates the picture of a wave zigzagging within the air
core but also conf irms the use of the GM finger dia-
gram in the analysis of PCF, because the photon wave
vector shown in Fig. 4(a) samples the Bloch wave vec-
tor only along the GM direction. For the TE-like or
the HE-like modes Eq. (1) does not work so well, a fact
that can be attributed to the presence of both the TE
component and the TM component. For the HE-like
mode we use Eq. (1) to estimate the propagation loss
for both the TE-polarized wave and the TM-polarized
wave. Comparing these results with those obtained
from the multipole moment method, we find that the
loss of the HE-like mode largely follows the estimate
that assumes a TM incident wave. In contrast, the
loss of the TE-like modes varies from the estimates for
which a TE-polarized wave is assumed and those for
which a TM-polarized wave is assumed.
References
1. P. Yeh, A. Yariv, and E. Marom, J. Opt. Soc. Am. 68,
1196 (1978).
2. R. F. Cregan, B. J. Mangan, J. C. Knight, T. A. Birks,
P. St. J. Russell, P. J. Roberts, and D. C. Allan, Science
285, 1537 (1999).
3. J. Broeng, S. E. Barkou, T. Søndergaard, and A.
Bjarklev, Opt. Lett. 25, 96 (2000).
4. T. P. White, R. C. McPhedran, L. C. Botten, G. H.
Smith, and C. M. de Sterke, Opt. Express 9, 721 (2001),
http://www.opticsexpress.com.
5. N. Venkataraman, M. T. Gallagher, D. Müller, C. M.
Smith, J. A. West, K. W. Koch, and J. C. Fajardo, pre-
sented at the European Conference on Optical Commu-
nications, Copenhagen, Denmark, September 12–18,
2002.
6. Y. Xu, R. K. Lee, and A. Yariv, Opt. Lett. 25, 1756
(2000).
7. S. D. Hart, G. R. Maskaly, B. Temelkuran, P. H.
Prideaux, J. D. Joannopoulos, and Y. Fink, Science
296, 510 (2002).
8. Y. Xu, A. Yariv, J. G. Fleming, and S.-Y. Lin, Opt.
Express 11, 1039 (2003), http://www.opticsexpress.com.
9. T. P. White, B. T. Kuhlmey, R. C. McPhedran, D.
Maystre, G. Renversez, C. M. de Sterke, and L. C.
Botten, J. Opt. Soc. Am. B 19, 2322 (2002).
10. P. R. McIssac, IEEE Trans. Microwave Theory Tech.
MTT-23, 421 (1975).
11. G. H. Smith, L. C. Botten, R. C. McPhedran, and N. A.
Nicorovici, Phys. Rev. E 66, 056604 (2002).


presented at

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Loss analysis of air-core photonic crystal fibers

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