- Email: [email protected]
Analysis of single polarization single mode photonic crystal cladding with hexagonal and rectangular lattice
Accepted Manuscript Title: Analysis of Single Polarization Single Mode Photonic Crystal Cladding with Hexagonal and Rectangular Lattice Author: Shahiruddin Dharmendra K. Singh Naman Agarwal PII: DOI: Reference:
S0030-4026(15)00799-8 http://dx.doi.org/doi:10.1016/j.ijleo.2015.08.038 IJLEO 55962
To appear in: Received date: Accepted date:
22-7-2014 18-8-2015
Please cite this article as: Shahiruddin, D.K. Singh, Analysis of Single Polarization Single Mode Photonic Crystal Cladding with Hexagonal and Rectangular Lattice, Optik - International Journal for Light and Electron Optics (2015), http://dx.doi.org/10.1016/j.ijleo.2015.08.038 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Analysis of Single Polarization Single Mode Photonic Crystal Cladding with Hexagonal and Rectangular Lattice
ip t
Shahiruddin ([email protected]), Dharmendra K Singh
cr
([email protected]), Naman Agarwal ([email protected])
us
Department of Electronics and Communication Engineering, National Institute of Technology, Patna-800005, India
an
Corresponding Author: Telephone: +91 9835311885 (Shahiruddin)
M
Email address: [email protected]
d
ABSTRACT
te
Single-polarization single-mode (SPSM) operation of highly birefringent (HB)
Ac ce p
photonic crystal fibers (PCF) with rectangular and hexagonal lattice is investigated in detail by using a full-vector finite-element method (FEM) with anisotropic perfectly matched layers (PMLs). The position of the region of single polarization can be tuned freely by adjusting the size of the central enlarged air holes. The confinement loss and dispersion losses for standard single-mode fiber for particular SPSM PCFs are calculated and optimized. The proposed fibers are nonlinear SPSM, which may be useful for nonlinear optical applications or applications with a wide SPSM operating bandwidth requirement.
Page 1 of 17
Index Terms—Confinement loss, dispersion, finite-element method, photonic crystal fiber, single-polarization single-mode fiber.
ip t
INTRODUCTION
cr
Photonic crystal fibers (PCFs) have attracted significant attention recently [1]–[4]. A
us
PCF consists of a central core of air or pure silica surrounded by an array of air holes running along its length. PCFs may be divided into two categories: the index [2], [3]
an
guiding PCF, where light is guided by modified total internal reflection
, and
the bandgap guiding PCF, where light guiding is based on the effect of photonic
M
bandgap (PBG) within which light propagation is prohibited [4].
d
SPSM fibers were successfully designed by using a highly elliptical core or a bow
te
tie structure or by introducing absorbent material along the core. The broadest
Ac ce p
single polarization bandwidth is about 180 nm
[5]
, which may not be enough for
some practical applications. To make this kind of fiber, a very large refractiveindex difference between the core and cladding region is required. This may be the reason why it is difficult to realize wideband SPSM operation in conventional fibers. In recent years, photonic crystal fibers (PCFs)
[6], [7]
consisting of a core
surrounded by a cladding region with multiple air holes have attracted significant attention. This kind of optical fiber possess many unique properties such as a wide single-mode wavelength range
[8]
and anomalous group velocity dispersion
[9]–[12]
Page 2 of 17
compared with traditional fibers. By having different air-hole diameters along the two orthogonal axes or changing the air-hole arrangement in the cladding region,
contrast is higher than that in conventional fibers
[13]
ip t
PCFs with high modal birefringence can be easily achieved, because the index . Up to now, various types of [14], [15]
cr
highly birefringent PCFs with a modal birefringence have been proposed
.
us
Recently, it has been shown that highly birefringent PCFs have the potential to
an
achieve wideband SPSM fibers.
In this paper, we numerically explored the single polarization operation for two
M
different HB PCFs, one with rectangular lattice and other one with hexagonal
d
lattice structure. The effects of varying PCF parameters on the characteristics of
te
SPSM operation are discussed and general design criteria are given in Section II. A
Ac ce p
full-vector finite-element method (FEM) with perfectly matched layers (PMLs) as absorbing boundary conditions is used to analyze the polarization-dependent loss. DESIGN OF SPSM PCF
The structure of the hexagonal lattice HB PCF to achieve single-polarization operation is assumed to be as shown in Figure 1. It should be noticed that PCF with similar cross-sections can be HB because of different air hole diameters along the orthogonal axis
[6], [7]
. It is thus meaningful to find the proper PCF parameters that
support only a single linearly polarized mode. The background refractive index
Page 3 of 17
was obtained from the Sellmeier equation for silica
[19]
. Designed single
polarization fibre has pitch = 2.2 µm, hole diameter to pitch ratio for smaller holes
d
M
an
us
cr
ip t
to be 0.40 and for larger holes to be 0.95.
Ac ce p
te
Figure 1: Field intensity distribution of the SPSM PCF.
Figure 2: Effective index curve as a function of wavelength with ʌ =2.2µm, d1/ʌ=0.40 and d2/ʌ=0.95.
Page 4 of 17
Figure 2 shows the effective index of the x- and y-polarization mode as a function of wavelength for the PCF with ʌ = 2.2 μm, d1/ʌ = 0.40, and d2/ʌ = 0.95. Also
filling mode (SFM)
[3]
ip t
shown in Figure 2 is the effective index (cladding index) of the fundamental space , which is evaluated by applying FEM to the elementary
cr
piece. Polarization cutoff occurs when the effective index of one of the polarization
us
states falls below that of the SFM [14].
an
We know that it is possible to tune the wavelength range for single-polarization operation by varying the fiber structural parameters: d1/ʌ, d2/ʌ, and ʌ. Therefore, it
M
is worthwhile to examine the dispersion properties under the influence of these
Ac ce p
te
0.75 as shown in Figure 3.
d
parameters. First, we fixed d1/ʌ at 0.4 and ʌ at 2 μm and changed d2/ʌ from 0.95 to
Figure 3: Cut-off wavelength as a function of hole diameter to pitch ratio. Tuning the cutoff wavelength by changing the d2/ʌ ratio is possible but at the cost of the reduction of single-polarization wavelength range from 220 to 135 nm for d2/ʌ = 0.95 to 0.75. The single polarization wavelength length reduction can be
Page 5 of 17
attributed to the reduced birefringence induced by decreasing the diameter of the two big air holes. Therefore, d2/ʌ should be set as high as possible to attain a wider
ip t
single-polarization wavelength range. d2/ʌ = 0.95 is a reasonable parameter for practical PCFs. The hole pitch Λ is then varied from 1.6 to 2.4 μm in steps of 0.4
cr
μm, where d1/ʌ and d2/ʌ are fixed at 0.40 and 0.95. The dispersion curves are
us
shown in Figure 4. The cutoff wavelength for x- and y-polarization shifts to longer wavelength with the increase of pitches, and the single-polarization wavelength
te
d
M
an
range increases slightly.
Ac ce p
(a): d1/ʌ=0.40 and d2/ʌ=0.95 for ʌ= 1.6 µm
(b): d1/ʌ=0.40 and d2/ʌ=0.95 for ʌ= 2 µm
(c): d1/ʌ=0.40 and d2/ʌ=0.95 for ʌ= 2.4 µm
Figure 4: Cut-off wavelength as a function of pitch with d1/ʌ=0.40 and d2/ʌ=0.95 for ʌ= 1.6 µm, 2 µm and 2.4 µm.
Page 6 of 17
The structure of the rectangular lattice HB PCF to achieve single-polarization operation is assumed to be as shown in Figure 5. The reported structure is a
ip t
rectangular lattice where six enlarged central air holes are used, three holes each in both rows around the core of the fibre. Here two different pitches are used, i.e.,
cr
hole to hole spacing is different in x and y direction, ʌ is pitch or hole to hole
Ac ce p
te
d
M
an
diameter is d1 and enlarged air hole diameter is d2.
us
spacing in x-direction and ʌy is hole to hole spacing in y direction. Small hole
Figure 5: Field intensity distribution of the SPSM PCF.
The effective refractive index graph for orthogonal polarization as a function of wavelength for a photonic crystal fibre have been shown in Figure 6 with ʌ =2.2µm, ʌ y =0.8ʌ, d1/ʌ= 0.50 and d2/ʌ=0.85.
Page 7 of 17
There are two separate curves for x-polarization FSM and y-polarization FSM because air holes are arranged in rectangular lattice and the two FSM’s are no
an
us
cr
ip t
longer degenerate.[15]
Figure 6: Effective index curve as a function of wavelength with ʌ = 2.2µm,
te
CONFINEMENT LOSS
d
M
ʌy= 0.8ʌ, d1/ʌ=0.50 and d2/ʌ=0.85.
Confinement loss can be defined as the light confinement ability within the core
Ac ce p
region and it usually occurs in single material fibres. Since the cladding of the photonic crystal fibre is obviously made of a limited number of air holes rings, the guided modes in fibre have a inherent property of leaking out. The imaginary part of the effective refractive index leads to the calculation of the confinement loss by using equation (i). (i)
Page 8 of 17
The Hexagonal fibre is analyzed at 1.3 & 1.5 µm, single polarization window. The photonic crystal fibre parameters are varied and confinement loss calculated for
ip t
each of them and plots are generated which are shown in Figure 7 and Figure 8. In the given plots, different air fill fractions and different pitches are used and their
M
an
us
cr
confinement loss is given.
(b): d1/ʌ=0.40 & pitch = 2.1 µm
Ac ce p
te
d
(a): d1/ʌ=0.50 & pitch = 1.2 µm
(c): d1/ʌ=0.35 & pitch = 3.2 µm
(d): d1/ʌ=0.30 & pitch = 9 µm
Figure 7: Confinement Loss of SPSM PCF for x & y polarization at 1.3 µm with different pitch and hole diameter ratio.
Page 9 of 17
ip t
(b): d1/ʌ=0.40 & pitch = 2.48 µm
an
us
cr
(a): d1/ʌ=0.50 & pitch = 1.46 µm
(d): d1/ʌ=0.30 & pitch = 10.8 µm
M
(c): d1/ʌ=0.35 & pitch = 3.85 µm
d
Figure 8: Confinement Loss of SPSM PCF for x & y polarization at 1.5 µm
te
with different pitch and hole diameter ratio.
Ac ce p
Now, confinement loss of the rectangular lattice single mode single polarization photonic crystal fibre is investigated. SPSM bandwidth can be increased by increasing the pitch or the crystal constant in the photonic crystal fibre. Single polarization over the whole communication band can be achieved by having higher pitch and air filling ratio. Thus, the impact of pitch variations and hole diameter of enlarged holes are studied. The different pitches analyzed are 2.5, 2.6 and 2.7 µm. Different d2/ʌ ratio’s investigated are
and is shown in Figure 9
and Figure 10. It can be observed that if d2/ʌ decreases then confinement loss of xpolarization also decreases.
Page 10 of 17
ip t cr
(a): x-polarization
us
(b): y-polarization
Figure 9: Influence of Variation of d2/ʌ =0.91, 0.90 and 0.89 for x-polarization
Ac ce p
te
d
M
an
and y-polarization.
(a): x-polarization
(b): y-polarization
Figure 10: Influence of variation of pitch with ʌ = 2.5, 2.6 and 2.7 µm for x and y-polarization.
DISPERSION ANALYSIS
Dispersion is the name given to broadening of pulse while propagating through the fibre. The unit of dispersion is ps/(nm-km), that shows elongation in pulse time with certain spectral width, in nm after a propagation length of 1 km.
Page 11 of 17
Dispersion can be calculated by using equation (ii)
ip t
HEXAGONAL LATTICE: The graph showed in Figure 11 is the dispersion vs
cr
wavelength graph for the hexagonal lattice single mode single polarization photonic crystal fibre with pitch = 2.2 µm, hole diameter to pitch ratio of smaller
Ac ce p
te
d
M
an
that we have very low dispersion at 1.55 µm.
us
holes to be 0.40 and 0.95 for the enlarged holes. From the graph, it can be observed
Figure 11: Dispersion graph for ʌ =2.2
, d1/ʌ = 0.4 and d2/ʌ= 0.95.
We further reduced the air filling fraction of the enlarged central air holes to study the impact of variation on the dispersion behavior of the SPSM PCF. The hole diameter to pitch ratio for smaller holes is kept at 0.4 and pitch is 2.2 µm. The d2/ʌ ratio is reduced to 0.75. It is observed from Figure 12 that small negative dispersion can be achieved using the above mention crystal parameters for the
Page 12 of 17
single polarization operation. The negative dispersion counter effects the
an
us
cr
ip t
dispersion losses.
, d1/ʌ = 0.4 and d2/ʌ= 0.75.
M
Figure 12: Dispersion graph for ʌ =2.2
Now, the air fill fraction of the enlarged air holes is kept at 0.95 but for smaller
d
holes, it is changed to 0.5, thereby increasing the air filling fraction of the cladding.
te
Pitch is reduced to 1.25 µm. It is observed that very low dispersion can be
Ac ce p
achieved at 1.3 and 1.6µm. Thus, single polarization operation can be achieved with very low dispersion as shown in Figure 13.
Figure 13: Dispersion graph for ʌ =1.25
, d1/ʌ = 0.5 and d2/ʌ= 0.95.
Page 13 of 17
RECTANGULAR LATTICE The graph showed in the Figure 14 is the dispersion vs wavelength graph for the
, ʌy =0.8 ʌ, d1/ʌ = 0.5 and d2/ʌ= 0.85.
M
Figure 14: Dispersion graph for ʌ =2.2
an
us
cr
ip t
square lattice single mode single polarization photonic crystal fibre.
d
It can be easily observed that very low dispersion values can be achieved in the
te
single polarization single mode wavelength range of the given photonic crystal
Ac ce p
fibre. Low values of dispersion are found from 1.35 µ to 1.8 µm. Now the pitch is increased to 2.3 µm and air filling fraction of larger holes is increased to 0.92. Other parameters are kept unchanged. The effect of increased pitch and air filling fraction on dispersion behavior of the fibre are investigated. It can be observed from Figure 15 that low dispersion values are achieved at 1.65 µm to higher wavelengths.
Page 14 of 17
ip t cr , ʌy =0.8 ʌ, d1/ʌ = 0.5 and d2/ʌ= 0.92.
an
CONCLUSIONS
us
Figure 15: Dispersion graph for ʌ =2.3
M
The properties of the photonic crystal fibres are very flexible. By introducing very small variations in the design parameters in the structure of the fibre, the impact is
d
immediately visible in the confinement loss and dispersion behavior of the
te
photonic crystal fibre. Thus, it can be concluded that by bringing small variations
Ac ce p
in the PCF structure, desired dispersion profiles with low confinement loss can be obtained and implemented in many areas with specific dispersion requirements. REFERENCES
[1] J.C. Knight, P.S.J. Russell, Photonic Crystal Fibers: New Way to Guide Light, Science 296 (2002) 276–277.
[2] J.C. Knight, T.A. Birks, P.S.J. Russell, D.M. Atkin, All-silica Single-mode Optical Fiber with Photonic Crystal Cladding, Opt. Lett. 21 (1996) 1547–1549.
Page 15 of 17
[3] T.A. Birks, J.C. Knight, P.S.J. Russell, Endlessly Single-mode Photonic Crystal Fiber, Opt. Lett. 22 (1997) 961–963.
ip t
[4] J.C. Knight, J. Broeng, T.A. Birks, P.S.J. Russell, Photonic Band gap Guidance in Optical Fibers, Science 282 (1998) 1476–1478.
cr
[5] M.J. Messerly, J.R. Onstott, R.C. Mikkelson, A Broad-band Single Polarization
us
Optical Fiber, Journal of Lightwave Technology 9 (1991) 817–820.
[6] S. Furukawa, T. Fujimoto, T. Hinata, Propagation Characteristics of a Single-
an
polarization Optical Fiber with an Elliptic Core and Triple-clad, Journal of
M
Lightwave Technology 21 (2003) 1307–1312.
[7] M.J. Gander, R. McBride, J.D.C. Jones, D. Mogilevtsev, T.A. Birks, J.C.
d
Knight, P.S.J. Russell, Experimental Measurement of Group Velocity Dispersion
te
in Photonic Crystal Fibre, Electron. Lett. 35 (1999) 63–64.
Ac ce p
[8] W. Reeves, J. Knight, P. Russell, P. Roberts, Demonstration of Ultraflattened Dispersion in Photonic Crystal Fibers, Opt. Express 10 (2002) 609–613. [9] K. Saitoh, N. Florous, M. Koshiba, Ultra-flattened Chromatic Dispersion Controllability using a Defected-core Photonic Crystal Fiber with low Confinement Losses, Opt. Express 13 (2005) 8365–8371. [10] A. Ortigosa-Blanch, J.C. Knight, W.J. Wadsworth, J. Arriaga, B.J. Mangan, T.A. Birks, P.S.J. Russell, Highly Birefringent Photonic Crystal Fibers, Opt. Lett. 25 (2000) 1325–1327.
Page 16 of 17
[11] K. Suzuki, H. Kubota, S. Kawanishi, M. Tanaka, M. Fujita, Optical Properties of a Low-loss Polarization-maintaining Photonic Crystal Fiber, Opt. Express 9
ip t
(2001) 676–680. [12] K. Saitoh, M. Koshiba, Single-polarization Single-mode Photonic Crystal
cr
Fibers, IEEE Photon. Technol. Lett. 15 (2003) 1384–1386.
us
[13] H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, S. Yamaguchi, Absolutely Single Polarization Photonic Crystal Fiber, IEEE Photon. Technol. Lett. 16 (2004)
an
182–184.
M
[14] F. Zhang, M. Zhang, X. Liu, P. Ye, Design of Wideband Single-Polarization
1184-1189.
d
Single-Mode Photonic Crystal Fiber, Journal of Lightwave Technology 25 (2007)
te
[15] J. Ju, J. Wei, M.S. Demokan, Design of Single-Polarization Single-Mode
Ac ce p
Photonic Crystal Fiber at 1.30 and 1.55 μm, Journal of Lightwave Technology 24 (2006) 825-830.
Page 17 of 17
S0030-4026(15)00799-8 http://dx.doi.org/doi:10.1016/j.ijleo.2015.08.038 IJLEO 55962
To appear in: Received date: Accepted date:
22-7-2014 18-8-2015
Please cite this article as: Shahiruddin, D.K. Singh, Analysis of Single Polarization Single Mode Photonic Crystal Cladding with Hexagonal and Rectangular Lattice, Optik - International Journal for Light and Electron Optics (2015), http://dx.doi.org/10.1016/j.ijleo.2015.08.038 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Analysis of Single Polarization Single Mode Photonic Crystal Cladding with Hexagonal and Rectangular Lattice
ip t
Shahiruddin ([email protected]), Dharmendra K Singh
cr
([email protected]), Naman Agarwal ([email protected])
us
Department of Electronics and Communication Engineering, National Institute of Technology, Patna-800005, India
an
Corresponding Author: Telephone: +91 9835311885 (Shahiruddin)
M
Email address: [email protected]
d
ABSTRACT
te
Single-polarization single-mode (SPSM) operation of highly birefringent (HB)
Ac ce p
photonic crystal fibers (PCF) with rectangular and hexagonal lattice is investigated in detail by using a full-vector finite-element method (FEM) with anisotropic perfectly matched layers (PMLs). The position of the region of single polarization can be tuned freely by adjusting the size of the central enlarged air holes. The confinement loss and dispersion losses for standard single-mode fiber for particular SPSM PCFs are calculated and optimized. The proposed fibers are nonlinear SPSM, which may be useful for nonlinear optical applications or applications with a wide SPSM operating bandwidth requirement.
Page 1 of 17
Index Terms—Confinement loss, dispersion, finite-element method, photonic crystal fiber, single-polarization single-mode fiber.
ip t
INTRODUCTION
cr
Photonic crystal fibers (PCFs) have attracted significant attention recently [1]–[4]. A
us
PCF consists of a central core of air or pure silica surrounded by an array of air holes running along its length. PCFs may be divided into two categories: the index [2], [3]
an
guiding PCF, where light is guided by modified total internal reflection
, and
the bandgap guiding PCF, where light guiding is based on the effect of photonic
M
bandgap (PBG) within which light propagation is prohibited [4].
d
SPSM fibers were successfully designed by using a highly elliptical core or a bow
te
tie structure or by introducing absorbent material along the core. The broadest
Ac ce p
single polarization bandwidth is about 180 nm
[5]
, which may not be enough for
some practical applications. To make this kind of fiber, a very large refractiveindex difference between the core and cladding region is required. This may be the reason why it is difficult to realize wideband SPSM operation in conventional fibers. In recent years, photonic crystal fibers (PCFs)
[6], [7]
consisting of a core
surrounded by a cladding region with multiple air holes have attracted significant attention. This kind of optical fiber possess many unique properties such as a wide single-mode wavelength range
[8]
and anomalous group velocity dispersion
[9]–[12]
Page 2 of 17
compared with traditional fibers. By having different air-hole diameters along the two orthogonal axes or changing the air-hole arrangement in the cladding region,
contrast is higher than that in conventional fibers
[13]
ip t
PCFs with high modal birefringence can be easily achieved, because the index . Up to now, various types of [14], [15]
cr
highly birefringent PCFs with a modal birefringence have been proposed
.
us
Recently, it has been shown that highly birefringent PCFs have the potential to
an
achieve wideband SPSM fibers.
In this paper, we numerically explored the single polarization operation for two
M
different HB PCFs, one with rectangular lattice and other one with hexagonal
d
lattice structure. The effects of varying PCF parameters on the characteristics of
te
SPSM operation are discussed and general design criteria are given in Section II. A
Ac ce p
full-vector finite-element method (FEM) with perfectly matched layers (PMLs) as absorbing boundary conditions is used to analyze the polarization-dependent loss. DESIGN OF SPSM PCF
The structure of the hexagonal lattice HB PCF to achieve single-polarization operation is assumed to be as shown in Figure 1. It should be noticed that PCF with similar cross-sections can be HB because of different air hole diameters along the orthogonal axis
[6], [7]
. It is thus meaningful to find the proper PCF parameters that
support only a single linearly polarized mode. The background refractive index
Page 3 of 17
was obtained from the Sellmeier equation for silica
[19]
. Designed single
polarization fibre has pitch = 2.2 µm, hole diameter to pitch ratio for smaller holes
d
M
an
us
cr
ip t
to be 0.40 and for larger holes to be 0.95.
Ac ce p
te
Figure 1: Field intensity distribution of the SPSM PCF.
Figure 2: Effective index curve as a function of wavelength with ʌ =2.2µm, d1/ʌ=0.40 and d2/ʌ=0.95.
Page 4 of 17
Figure 2 shows the effective index of the x- and y-polarization mode as a function of wavelength for the PCF with ʌ = 2.2 μm, d1/ʌ = 0.40, and d2/ʌ = 0.95. Also
filling mode (SFM)
[3]
ip t
shown in Figure 2 is the effective index (cladding index) of the fundamental space , which is evaluated by applying FEM to the elementary
cr
piece. Polarization cutoff occurs when the effective index of one of the polarization
us
states falls below that of the SFM [14].
an
We know that it is possible to tune the wavelength range for single-polarization operation by varying the fiber structural parameters: d1/ʌ, d2/ʌ, and ʌ. Therefore, it
M
is worthwhile to examine the dispersion properties under the influence of these
Ac ce p
te
0.75 as shown in Figure 3.
d
parameters. First, we fixed d1/ʌ at 0.4 and ʌ at 2 μm and changed d2/ʌ from 0.95 to
Figure 3: Cut-off wavelength as a function of hole diameter to pitch ratio. Tuning the cutoff wavelength by changing the d2/ʌ ratio is possible but at the cost of the reduction of single-polarization wavelength range from 220 to 135 nm for d2/ʌ = 0.95 to 0.75. The single polarization wavelength length reduction can be
Page 5 of 17
attributed to the reduced birefringence induced by decreasing the diameter of the two big air holes. Therefore, d2/ʌ should be set as high as possible to attain a wider
ip t
single-polarization wavelength range. d2/ʌ = 0.95 is a reasonable parameter for practical PCFs. The hole pitch Λ is then varied from 1.6 to 2.4 μm in steps of 0.4
cr
μm, where d1/ʌ and d2/ʌ are fixed at 0.40 and 0.95. The dispersion curves are
us
shown in Figure 4. The cutoff wavelength for x- and y-polarization shifts to longer wavelength with the increase of pitches, and the single-polarization wavelength
te
d
M
an
range increases slightly.
Ac ce p
(a): d1/ʌ=0.40 and d2/ʌ=0.95 for ʌ= 1.6 µm
(b): d1/ʌ=0.40 and d2/ʌ=0.95 for ʌ= 2 µm
(c): d1/ʌ=0.40 and d2/ʌ=0.95 for ʌ= 2.4 µm
Figure 4: Cut-off wavelength as a function of pitch with d1/ʌ=0.40 and d2/ʌ=0.95 for ʌ= 1.6 µm, 2 µm and 2.4 µm.
Page 6 of 17
The structure of the rectangular lattice HB PCF to achieve single-polarization operation is assumed to be as shown in Figure 5. The reported structure is a
ip t
rectangular lattice where six enlarged central air holes are used, three holes each in both rows around the core of the fibre. Here two different pitches are used, i.e.,
cr
hole to hole spacing is different in x and y direction, ʌ is pitch or hole to hole
Ac ce p
te
d
M
an
diameter is d1 and enlarged air hole diameter is d2.
us
spacing in x-direction and ʌy is hole to hole spacing in y direction. Small hole
Figure 5: Field intensity distribution of the SPSM PCF.
The effective refractive index graph for orthogonal polarization as a function of wavelength for a photonic crystal fibre have been shown in Figure 6 with ʌ =2.2µm, ʌ y =0.8ʌ, d1/ʌ= 0.50 and d2/ʌ=0.85.
Page 7 of 17
There are two separate curves for x-polarization FSM and y-polarization FSM because air holes are arranged in rectangular lattice and the two FSM’s are no
an
us
cr
ip t
longer degenerate.[15]
Figure 6: Effective index curve as a function of wavelength with ʌ = 2.2µm,
te
CONFINEMENT LOSS
d
M
ʌy= 0.8ʌ, d1/ʌ=0.50 and d2/ʌ=0.85.
Confinement loss can be defined as the light confinement ability within the core
Ac ce p
region and it usually occurs in single material fibres. Since the cladding of the photonic crystal fibre is obviously made of a limited number of air holes rings, the guided modes in fibre have a inherent property of leaking out. The imaginary part of the effective refractive index leads to the calculation of the confinement loss by using equation (i). (i)
Page 8 of 17
The Hexagonal fibre is analyzed at 1.3 & 1.5 µm, single polarization window. The photonic crystal fibre parameters are varied and confinement loss calculated for
ip t
each of them and plots are generated which are shown in Figure 7 and Figure 8. In the given plots, different air fill fractions and different pitches are used and their
M
an
us
cr
confinement loss is given.
(b): d1/ʌ=0.40 & pitch = 2.1 µm
Ac ce p
te
d
(a): d1/ʌ=0.50 & pitch = 1.2 µm
(c): d1/ʌ=0.35 & pitch = 3.2 µm
(d): d1/ʌ=0.30 & pitch = 9 µm
Figure 7: Confinement Loss of SPSM PCF for x & y polarization at 1.3 µm with different pitch and hole diameter ratio.
Page 9 of 17
ip t
(b): d1/ʌ=0.40 & pitch = 2.48 µm
an
us
cr
(a): d1/ʌ=0.50 & pitch = 1.46 µm
(d): d1/ʌ=0.30 & pitch = 10.8 µm
M
(c): d1/ʌ=0.35 & pitch = 3.85 µm
d
Figure 8: Confinement Loss of SPSM PCF for x & y polarization at 1.5 µm
te
with different pitch and hole diameter ratio.
Ac ce p
Now, confinement loss of the rectangular lattice single mode single polarization photonic crystal fibre is investigated. SPSM bandwidth can be increased by increasing the pitch or the crystal constant in the photonic crystal fibre. Single polarization over the whole communication band can be achieved by having higher pitch and air filling ratio. Thus, the impact of pitch variations and hole diameter of enlarged holes are studied. The different pitches analyzed are 2.5, 2.6 and 2.7 µm. Different d2/ʌ ratio’s investigated are
and is shown in Figure 9
and Figure 10. It can be observed that if d2/ʌ decreases then confinement loss of xpolarization also decreases.
Page 10 of 17
ip t cr
(a): x-polarization
us
(b): y-polarization
Figure 9: Influence of Variation of d2/ʌ =0.91, 0.90 and 0.89 for x-polarization
Ac ce p
te
d
M
an
and y-polarization.
(a): x-polarization
(b): y-polarization
Figure 10: Influence of variation of pitch with ʌ = 2.5, 2.6 and 2.7 µm for x and y-polarization.
DISPERSION ANALYSIS
Dispersion is the name given to broadening of pulse while propagating through the fibre. The unit of dispersion is ps/(nm-km), that shows elongation in pulse time with certain spectral width, in nm after a propagation length of 1 km.
Page 11 of 17
Dispersion can be calculated by using equation (ii)
ip t
HEXAGONAL LATTICE: The graph showed in Figure 11 is the dispersion vs
cr
wavelength graph for the hexagonal lattice single mode single polarization photonic crystal fibre with pitch = 2.2 µm, hole diameter to pitch ratio of smaller
Ac ce p
te
d
M
an
that we have very low dispersion at 1.55 µm.
us
holes to be 0.40 and 0.95 for the enlarged holes. From the graph, it can be observed
Figure 11: Dispersion graph for ʌ =2.2
, d1/ʌ = 0.4 and d2/ʌ= 0.95.
We further reduced the air filling fraction of the enlarged central air holes to study the impact of variation on the dispersion behavior of the SPSM PCF. The hole diameter to pitch ratio for smaller holes is kept at 0.4 and pitch is 2.2 µm. The d2/ʌ ratio is reduced to 0.75. It is observed from Figure 12 that small negative dispersion can be achieved using the above mention crystal parameters for the
Page 12 of 17
single polarization operation. The negative dispersion counter effects the
an
us
cr
ip t
dispersion losses.
, d1/ʌ = 0.4 and d2/ʌ= 0.75.
M
Figure 12: Dispersion graph for ʌ =2.2
Now, the air fill fraction of the enlarged air holes is kept at 0.95 but for smaller
d
holes, it is changed to 0.5, thereby increasing the air filling fraction of the cladding.
te
Pitch is reduced to 1.25 µm. It is observed that very low dispersion can be
Ac ce p
achieved at 1.3 and 1.6µm. Thus, single polarization operation can be achieved with very low dispersion as shown in Figure 13.
Figure 13: Dispersion graph for ʌ =1.25
, d1/ʌ = 0.5 and d2/ʌ= 0.95.
Page 13 of 17
RECTANGULAR LATTICE The graph showed in the Figure 14 is the dispersion vs wavelength graph for the
, ʌy =0.8 ʌ, d1/ʌ = 0.5 and d2/ʌ= 0.85.
M
Figure 14: Dispersion graph for ʌ =2.2
an
us
cr
ip t
square lattice single mode single polarization photonic crystal fibre.
d
It can be easily observed that very low dispersion values can be achieved in the
te
single polarization single mode wavelength range of the given photonic crystal
Ac ce p
fibre. Low values of dispersion are found from 1.35 µ to 1.8 µm. Now the pitch is increased to 2.3 µm and air filling fraction of larger holes is increased to 0.92. Other parameters are kept unchanged. The effect of increased pitch and air filling fraction on dispersion behavior of the fibre are investigated. It can be observed from Figure 15 that low dispersion values are achieved at 1.65 µm to higher wavelengths.
Page 14 of 17
ip t cr , ʌy =0.8 ʌ, d1/ʌ = 0.5 and d2/ʌ= 0.92.
an
CONCLUSIONS
us
Figure 15: Dispersion graph for ʌ =2.3
M
The properties of the photonic crystal fibres are very flexible. By introducing very small variations in the design parameters in the structure of the fibre, the impact is
d
immediately visible in the confinement loss and dispersion behavior of the
te
photonic crystal fibre. Thus, it can be concluded that by bringing small variations
Ac ce p
in the PCF structure, desired dispersion profiles with low confinement loss can be obtained and implemented in many areas with specific dispersion requirements. REFERENCES
[1] J.C. Knight, P.S.J. Russell, Photonic Crystal Fibers: New Way to Guide Light, Science 296 (2002) 276–277.
[2] J.C. Knight, T.A. Birks, P.S.J. Russell, D.M. Atkin, All-silica Single-mode Optical Fiber with Photonic Crystal Cladding, Opt. Lett. 21 (1996) 1547–1549.
Page 15 of 17
[3] T.A. Birks, J.C. Knight, P.S.J. Russell, Endlessly Single-mode Photonic Crystal Fiber, Opt. Lett. 22 (1997) 961–963.
ip t
[4] J.C. Knight, J. Broeng, T.A. Birks, P.S.J. Russell, Photonic Band gap Guidance in Optical Fibers, Science 282 (1998) 1476–1478.
cr
[5] M.J. Messerly, J.R. Onstott, R.C. Mikkelson, A Broad-band Single Polarization
us
Optical Fiber, Journal of Lightwave Technology 9 (1991) 817–820.
[6] S. Furukawa, T. Fujimoto, T. Hinata, Propagation Characteristics of a Single-
an
polarization Optical Fiber with an Elliptic Core and Triple-clad, Journal of
M
Lightwave Technology 21 (2003) 1307–1312.
[7] M.J. Gander, R. McBride, J.D.C. Jones, D. Mogilevtsev, T.A. Birks, J.C.
d
Knight, P.S.J. Russell, Experimental Measurement of Group Velocity Dispersion
te
in Photonic Crystal Fibre, Electron. Lett. 35 (1999) 63–64.
Ac ce p
[8] W. Reeves, J. Knight, P. Russell, P. Roberts, Demonstration of Ultraflattened Dispersion in Photonic Crystal Fibers, Opt. Express 10 (2002) 609–613. [9] K. Saitoh, N. Florous, M. Koshiba, Ultra-flattened Chromatic Dispersion Controllability using a Defected-core Photonic Crystal Fiber with low Confinement Losses, Opt. Express 13 (2005) 8365–8371. [10] A. Ortigosa-Blanch, J.C. Knight, W.J. Wadsworth, J. Arriaga, B.J. Mangan, T.A. Birks, P.S.J. Russell, Highly Birefringent Photonic Crystal Fibers, Opt. Lett. 25 (2000) 1325–1327.
Page 16 of 17
[11] K. Suzuki, H. Kubota, S. Kawanishi, M. Tanaka, M. Fujita, Optical Properties of a Low-loss Polarization-maintaining Photonic Crystal Fiber, Opt. Express 9
ip t
(2001) 676–680. [12] K. Saitoh, M. Koshiba, Single-polarization Single-mode Photonic Crystal
cr
Fibers, IEEE Photon. Technol. Lett. 15 (2003) 1384–1386.
us
[13] H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, S. Yamaguchi, Absolutely Single Polarization Photonic Crystal Fiber, IEEE Photon. Technol. Lett. 16 (2004)
an
182–184.
M
[14] F. Zhang, M. Zhang, X. Liu, P. Ye, Design of Wideband Single-Polarization
1184-1189.
d
Single-Mode Photonic Crystal Fiber, Journal of Lightwave Technology 25 (2007)
te
[15] J. Ju, J. Wei, M.S. Demokan, Design of Single-Polarization Single-Mode
Ac ce p
Photonic Crystal Fiber at 1.30 and 1.55 μm, Journal of Lightwave Technology 24 (2006) 825-830.
Page 17 of 17