Transmission of CO2 Laser Radiation Through Glass Hollow Core Microstructured Fibers

In this chapter we would like to highlight and analyze the main problems of transmission and propagation of CO2 laser radiation in the hollow core microstructured fibers (HC MFs). It is well known that there is a strong need for the fiber delivery systems for 10.6 μm CO2 lasers due to a wide range of CO2 laser applications in medicine, spectrometry, industry, military applications and in other fields of science and technology. Research into the possibility of the mid IR laser radiation transmission (especially, CO and CO2 lasers) with the help of optical fibers as well as with crystalline or glass cores made of different materials has been going hand with hand with the technological development. However, until recently these fibers haven’t been used for industrial applications due to a relatively high level of optical losses at the lasers wavelengths and certain physicochemical properties of the fiber materials. These problems mainly arise from a low laser damage threshold, low melting temperatures of most IR transmitting materials and their high nonlinearity.


Introduction
In this chapter we would like to highlight and analyze the main problems of transmission and propagation of CO 2 laser radiation in the hollow core microstructured fibers (HC MFs).It is well known that there is a strong need for the fiber delivery systems for 10.6 µm CO 2 lasers due to a wide range of CO 2 laser applications in medicine, spectrometry, industry, military applications and in other fields of science and technology.Research into the possibility of the mid IR laser radiation transmission (especially, CO and CO 2 lasers) with the help of optical fibers as well as with crystalline or glass cores made of different materials has been going hand with hand with the technological development.However, until recently these fibers haven't been used for industrial applications due to a relatively high level of optical losses at the lasers wavelengths and certain physicochemical properties of the fiber materials.These problems mainly arise from a low laser damage threshold, low melting temperatures of most IR transmitting materials and their high nonlinearity.
Hollow waveguides are exempt from many problems that are common to all types of the solid waveguides in this spectral region and thus can serve as much more reliable delivery systems.Glass hollow waveguides, crystalline hollow waveguides, dielectric -coated cylindrical hollow waveguides, polycrystalline fibers are well known examples of such systems.Here we consider only the glass HC MFs with their characteristics and physical phenomena laying the basis for their waveguide mechanisms.In particular, we propose a new type of HC MF for CO 2 laser radiation delivery with the cladding consisting of one row of the glass capillaries.We show that due to complicated boundary conditions and optical properties of an individual capillary it is possible to obtain low loss waveguide regimes for CO 2 laser radiation.Moreover, we show that the HC MFs with a determined symmetry type of a capillary arrangement in the cladding exhibit low bend losses when such low loss waveguide regimes occur.
The chapter is organized as follows.In Section 2 we consider all types of proposed hollow waveguides for CO 2 laser radiation transmission and give a short historical overview to highlight the problem of CO 2 laser beam delivery in the hollow core waveguides.In Section 3 we consider physical mechanisms of the light guiding in the glass HC MFs with cladding consisting of capillaries due to which, in our opinion, it becomes possible to guide the light in the mid IR including CO 2 laser radiation.In Section 4 we offer some numerical analyses core materials with a refractive index less than 1.Due to this fact, the angle of incident radiation in the core is greater than the critical angle and the light experiences the total internal reflection.An example of such fiber operating at the CO 2 laser wavelength is the sapphire fiber with a refractive index of n = 0.67 [Harrington & Gregory, 1990].
For our part, in the following subsections we will consider the second group of the circular cross section hollow waveguides, previously classified as leaky waveguides, in particular, HC MFs.This group of hollow waveguides has inner wall surface with refractive indices greater than 1.Leaky hollow waveguides have metallic or dielectric layers deposited on the inside metallic, plastic or glass tubing to enhance their surface reflectivity.Creation of such dielectric coated cylindrical hollow waveguides presented a complicated technological problem since high quality reflective coatings are not compatible with circular cross section geometry.Traditional vapor deposition techniques don't produce good quality coatings on the inside of a capillary.The theoretical calculated attenuation of a dielectric coated hollow waveguide for the IR region was obtained in [Miyagi & Kawakami, 1984].The authors have shown that the attenuation is very sensitive to the material and geometry of a dielectric film.Also, the attenuation is very sensitive to the properties of the metal under the dielectric film.A metal should have a low refractive index and a very high extinction coefficient.For example, these can be silver, nickel, copper.A dielectric should be selected with a maximum refractive index, for example, KCl, ZnSe, ZnS etc.The first demonstration of a dielectric coated cylindrical hollow waveguide was perfomed by Prof. Miyagi's group [Miyagi et. al., 1983] in 1983.The 1.2 m -long mandrel of polished aluminum tubing was coated with approximately 0.45 µm of germanium.Then, a layer of nickel up to 200 µm was deposited on the top of germanium before the aluminum mandrel was removed by leaching.The final structure was a nickel tube with optically thick dielectric layers on the inner wall.The fabricated waveguide had a core diameter of 1.5 mm and length of about 1 m.The measured attenuation was ~ 0.4 dB/m in the straight waveguide but the bend loss was very high.
Authors in [Croitoru et. al., U.S. Patent, 1990] have used polyethylene and Teflon tubing as a substrate in which thin and flexible metallic layers of Al followed by AgI were deposited.They reported attenuation in a straight waveguide of about 0.6 dB/m at the bore size value of 4 mm [Croitoru et. al., 1990].Authors of [Morrow & Gu, 1994] reported a cylindrical hollow waveguide in which Ag and Ag halide coatings were deposited inside Ag tubes.The waveguide with 1 -mm -bore size had attenuation below 0.1 dB/m at λ = 10.6 µm which was still below 0.8 dB/m at a bend radius up to 25 cm.
At the end of this subsection, we will turn to the currently most popular hollow circular cross section waveguides for CO 2 laser radiation transmission.These are hollow glass waveguides developed initially by Prof. Harrington's group [Abel et. al., 1994].There are two main advantages of the glass tubing substrate.First, it is easier to make a long, uniform tubing from glass having considerably smoother wall surfaces than metal or plastic tubings.As a result, the scattered losses are less.The second advantage is that the technology of making glass capillary tubings is common and inexpensive.The authors fabricated a hollow glass waveguide using wet chemistry methods.First, an Ag layer was deposited on the inside of a silica glass tubing.Then, an AgI layer was formed over the metallic film.The thickness of the layer was optimized to obtain a high reflectivity at the required wavelength.A straight waveguide with a bore of 530 µm demonstrated a loss of 0.3 dB/m at CO 2 laser wavelength.This fiber could maintain a loss level under 2 dB/m at a bend radius as small as 5 cm.Hollow glass waveguides have been used successfully for a modest CO 2 laser power delivery below ~ 80 W. For the higher power delivery it is necessary to place a watercooled jacket around the guides.The highest CO 2 laser power delivered through the watercooled hollow glass waveguide with 700 µm bore was 1040 W [Nubling & Harrington, 1996].This is comparable to CO 2 laser power delivered through the water -cooled hollow metallic waveguide with 1800 µm bore which was 2700 W [Hongo et. al., 1992].

CO 2 laser radiation transmission through hollow core microstructured fibers
In this subsection we will consider a new approach to solving the problem of the mid IR transmission (in particular, CO 2 laser radiation) through the glass hollow core microstructured fibers (HC MFs).The possibility of the light confinement in the air core of HC MFs with the cladding consisting of two dimensional periodic array of air holes was predicted by Russell at the beginning of 1990s and theoretically demonstrated by Birks et.al. [Birks et. al., 1995].The most advanced HC MFs are hollow core photonic crystal fibers (HC PCFs).HC PCFs in turn can be divided into two main groups.The HC PCFs from the first group guide the light by virtue of photonic band gap (BG HC PCFs).The HC PCFs from the second group have no band gaps and guide the light due to an inhibited coupling between the core guided modes and modes associated with a cladding [Benabid et. al., 2002].They are called inhibited coupling HC PCFs (IC HC PCFs).Both types of HC PCFs have the claddings with very little solid material, usually, with a filling fraction less than 10%.
The guidance mechanism for BG HC PCFs is based on the concept of 'out of plane' band gap.The microstructure of BG HC PCF cladding consists of air holes packed in a triangular arrangement.It gives rise to a full two dimensional photonic band gap [Birks et. al., 1995].As a result, forbidden frequencies occur for optical waves whose wave vector (axial) component is not equal to zero.Such frequency ranges constitute bands.The first experimental demonstration of light transmission in the BG HC PCF was made in 1999 [Gregan et. al., 1999].Up to now, considerable efforts have been put forth in experimental and theoretical studies of BG HC PCFs made of silica glass [Humbert et. al., 2004;Benabid et. al., 2004].This special interest can be partly explained by a need to find a way of yielding a loss level less than 0.2 dB/km for telecommunication spectral region.So far, the BG HC PCFs loss was reduced only to 1.2 dB/km due to intrinsic roughness of the air -glass interfaces in the structure [Roberts, Couny et. al., 2005].
As it was mentioned above, BG HC PCFs made of silica glass have claddings with very little solid material.The bandgap located between 4 th and 5 th bands is used for guiding in HC PCFs with such high air -filling fraction ( ≥ 80%) [Humbert et. al., 2004].The number of each band is counted from the band with the largest value of the propagation constant of the air core mode.However, there is an important need for BG HC PCFs which can be used in the mid and far IR.BG HC PCF made of silica glass with a core diameter of 40 µm demonstrated single mode waveguide regime in a narrow transmission window near the wavelength of λ = 3.14 µm with an attenuation of ~ 2.6 dB/m [Shephard et. al., 2005].But silica glass BG HC PCFs cannot be used for CO 2 laser radiation transmission due to a very high material loss of silica.Transmission of light in the mid IR region becomes possible with B G H C P C F s m a d e o f g l a s s e s w h i c h a r e t r a n s p a r e n t i n t h i s s p e c t r a l r e g i o n s u c h a s www.intechopen.comchalcogenide glasses.These glasses are composed of the chalcogen elements Se and Te with an addition of such elements as Ge, As, Sb.The transparency windows of these glasses correspond approximately to the mid IR region 2 -25 µm.
Modeling of BG HC PCFs made of nonsilica glasses was performed by a number of authors [Shaw et. al., 2003;Pottage et. al., 2003;Pearce et. al., 2005].In this paper [Shaw et. al., 2003] BG HC PCFs made of As -S (refractive index ~ 2.4) and As -Se (refractive index ~ 2.8) were analyzed.It was shown that there exist several spectral regions with bandgaps for an air filling fraction > 40% in both As -S and As -Se BG HC PCFs.These BG HC PCFs have large bandgap widths at the air filling fraction of 45% to 60%.BG HC PCFs with high air filling fractions > 80% also exhibited large bandgap widths.In the author' opinion, all these results show a possibility of a light transmission in the mid IR using halcogenide BG HC PCFs.The authors of [Pottage et. al., 2003] have carried out a numerical analyses of BG HC PCFs for a wide range of refractive indices from n = 1.5 to n =3.6 and for different values of air filling fractions from 33% to 87%.They discovered a new type of the bandgap which was called type 2 bandgap at an air filling fraction ~ 60% for any glass index beyond 2. The results showed a possibility of obtaining a satisfactory guidance in such BG HC PCFs.
Another important aspect of the problem of the mid IR radiation transmission was discussed in [Pearce et. al., 2005].Apart from a limitation to attaining a low loss guidance in BG HC PCFs connected with the intrinsic roughness of the air glass interfaces, there is another problem connected with an existence of surface guided modes that are trapped in the core surroundings.Experimental and theoretical studies [Smith et. al., 2003;Humbert et. al., 2004;West et. al., 2004;Saitoh et. al., 2004] carried out for silica BG HC PCFs have shown that the anticrossing between dispersion curves of the surface modes and the air core modes is the main factor leading to a transmission loss in BG HC PCFs.Several methods were proposed to suppress the surface modes.The first method is used to reduce the distortion of the core by including 'fingers' of glass [West et. al., 2004].The second method is to use thin core walls [Saitoh et. al., 2004] and the third one is to use 'antiresonance' walls [Roberts, Williams et. al., 2005].The authors of [Pearce et. al., 2005] modeled a realistic design of distorted cores for BG HC PCFs which can guide the light in the type 2 bandgap.They have demonstrated that BG HC PCFs made of high index glass can guide a fundamental air core mode with a fraction of power in the air of up to 98%.
In their paper [Hu & Menyuk, 2007] the authors analyzed BG HC PCFs for refractive indices between 1.4 and 2.8.They found two maxima of the relative bandgap as a function of the air filling fraction and refractive index.The authors also found that the relative bandgap and the level of loss are interrelated.When the relative bandgap increases the loss decreases and vice versa.
Despite the promising results of modeling obtained in the above listed works a practical realization of BG HC PCFs made completely of chalcogenide glass for the mid IR spectral region has not been reported.The only successful realization of a photonic band gap hollow core fiber for the CO 2 laser radiation transmission was 'Omniguide' fiber where the cladding is a Bragg reflector (hollow core Bragg fiber) made of soft glass and polymer [Temelkuran et. al., 2002].The authors of [Bowden & Harrington, 2009] have studied low and high index chalcogenide glasses for their potential use in the fabrication of all glass hollow core Bragg fiber.
The first work devoted to the fabrication and experimental investigation of BG HC PCF for CO 2 laser radiation transmission has appeared only in 2010 [Deseveday et. al., 2010].The authors designed BG HC PCF made of chalcogenide glass to guide the light in the air core at λ = 9.3 µm.They also fabricated two BG HC PCFs which could potentially guide CO 2 laser radiation but no guidance was observed.The authors explained this fact by technological difficulties in the fabrication process.They hope to improve the process by avoiding air tightness anomalies and by decreasing the core's wall thickness.
In the next section we will represent our approach to solving the problem of CO 2 laser radiation transmission through the glass HC MFs.

Mechanisms of CO 2 laser radiation transmission through the glass hollow core microstructured fiber with the cladding consisting of capillaries
In this section, we will consider physical mechanisms and principles which enable, in our opinion, to obtain a loss level much lower than the material loss of the glass of HC MFs with a negative curvature of the core boundary [Pryamikov et. al., 2011].The negative curvature of the core boundary is obtained by the cladding consisting of one or several rows of glass capillaries.Such microstructure design leads to a significant complication of the boundary conditions for the air -core modes.To justify our assumptions it will be necessary to consider a plane wave scattering on a cylindrical surface to show an analogy between this phenomenon and the light scattering on the plane optical diffraction grating.An analogy between discrete rotational symmetry of the capillary arrangement in the cladding and the plane diffraction grating will also be outlined.We will also consider the second main factor leading to a loss reduction of the air core modes of the HC MFs and to an increase in the width of transmission regions.It is connected with the geometry parameters of an individual capillary and the glass refractive index.In the end, we will try to justify the statement how these factors can result in the loss reduction of the air core mode in HC MF with the cladding consisting of capillaries with respect to the BG HC PCFs and kagome lattice IC HC PCFs.

The cylindrical surface as a diffraction grating
In this subsection, we will offer a reason which, in our opinion, lies behind the low loss waveguide regimes for the glass HC MFs with negative curvature of the core boundary.For the first time, an effect of the loss level decrease resulting from the negative curvature of the core boundary was observed for a large pitch kagome -lattice IC HC PCF with a hypocycloid -shaped core structure (the second group of PCFs) [Wang et. al., 2011].We have used a simple cladding structure of the HC MF consisting of eight silica capillaries (Fig. 1(left)).Such HC MF guided light in the mid IR up to 4 µm despite of very high material losses of silica in this spectral region.In this case, the negative curvature of the core boundary was created by the capillary surfaces.Of course, such long a wavelength guiding is determined by not only the negative curvature of the core boundary but (may be to a greater extent) also by the optical properties of an individual capillary of the cladding.For example, the simple cladding structure consisting of one row of the capillaries has a lower density of eigenstates with respect to the cladding consisting of the solid rods (Fig. 1(right)).In [Pryamikov et. al., 2011] the authors have made a calculation mistake when they were trying to justify their assumptions on the role of the negative curvature of the core boundary.The mistake was made when calculating the waveguide loss for a dielectric tube using an analytical formula from [Marcatili & Schmeltzer, 1964].Indeed, the loss levels for the dielectric tube and silica HC MF with the cladding consisting of solid rods with equal air core diameters are approximately the same.Despite this fact, the main conclusion of the paper remains accurate, i.e. that to obtain a long wavelength waveguide regime in the mid IR with silica HC MF it is necessary to combine two factors, namely, the negative curvature of the core boundary and the low density of eigenstates of the individual elements of the cladding.
To clarify a role of the negative curvature of the core boundary we should consider a plane wave scattering on the curved surface, for example, of a solid rod.This problem was solved by many authors [Wait, 1955;Lind & Greenberg, 1966].Depending on the polarization state of the incident plane wave z -component of the electric (TM polarization) or magnetic field (TE polarization) is parallel to the incident plane.Suppose the refractive index of the solid rod is n 1 and the outer space is n 2 .In the following, only TE polarized plane wave will be considered.Its z -component of the magnetic field can be expanded according to addition theorem for Bessel functions and taking into account the temporal dependence as it e ω one can obtain: where θ is an angle of incidence, ϕ and r are an azimuthal and radial cylindrical = is its axial component.In such a way, the incident plane wave is expanded into an infinite number of cylindrical harmonics due to the curvature of the cylinder surface.The z -components of the scattered field and field inside the cylinder can be expressed in the same manner: where n Hr is a transverse components of the wavevector kn 11 2π λ = for the field inside the rod.On the basis of these expressions for zcomponents of the incident, scattered and inside fields it is possible to calculate ϕ and rcomponents of the fields [Adler, 1952].In other words, the field of the incident wave is represented by an infinite number of space 'channels' (harmonics) through which the energy of the incident wave is transferred to the scattered fields and the fields inside the dielectric rod.It is seen, that such sets of cylindrical harmonics (1) -( 2) have a mode structure and can be considered as radiation or continuous modes of ITE (incident transverse electric) type of the individual dielectric rod [Snyder, 1971].
To calculate the coefficients ss nnnn baba ,,, it is necessary to apply boundary conditions for z and ϕ -components of the incident, scattered fields and the fields inside the cylinder.
Because of the mode structure of the total field it is not necessary to solve an infinite set of simultaneous linear equations.To obtain the nth order coefficients one needs to solve 4*4 inhomogeneous system of linear equations.For a solid cylinder rod it is possible to obtain analytical expression for the coefficients which includes such terms as nn Ja Ja ' 11 () / () λλ [ Wait,  1955].These terms have resonances (poles) corresponding to zeros of n Ja 1 () λ .It is necessary to point out that the resonances of the nth order coefficients of the scattered field are determined not only by nth order functions n Ja 1 () λ but also by n Ja 11 () λ + and n Ja 11 () λ − due to recurrent relations for the derivatives of the Bessel functions.In other words, the different diffraction orders are coupled between each other.Due to this fact, it is possible to observe not only Lorentzian -like resonances for the spectral dependencies of absolute values of the amplitudes ss nn ba , but also Fano type resonances [Fano, 1961].For example, Fano type resonances were analysed in the case of all solid band gap fibers [Steinvurzel et. al., 2006] with the cladding consisting of solid dielectric cylinders with a refractive index higher than the background.
A similar phenomenon occurs when the plane wave is scattered on optical diffraction grating.A short analysis can be carried out based on the work [Hessel & Oliner, 1965].
Suppose the diffraction grating extends infinitely in the y and z directions, with the period d in the z direction only.The refractive index of the outer medium is again 2 n .The magnetic field of this incident wave of S polarization is represented as: Then, due to periodicity the scattered magnetic field can be represented as [Hessel & Oliner, 1965]: where n A () β are the amplitudes of the various spectral orders and n x n kk d transverse components of the wavevector of the scattered field.Applying boundary conditions one obtains an infinite inhomogeneous set of simultaneous linear equations [Hessel & Oliner, 1965] for the scattered field amplitudes n A () β .These amplitudes are coupled between each other as in the case of the plane wave scattering on the rod and a set of resonances Lorentzian and Fano types known as Wood anomalies [Wood, 1902;Lord Rayleigh, 1907;Hessel & Oliner, 1965] also occur.In the case of optical grating the conservation of light momentum for the scattered light in the z -direction looks like: In the case of the plane wave scattering on a dielectric rod there is also a conservation of the angular momentum of the light.The fields of the source plane wave are expanded into a set of the 'space' channels with an angular momentum determined by the number of n in in e φ − (1) and each continuous (radiation) mode of the dielectric rod has the same angular momentum (1, 2).Note, that this is (4) of the same form as an expression (2a) for the scattered field s z H in the case of the dielectric rod.Comparing expressions (2a) and ( 4) one can state that the process of the plane wave scattering on the rod can be represented as scattering on the diffraction quasi -grating.The surface of the dielectric rod can be considered as an azimuthal diffraction grating with periodicity in ϕ -direction and conservation of the angular momentum of the light.But instead of one incident plane wave as in the case of the plane diffraction grating (3) there is an infinite set of incident cylindrical harmonics with their own angular momentums of the light.It can also be shown that the spectral dependencies of ss nn ba , in expansions (2a) demonstrate the resonance behaviour analogous to Wood anomalies as in the case of the plane diffraction grating.This problem and its application to the waveguide mechanism in all solid band gap fibers will be considered by us thoroughly in our future publication.In the same way, it is possible to consider a cylindrical surface of the capillary as a diffraction quasi -grating in the ϕ direction instead of z -direction in the case of the plane diffraction grating.
The main conclusion to draw is that the plane wave incidence on a curved cylindrical surface leads to the appearance of an infinite set of cylindrical harmonics with different angular momentums of the light.As the source of the incident wave is outside of a solid cylinder or a capillary one obtains an inhomogeneous set of linear equations for determining amplitudes of the scattered field harmonics.These scattered field harmonics have the same values of the angular momentums of the light as the harmonics of the incident field.Different orders of these harmonics (diffraction orders) are coupled between each other due to the properties of the Bessel functions.It leads to appearance of Wood anomalies in the spectral dependencies of their amplitudes.As a consequence, the curved cylindrical surface can be considered as an azimuthal diffraction grating with modulation in ϕ -direction [Pryamikov, to be prepared].This fact leads to a loss of simplicity of the boundary conditions for the air core modes of the HC MF with the cladding consisting of capillaries compared with BG HC PCF, for example.Similar situation occurs if one considers the boundary conditions for the core modes of all solid photonic band gap fibers [White et. al., 2002] and Bragg fibers [Yeh et. al., 1978].

Discrete rotational symmetry of the core boundary
Another aspect of the problem of complicated boundary conditions for HC MF with the cladding consisting of N capillaries is that the capillary location in the cladding is also periodic in the azimuthal coordinate ϕ (Fig. 1).As a result, the boundary conditions for the air core modes are also periodic.It is known that if the system transforms into itself for a set of discrete rotations kN 2/ δϕ π = , kN (0, 1) ∈− around axial vector z  the eigenfunctions of the system (in other words, the air core modes) can be represented as [Skorobogatiy & Yang, 2009]: and their eigenvalues are: These eigenvalues are the same for any nnN m =+ , where m is an integer and the eigenfunctions characterized by an integer n are degenerate ones [Skorobogatiy & Yang, 2009].As a consequence, such eigenstates of the system with a discrete rotational symmetry can be expressed as a superposition of all the degenerate states: where an expression under the sum sign is a periodic function in ϕ with a period N 2π and nN [0, 1] ∈− .Note that this is (6) of the same form as an expression (2a) and (4) for the scattered field s z H in the case of the dielectric rod and the plane diffraction grating.The discrete symmetry of the core boundary gives one more type of azimuthal diffraction grating in the considered HC MFs.
In such a way, the boundary conditions for the air core modes of the HC MF with the cladding consisting of capillaries are complicated not only by the curvature of the cylindrical surface of an individual capillary but also by the discrete rotational symmetry of the core boundary.

Complicated boundary conditions for the air core mode of HC MF with the cladding consisting of capillaries and low loss guidance
Summarizing the conclusions of the two previous subsections one can state that the air core modes of HC MFs with the cladding consisting of capillaries are formed by a superposition (interference) of several space cylindrical harmonics originated from the azimuthal diffraction quasi -gratings occurred in the cladding.The main factors affecting the air core mode formation are geometry parameters of an individual capillary and the type of a periodic arrangement of the capillaries in the cladding.All these space harmonics interact with the capillary walls in stronger or weaker ways depending on their radial and azimuthal distribution.In this case, the material loss of the capillary walls doesn't have the same affect on the attenuation of each harmonic.In such a way, the energy of the air core m o d e a t t h e c o r e b o u n d a r y f o r H C M F w i t h t h e c l a d d i n g c o n s i s t i n g o f c a p i l l a r i e s i s distributed in much more complicated way than in the case of BG HC PCFs or kagome lattice IC HC PCFs.
BG HC PCFs and kagome lattice IC HC PCFs do not have such complicated mechanism of the air core modes formation because of a quasi continuous rotational symmetry of the core boundary and the boundaries of succeeding layers in the cladding.Kagome lattices IC HC PCFs sometimes have the discrete symmetry of the core boundary but just with a polygonal shape of the core without a negative curvature of the core boundary.The authors in [Pearce et al., 2007] have shown that the loss behavior of a hollow core fiber with the cladding consisting of concentric glass rings or hexagons explain the qualitative features in the loss curves associated with kagome lattice IC HC PCF.The hollow core photonic band gap fiber with the cladding consisting of concentric glass rings has a continuous rotational symmetry of the core modes and is analogous to solid Bragg fibers [Fevrier et. al., 2006].In this case, the air core is formed by the boundaries of the concentric glass rings which cannot play the role of the azimuthal diffraction gratings.Applying boundary conditions to each concentric ring of the cladding one obtains a homogeneous system of linear equations determinant of which is a dispersion relation for the air core modes.Each air core mode is formed and described by only one space cylindrical harmonic with the propagation constant β determined from the corresponding dispersion relation.These space harmonics don't interact with each other's as in the case of all solid band gap fibers and Fano type resonances cannot be observed (an exception can be kagome lattice IC HC PCF with a polygonal shape of the air core).As a consequence, all energy of each air core mode is concentrated in one 'space' channel (cylindrical harmonic) which interacts with glass rings of the cladding in the same way according to its azimuthal angular dependence of in e ϕ − .This air core mode is leaky and its imaginary part of β is determined from the dispersion relation mentioned above.All resonances in high index layers for this mode are radial and can be described by the ARROW model [Litchinitser et. al., 2002].
The other very important moment is that the geometry parameters of the capillaries ins out dd , and the glass refractive index at the determined value of D core (Fig .1(left)) should be chosen in such a way so as to excite leaky modes with high quality factor.These modes usually have high radial and azimuthal indices.To this end, the capillary wall thickness must be thin enough and comparable with the wavelength.In this case, it is possible to obtain large widths of the transmission regions for the air core modes.Moreover, it seems possible to choose all parameters of the considered HC MF including glass refractive index in such a way so as to obtain a very weak interaction (coupling) of each cylindrical harmonic constituting the air core mode with capillary walls inside the transmission regions.All these factors can give rise to a low loss waveguide regime for CO 2 laser radiation.Several examples of such waveguide regimes will be given in the next section.

Numerical modelling of HC MFs with different types of discrete rotational symmetry of the core boundary and glass composition of the capillaries
In the following subsections four types of HC MFs with the claddings consisting of 6, 8, 10 and 12 capillaries will be considered.The calculations will be carried out for two values of the glass refractive indices n = 2.4, 2.8 and with three values of ins out dd / = 0.8, 0.85, 0.9.All calculations will be made in the narrow spectral region near λ = 10.6 µm.This fact is connected with high density of the individual capillary eigenstates (leaky modes with high azimuthal indices [Vincetti & Setti, 2010]) which occur at such high values of the glass refractive indices and individual capillary dimensions with respect to the wavelength.We will show that the way of obtaining a low loss waveguide regime in the glass HC MFs with the cladding consisting of capillaries presents a complicated multiparameter task.Unusual behaviour of the bend loss depending on the bend radii for high index glass HC MFs with the cladding consisting of capillaries will be demonstrated.

Loss dependencies for HC MF with different number of capillaries in the cladding
First, we will consider the loss dependencies for HC MF with six capillaries in the cladding.As was mentioned above, all loss dependencies are calculated in the narrow spectral range from 10.59 µm to 10.61 µm with a wavelength step equals to 1 nm.The calculations will be made for two values of the ratio of ins out dd /0 .8 5 = and 0.9 because the losses are very high at lower values of the one.In Fig. 2  As one can see from Fig. 2(a) HC MF with n = 2.4 and ins out dd / = 0.9 has the minimal loss level in the considered spectral range.In our opinion, it can be explained by the minimal value of density of individual capillary states.In this case, the capillary has a minimal capillary wall thickness with respect to the wavelength and the lowest refractive index.The loss dependence for HC MF with n = 2.8 and ins out dd / = 0.9 is relatively inhomogeneous and has a strong peak at λ = 10.601µm caused by the excitation of a capillary leaky mode with high azimuthal and radial indices.This mode is shown in Fig. 3

(left).
Other curves in Fig. 2      The loss dependencies in Fig. 5 show that a low loss regime for the high index glass HC MFs with the cladding consisting of capillaries can be obtained by the right selection of many parameters characterizing HC MF including D core .
At the end of this subsection, the loss dependencies for HC MFs with the cladding consisting of 12 capillaries will be shown (Fig. 6).

Bend loss dependencies on a bend radius for HC MF with a different number of capillaries in the cladding
In this subsection we will consider characteristics of the bend loss behaviour for HC MFs with a different number of capillaries in the cladding.To reveal the special features of the bend loss one analyses the bend loss behaviour for HC MFs with optimal waveguide regimes found in the previous subsection and, correspondingly, with minimal waveguide losses.
In Fig. 7 the bend loss dependencies for two low loss waveguide regimes in the case of HC MF with the cladding consisting of 6 capillaries and 8 capillaries are shown.Such resonance behaviour of the bend loss occurs due to a very high density of dielectric modes (eigenstates) of an individual capillary at such values of n, d ins , d out .The energy of the air core mode of the HC MFs is tunnelled by these dielectric modes into the capillary 'airy' modes.The higher the values of n, d ins , d out with respect to the wavelength the more effective tunnelling is observed and the excited 'airy' modes of the individual capillary have higher quality factor.For example, the bend loss dependence for HC MF with the cladding consisting of 8 capillaries and D core = 220 µm has no resonance peaks due to suppressing the tunnelling through the capillary walls due to a decrease in the values of d ins , d out or n.
To confirm the above conclusions, bend losses for HC MFs with the cladding consisting of 10 and 12 capillaries were calculated (Fig. 9).As in the case of Fig. 7, HC MFs with the lowest waveguide losses were taken for the bend loss calculations.All curves in Fig. 9 have no resonance peaks except for the HC MF with D core = 320 µm, ins out dd / = 0.85 and n = 2.4.In this case, the dielectric capillary mode with a high azimuthal index (Fig. 3(left)) was excited at the bend radius R = 18 cm and a weak tunnelling process into the 'airy' mode occurred.The level of the bend losses for all considered bent HC MFs (Fig. 9) is very close to that of the straight HC MFs.
In conclusion, one can state that the optimal waveguide regime in the spectral region near λ = 10.6 µm for HC MFs made of high index glass (n > 2) is possible at N > 8, where N is a number of capillaries in the cladding.In this case, the process of tunnelling of the air core modes of HC MF into the 'airy' modes of an individual capillary is suppressed due to low quality factor of the 'airy' modes and thus a low loss waveguide regime for a bend becomes possible.

Conclusion
The guidance of CO 2 laser radiation in HC MFs with the cladding consisting of capillaries was analysed.Two main factors affecting the waveguide mechanism in these waveguide structures were proposed.The first factor is connected with the representation of the curved air core boundary of the HC MFs as azimuthal diffraction quasi -gratings.These azimuthal diffraction gratings occur due to the cylindrical surface of an individual capillary of the cladding as well as to a discrete rotational symmetry of their arrangement in the cladding.In this way, the air core mode of the HC MF is formed by the interference of space cylindrical harmonics originated from the light scattering on these quasi -diffraction gratings.The process of the air core mode formation is much more complicated compared to the one for the HC MFs with continuous rotational symmetry of the air core boundary.The interaction of cylindrical harmonics forming the air core mode of the HC MF with the capillary walls is different from each other.It leads to a weakened material loss effect on the waveguide regime in comparison with the HC MF with a continuous rotational symmetry of the air core boundary, for example, IC HC PCFs.The second factor is connected with the optical properties of an individual capillary, in particular, with the density of its eigenstates (leaky modes).This factor determines the widths of the transmission regions and the level of waveguide losses.Numerical analyses have shown that a low loss waveguide regime in the spectral region near λ = 10.6 µm for the HC MFs with the cladding consisting of high index glass capillaries becomes possible.The optimisation of the HC MF structure to achieve these regimes is a complicated multiparameter task depending on all geometry parameters characterizing the HC MFs and the value of a glass refractive index.The bend loss of the HC MFs with the cladding consisting of capillaries made of high index glasses has a resonance character depending on the bend radius.To suppress such resonances it is necessary to increase the number of capillaries in the cladding.It leads to a decrease in the capillary sizes and to a decrease in the quality factor of the 'airy' modes of the capillaries.
In the end, we would like to outline the prospects of future investigations in this field.In our opinion, to improve the waveguide properties of the HC MFs it is necessary to study the process of the air core mode formation more carefully.An effect of different types of symmetries of the capillaries arrangements in the cladding and the value of D core on the level of waveguide loss should be investigated.The optical properties of an individual capillary made of high index glass, in particular, its optical eigenstates and the density of these eigenstates depending on the geometry parameters of the capillary and glass refractive indices should be studied.To achieve a low loss guidance it is necessary to perform an optimisation of the HC MFs structure.This complicated optimisation task can be performed by powerful numerical algorithms which have already been applied, for example, to optimisation of the Bragg fibers structures [Biriukov et. al., 2008].Also, it is necessary to improve the technology of making high index glasses with a low material loss and a technology of the HC MFs fabrication.It is worth mentioning that the early experiments demonstrated the possibility of obtaining waveguide regimes for such HC MFs made of high index chalcogenide glasses at CO 2 laser wavelengths [Kosolapov et. al., 2011].As the light was well localized in the core, such fibers hold much promise for the delivery of CO 2 laser radiation.

Acknowledgment
The authors thank Alexandra Nikolskaya for her assistance in translating this chapter into English.

Fig. 1 .
Fig. 1. (left) HC MF with the cladding consisting of eight capillaries (right) HC MF with the cladding consisting of eight solid rods.
coordinates, H 0 is an amplitude of the incident plane wave and J n (λ 2 r) are the Bessel functions of first kind .If kn 22 2π these dependencies are shown for two values of the air core diameter core D 220 = µm and 320 µm.
have a higher loss level due to thicker capillary walls compared to the previous case.In Fig.2(b) HC MF with n = 2.8 and ins out dd / = 0.85 has the maximal loss due to the excitation of the second type of the capillary leaky modes.www.intechopen.comThesecapillary modes have a lower azimuthal index and a different radial dependence (Fig.3(right)).It is worth pointing out that these two types of the individual capillary leaky modes are the main reason for occurring high loss regions for the considered HC MFs made of high index glasses.

Fig. 3 .
Fig. 3. (left) a typical Pointing vector distribution for a capillary leaky mode in the cladding with a high azimuthal index; (right) the Pointing vector distribution for the second type of the capillary modes with a lower azimuthal index.Further, we will consider HC MF with eight capillaries in the cladding.The air core diameters and the glass refractive indices will be the same as in the case of HC MFs with six capillaries in the cladding.In Fig. 4 the loss dependencies for this HC MF are shown.As one can see from Fig. 4(a) HC MF with ins

Fig. 8 .
Fig. 8. (left) the air core mode of HC MF with 8 capillaries in the cladding begin to couple with an 'airy' mode of an individual capillary of the cladding at a certain value of the bend radius; (right) the resonance excitation of the 'airy' mode of an individual capillary occurs at a lower value of the bend radius.