Broadband transmission in hollow-core Bragg fibers ... - OSA Publishing

Broadband transmission in hollow-core Bragg fibers with geometrically distributed multilayered cladding Dora Juan Juan Hu,1,* Gandhi Alagappan,2 Yong-Kee Yeo,1 Perry Ping Shum3 and Ping Wu2 1 Institute for Infocomm Research, Agency for Science, Technology and Research (A*STAR), Singapore Institute of High Performance Computing, Agency for Science, Technology and Research (A*STAR), Singapore 3 School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore *jjhu@i2r.a-star.edu.sg

2

Abstract: For the first time, the quasiperiodic Bragg fibers with geometrically distributed multilayered cladding are proposed and analyzed. We demonstrate that hollow-core Bragg fibers with quasiperiodic dielectric multilayer cladding can achieve low loss transmission over a broadband wavelength range of more than an octave (from 0.81 µ m to 1.7 µ m ). The periods of the Bragg blocks follows a geometrical progression with a common ratio rnL) and the refractive index of the hollow core is n0 = 1. There are m numbers of Bragg blocks and the period for k-th block is pk (pk-1< pk< pk + 1). N unit cells exist in each block, i.e. N pairs of two dielectric materials are alternating in the block. The periods of the Bragg blocks are arranged in a geometrical distribution, i.e. the common ratio of the periods of two adjacent blocks r = pk + 1/ pk. For instance, the fiber structure with m = 2, N = 4 and its refractive index profile in radial direction are shown in Figs. 1(c) and 1(d). The thickness of each dielectric layer is governed by the quarter-wave condition along the light line of air [9]. First of all, the effective transverse wavelength λt ,i in i-th layer can be estimated by

kt2,i = ki2 − k z2

(1)

where kt ,i = 2π λt ,i , ki = 2π λ ⋅ ni , and k z = 2π λ ⋅ neff . Here λ is the targeted operating

wavelength in middle of the bandgap,

ni is the refractive index of the material in i-th layer,

and neff is the effective mode index which is less than but very close to n0. For Bragg fibers with a large core size compared to the wavelength which, most of the mode energy resides in air and it is reasonable approximation to use neff = n0 . The optimum layer thickness can be computed as

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Received 13 Jan 2010; revised 24 Jan 2010; accepted 30 Jan 2010; published 18 Aug 2010

30 August 2010 / Vol. 18, No. 18 / OPTICS EXPRESS 18672

li =

λt ,i 4

+s

λt ,i

, s = 0,1, 2...

2

(2)

When s = 0, the fundamental Bragg condition is satisfied. Other s values correspond to higher-order Bragg conditions. We chose s = 0 in this work, so that we have an explicit expression to estimate lH / L , i.e. Equation (3) shows the optimal layer thickness of the two materials with high/low refractive indices nH/L in the bi-layer PC cladding. lH / L =

λ 4 n

2 H /L

−n

2 0

=

λ 2 H /L

4 n

−1

(3)

Fig. 1. (a) Schematic diagram of the hollow-core Bragg fiber cross-section. (b) The quasiperiodic cladding is formed by m numbers of Bragg blocks; there are N unit cells in each block, i.e. N pairs of two dielectric materials alternating in the block. The period for k-th Bragg block is pk. The ratio of the periods in two adjacent Bragg block is r = pk/ pk-1. (c) An example of a quasiperiodic Bragg fiber with m = 2, N = 4, r = 1.5, and (d) its refractive index profile in radial direction.

The bandgap generated by the multilayer cladding in the Bragg fiber, can be approximated to that of a planar multilayer structure. The bandgap of the 1D PC with an infinite number of unit cells can be calculated by the plane wave expansion method [10] or transfer matrix method [11]. The 1D PC can be formed by finite number of unit cells, or a series of PC blocks comprising a finite number of unit cells. In order to treat the structure in a more generic manner, we rely on TMM technique. The transfer matrix for the bilayer of two materials with high/low refractive index can be written as [12]:

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MT =

where ρT =

1 1 − ρT2

 e j (δ H +δ L ) − ρT2 e j (δ H −δ L )   2 j ρT e jδ H sin δ L

−2 j ρT e − jδ H sin δ L e

− j (δ H +δ L )

2 − j ( δ H −δ L ) T

−ρ e

  

(4)

nHT − nLT , δ H = k H lH cos θ H , δ L = k L lL cos θ L , k H = 2π λ ⋅ nH , k L = 2π λ ⋅ nL , nHT + nLT

and

nHT

 nH  nL TM polarization   =  cos θ H , nLT =  cos θ L n cos θ n cos θ TE polarization  H  L H L

(5)

θ H and θ L are angle of refraction in the layer of high/low refractive index, and they can be obtained by using Snell’s law, sin θ H / L =

n0 sin θinc sin θ inc = nH / L nH / L

(6)

θinc is the angle of incidence, which can vary over 0° ≤ θinc ≤ 90° . It is to be noted that 1 − sin 2 θ H / L or

cos θ H / L can be expressed as

1−

sin 2 θ inc . The eigenvalues of M T , i.e. nH2 / L

λ± = e jKl , can be either both real-valued or both unit-magnitude complex-valued. K is the Bloch wavenumber, where K = cos−1 ( a ) l , l = lH + lL , a is the trace of M T :

a=

cos (δ H + δ L ) − ρT2 cos (δ H − δ L )

(7)

1 − ρT2

The multilayer structure is primarily reflecting if K is imaginary and the eigenvalues λ± are real-valued, and it is primarily transmitting if K is real and the eigenvalues λ± are pure phases. Therefore, the condition a = −1 determines the bandedge wavelengths of the high reflection bands, which is equivalent to:  δ + δL cos 2  H  2

 2 2  δH − δL   = ρT cos  2    

(8)

Notice that we can define the parameters L± = nH lH cos θ H ± nL lL cos θ L , so that

δH ± δL 2

=

( kH lH cos θ H ± kL lL cos θ L ) 2

=

kL± 2

(9)

where k = 2π λ , is free space wavenumber. Taking the square root of Eq. (8) on both sides, we have

π  π  cos  L+  = − ρT cos  L−  λ λ  1   1  π  π  cos  L+  = ρT cos  L−   λ2   λ2 

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(10)



Here λ1 and λ2 refer to the lower and upper bandgap edge wavelength respectively. The coefficients L± and ρT are dependent on polarization state (TE or TM) and the incidence angle θinc . Equation (10) is the general expression to obtain the solution of bandgap edge wavelengths for all incidence angles, i.e. 0° ≤ θ inc ≤ 90° , and for both TM and TE polarizations. They can be solved numerically. Note that at grazing incidence or incidence angle near 90° , which is the case for the propagating light incident on the core/cladding interface inside the Bragg fiber, if we employ the fiber design of quarter-wave condition along the light line of air [9], it is fairly accurate to set L− = 0 . Therefore, we can have an approximate analytical solution for Eq. (10). Note that the lower and upper bandgap edges of the k-th PC block with period of pk in the Bragg fiber with hetero-structured cladding shown in Fig. 1 can be denoted as λk , − and λk , + . The approximate analytical solution is given in Eq. (11). On the other hand, L− ≠ 0 for other θinc values, no analytical solution is available and Eq. (10) has to be solved numerically.

λk , − = λ1 = λk , +

π L+ cos

−1

(− ρ ) T

π L+ = λ2 = cos −1 ( ρT

)

=

=

π L+

π 2 + sin −1 ( ρT π L+

π 2 − sin −1 ( ρT

)

(11)

)

In order to verify the accuracy of the Eq. (10), we also consider the special case with the same 1D PC structure discussed in [8]. When θinc = 0° , and the 1D PC is designed with quarter wavelength thickness at normal incidence on the planar multilayer structure, i.e. lH =

pk nL pn , lL = k H nH + nL nH + nL

(12)

nH − nL . We nH + nL observe that Eq. (11) is still the analytical solution in this special case. It is also equivalent to the expression derived in [8] at θinc = 0° . Having obtained the expressions for bandgap edge wavelengths of the 1D PC, we are ready to employ the idea of creating large bandgap by hetero-structured PC blocks with periods in geometrical distribution in the Bragg fiber. The critical common ratio, rc, would lead to the threshold condition of bandgap of adjacent blocks overlapping, i.e. λk −1, + = λk , − . From Eq. (11), rc is defined as pk is the period of the PC. We can derive L+ = nH lH + nL lL , L− = 0 , ρT =

rc =

π 2 + sin −1 ( ρT ) π 2 − sin −1 ( ρT )

(13)

Equation (13) is the general expression of the critical common ratio for all incidence angles, and for both TE and TM polarizations. The stretched bandgap of the hetero-structured cladding, is determined by the lower band edge wavelength of the first Bragg block λ1, − , and the upper band edge wavelength of the last Bragg block λm, + . The common ratio r, is r = ( pm p1 )

1 m −1

. The condition, r < rc, has to be satisfied in the hetero-structured cladding

design [8]. Figure 2 shows the critical common ratio rc, for the quasiperiodic cladding with materials of refractive index nL = 1.625, nH = 2.896, 3.6, and 4.6 respectively. There are a few

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observations from Fig. 2. First of all, the rc values for TM polarization are always smaller than that for TE polarization. Second, the minimum rc value occur at θinc = 0° , 90° for TE and TM polarization respectively. This is because for TE polarization, the bandgap is getting broader with increasing θinc , whereas for TM polarization, the light line lies above the point where the TM bandgap becomes zero [8]. Third, the rc curves are shifted to higher values when the refractive index difference between the two materials are larger. TM TE

2.2

2

nH=4.6

1.8 rc

nH=3.6 1.6 nH=2.896 1.4

1.2

1 0

10

20

30

40

50

60

70

80

90

θinc (oC)

Fig. 2. rc as a function of incidence angel θ inc and polarization. The refractive index nL = 1.625, nH = 2.896, 3.6, 4.6 respectively.

Fig. 3. Projected band structure of the planar periodic multilayer structure. The refractive index of the two materials in the Bragg cladding nL = 1.625, nH = 2.896. The solid line is the light line. The gray and black regions correspond to TE allowed band, and TM allowed band respectively. The white regions correspond to the total bandgaps.

Next, we set nH = 2.896 , nL = 1.625 , and assume the Bragg cladding has infinite number of unit cells of the two materials. The projected cladding bandgap map, which is approximated by the planar Bragg stack in the limit of large core radius [9], is calculated by TMM technique and shown in Fig. 3. β is the propagation constant. The TE allowed band and TM allowed band are plotted in gray and black regions, and the total bandgap is plotted in the white regions. It is clear that TM bandgap is narrower than TE bandgap. The omni-directional reflection at all angles and for both polarizations can occur within the range plotted by dashdot lines. #122629 - $15.00 USD

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The bandgap map of the quasiperiodic cladding cannot be obtained by assuming the structure has the infinite unit cells of uniform periodicity. Instead, by using transfer matrix method (TMM), we can treat the realistic hetero-structured cladding with finite layers, consisting of several Bragg blocks with different periods. Figure 4 plots the reflection factor distribution as a function of wavelength and incidence angle for both TE and TM polarizations. The quasiperiodic Bragg fiber has parameters m = 5, N = 6, nH = 2.896 , nL = 1.625 . Figure 4(a) is plotted with p1 = p2 = ⋅⋅⋅ = pm = 287 nm, corresponding to the periodic 1D PC, with 30 unit cells. Figure 4(b) is plotted with p1 = 287 nm, pm p1 = 1.2, corresponding to the quasiperiodic Bragg cladding. Notice that in the fundamental bandgap for both polarizations, the wavelength range enclosed by the dash-dot lines shown in (a) for illustration, can be used for omni-directional reflection for all incidence angles and for both polarizations. A broadened bandgap is observed in the quasiperiodic Bragg cladding.

Fig. 4. The reflection R versus wavelength λ and incidence angle θ for both TE and TM polarization. The Bragg cladding has the following parameters: nL = 1.625, nH = 2.896, m = 5, N = 6, (a) periodic 1D PC, p1 = p2 = ⋅⋅⋅ = pm = 287 nm. (b) quasiperiodic Bragg cladding, p1 = 287 nm, pm p1 = 1.2.

Figure 5(a) plots the bandedge wavelengths of the fundamental bandgap for periodic 1D PC (upper), quasiperiodic 1D PC (lower). The omni-reflection range is marked by the soliddash lines. The transmission spectrum at incidence angle θ = 90° for TE (dashed) and TM (solid) polarizations are plotted in Fig. 5(b). Notice that the lower bandgap edge wavelength is determined by the first block which has the same period in both periodic PC and quasiperiodic PC, so that it is the same in both cases. The upper bandgap edge wavelength is determined by the last block, so that we can observe an apparent red shift in quasiperiodic PC for both polarizations. Therefore, for each polarization states, the bandgap range is broadened in quasiperiodic 1D PC.

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90

1 TE TM

o θ ( C)

Transmission

60

30

TE TM

0.8 0.6 0.4 0.2

0 600

800

1000

1200

1400

1600

1800

0

2000

500

1000

λ (nm)

1500

2000

λ (nm)

90

1 TE TM

TE TM

0.8

o θ ( C)

Transmission

60

30

0.6 0.4 0.2

0 600

800

1000

1200

1400

1600

1800

2000

0

500

λ (nm)

1000

1500

2000

λ (nm)

(a)

(b)

Fig. 5. (a) The bandedge wavelengths of the fundamental bandgap for the periodic 1D PC (upper), and the quasiperiodic 1D PC (lower). There is obvious broadening in omni-reflection range marked by the solid-dash lines. (b) The transmission at incidence angle θ = 90° for TE (dashed) and TM (solid) polarizations.

In the next section, we will analyze the guided modes in the periodic and quasiperiodic Bragg fiber, and we will investigate the broadening effect of transmission spectrum in the quasiperiodic Bragg fiber with hetero-structured cladding. 3. Guided modes in the quasiperiodic Bragg fibers

The modal characteristics are calculated by transfer matrix method (TMM) [1]. We consider the wave equation in the i-th layer which is homogeneous material region:  E ( r , θ )  ∇ t2 + kt2,i   z =0  H z ( r , θ ) 

(14)

where ∇t2 = ∇ 2 − ∂ 2 ∂z 2 is the, kt ,i = k 2 ni2 − β 2 transverse wave number, ni is the refractive index in the i-th layer, β is the modal propagation constant which is generally a complex number. The general solution for the field components is written as a linear combination of Bessel function of the first kind of order m which represents a standing wave and Hankel function of the first kind of order m which represents an outward-traveling wave.

Ez =  Am J m ( kt ,i r ) + Bm H m ( kt ,i r )  exp ( imθ ) H z = Cm J m ( kt ,i r ) + Dm H m ( kt ,i r )  exp ( imθ )

(15)

where Am , Bm , Cm , Dm are expansion coefficients. The field continuity through the interface between two dielectric layers can produce the transfer matrix equation which relates the field in the core to that in the outermost layer:

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 A1m   AmN   1  N  Bm   Bm   1  = M r1 M r 2 ⋅⋅⋅ M r ( N −1)  N  Cm  Cm   D1  DN   m  m

(16)

By enforcing zero outward-traveling wave in the innermost layer and zero standing wave in the outermost layer, β can be solved by Eq. (16), and subsequently we can obtain the field profile. The effective modal index neff = β k , and the confinement loss (in dB/km): CL = k

20 109 Im ( neff ln (10 )

)

(17)

We consider a hollow-core Bragg fiber with multilayer claddings of PES/As2Se3 materials [2]. The material refractive index profiles are given in Fig. 6 [13]. The hollow-core Bragg fibers with periodic 1D PC cladding and quasiperiodic cladding are modeled. The core radius rcore = 10 µ m , the period of the first block p1 = 287 nm, pm p1 = 1.2 , the layer thickness is 195 nm, and 92 nm for PES and As2Se3 respectively. In the following discussions, we focus on the TE01 mode. 3.6 PES As 2Se3

3.4 3.2 3

2.6 2.4 2.2

Imag (n)

Real (n)

2.8

-5

10

2 1.8

1

λ (µm) 1.5

2

1.6 0.6

0.8

1

1.2

1.4

1.6

1.8

2

λ (µm)

Fig. 6. Material refractive index of PES and As2Se3 [13]

First of all, we assume that the cladding materials are non-absorptive, which is the ideal case. The fiber modal loss is purely due to the radiation/leakage loss. The imaginary part of the material refractive index is zero. The material dispersion is considered in the calculation, i.e. the wavelength dependence of the real part of the refractive index is included. In Fig. 7 (a), the loss of TE01 mode of the hollow-core periodic Bragg fiber and the quasiperiodic fiber are calculated. The circle line represents the loss curve for the periodic Bragg fiber, with 5 Bragg blocks, i.e. m = 5, and each Bragg block has 3 unit cells, i.e. N = 3. The guiding range corresponding to the first photonic bandgap spans from 0.8 µ m to 1.3 µ m . The other three lines represent the loss curve for the quasiperiodic Bragg fibers. The ratio of periods of the last Bragg block to the first Bragg block, pm p1 = 1.2 . The other cladding parameters are N = 3, m = 5 (star), N = 6, m = 5 (triangle), and N = 3, m = 10 (square). We can observe that the upper transmission band edge wavelength is pushed to larger values in quasiperiodic Bragg fibers. This is in agreement with the calculation of the photonic bandgap of the heterostructured cladding. By increasing the number of unit cells N in each Bragg block, or/and

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number of Bragg blocks m in the cladding, the loss is substantially suppressed. As less than 1 dB/m, i.e. 103 dB/km transmission loss is required for most applications, we plot the horizontal dashed line at this loss level, below which we consider low loss guiding range. It is clear that the low loss guiding range is evidently broadened. The broadening factor of the range is approximately 75% in quasiperiodic Bragg fiber than that in periodic Bragg fiber, i.e. from approximately 0.4 µ m to 0.7 µ m . Since the bandgap edge wavelengths are determined by the first and the last Bragg block, the ratio of pm p1 is an important parameter in stretching the transmission bandwidth. In Fig. 7(b), we show the effect of increasing pm p1 on the modal loss of the fiber. When pm p1 varies from 1.2, 1.4 to 1.6, indicated by the loss curves with star, square and triangle respectively, the guiding range is stretched. Meanwhile, the loss level is increased with larger pm p1 values if m and N are the same. Since larger m and N of the hetero-structured cladding can significantly reduce the loss level, we increase the number of Bragg blocks i.e., m = 10, and plot the loss curve as the solid line with squares. Indeed, the quasiperiodic Bragg fiber with N = 6, m = 10 give the best feature of broadening the transmission spectrum as well as keeping the loss level substantially low. In summary, for non-absorptive cladding materials, we can effectively broaden the low loss transmission bandwidth by designing the fiber the quasiperiodic structure in the cladding. 8

8

10

10

pm/p1=1.2,N=6,m=5 pm/p1=1.4,N=6,m=5 6

6

10

10

pm/p1=1.6,N=6,m=5 pm/p1=1.2,N=6,m=10

4

4

10 Confinement Loss (dB/km)

Confinement Loss (dB/km)

10

2

10

0

10

-2

2

10

0

10

-2

10

10 pm/p1=1.0, N=3, m=5 pm/p1=1.2, N=3, m=5

-4

10

-4

10

pm/p1=1.2, N=6, m=5 pm/p1=1.2, N=3, m=10 -6

-6

10

10 0.5

1

λ (µm) (a)

1.5

2

0.5

1

1.5

2

λ (µm) (b)

Fig. 7. Loss of TE01 mode of the periodic Bragg fiber, and quasiperiodic Bragg fiber with hetero-structured cladding. The fiber parameters are: nL, nH are assuming zero absorption loss. Other fiber parameters are: (a) Periodic Bragg fiber, N = 3, m = 5 (circle). Quasiperiodic Bragg fiber, pm p1 = 1.2 , N = 3, m = 5 (star); N = 6, m = 5 (triangle); and N = 3, m = 10 (square); (b) Quasiperiodic Bragg fiber, pm p1 = 1.2 , N = 6, m = 5 (star); pm p1 = 1.4 , N = 6, m = 5 (circle); pm p1 = 1.6 , N = 6, m = 5 (triangle); and N = 6, m = 10 (square).

If the cladding materials have absorption loss, we need to include the imaginary part of the refractive index of the materials. In the following calculation, both the material dispersion and the material absorption loss are considered. The azimuthal electrical field Eθ profile of TE01 mode at wavelength λ = 1.0, 1.4 and 1.6 µ m in the Bragg fiber with periodic cladding and quasiperiodic cladding are shown in Figs. 8(a) and 8(b) respectively. We can observe that the

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1

0.5

0.5

E

θ

1

E

θ

field penetration into the lossy cladding in Fig. 8(a) is more than that shown in Fig. 8(b). In other words, the modal power distributed within the hollow core of the Bragg fiber with periodic cladding is less than that of the Bragg fiber with quasiperiodic cladding. We calculate the percentage of the modal power distributed in the hollow core region. The periodic Bragg fiber has 99.98%, 97.89% and 97.56% of the modal power in the hollow core at three wavelengths. Comparatively, about 99.98%, 99.49%, 98.6% of the modal power is distributed in the hollow core of the quasiperiodic Bragg fiber at three wavelengths. The quasiperiodic Bragg fiber has more power within the hollow core; hence results lower loss than the periodic Bragg fiber, especially at longer wavelengths.

0

0 0

5

15

θ

0.5 0 5

10 r (µm)

10 r (µm)

15

0

5

10 r (µm)

15

0

5

10 r (µm)

15

0.5

15 1

0.5

0.5

θ

1 E

θ

5

0 0

E

0 1 E

E

θ

1

10 r (µm)

0

0 0

5

10 r (µm) (a)

15

(b)

Fig. 8. The azimuthal electric field profile Eθ of TE01 mode of the hollow Bragg fiber at

λ = 1.0, 1.4, and 1.6 µ m . (a) The cladding is periodic PC. (b) The cladding is quasiperiodic PC.

The loss characteristics of the periodic and quasiperiodic Bragg fiber are investigated theoretically. The absorption loss due to the material properties are considered in the mode loss calculation using Eq. (17). The loss profiles of the periodic (solid line) and quasiperiodic (solid circle line) Bragg fiber are shown in Fig. 9. The shape of the loss profiles are different from Fig. 7(a) in which the material absorption loss is neglected. Specifically, we can observe a much higher loss within the photonic bandgap range from 0.9 µ m to 1.3 µ m , the loss level is in the order of 102 dB/km. Due to the overall increase in the loss level, we plot the reference level at 105 dB/km denoted by the horizontal dotted line. It is clear that the transmission range of the quasiperiodic Bragg fiber below the horizontal dotted line is evidently broadened. Compared with the guiding range of the periodic Bragg fiber, i.e. approximately from 0.82 µ m to 1.28 µ m , the guiding range of the quasiperiodic Bragg fiber is stretched approximately by two times, i.e. from 0.81 µ m to 1.7 µ m , with modal loss lower than 105 dB/km. The lower bandgap edge is less steep in quasiperiodic Bragg fiber. Within the bandgap region of the periodic Bragg fiber, the loss values are almost the same for the periodic and quasiperiodic Bragg fiber. The upper bandgap edge is less steep in the quasiperiodic Bragg fiber, and pushed to longer wavelength from 1.3 µ m to 1.35 µ m .

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Notice that the upper bandgap edge of the quasiperiodic 1D PC is about 1.88 µ m shown in Fig. 5(a), which is different from that in the transmission spectrum of the quasiperiodic Bragg fiber. This can be due to the fact that the ‘ripples’ in the transmission spectrum of the quasiperiodic 1D PC cladding leads to abrupt ‘building-up’ of the modal loss of the Bragg fiber. To confirm it, we plot the magnified figures of the normalized transmission spectrum of the 1D quasiperiodic PC cladding on the inset of Fig. 9(a). There is a ‘peak’ value at about 1.354 µ m , and 1.459 µ m in the inset figures. Consequently, there are corresponding ‘peak’ values in the fiber modal loss at the wavelengths. Beyond 1.35 µ m , the loss of TE01 mode in quasiperiodic Bragg fiber is greatly reduced compared to that of periodic Bragg fiber, and it exhibits band behavior. Effectively, the upper edge of the TE01 mode for quasiperiodic Bragg fiber is pushed to about 1.7 µ m . It should be noted that large material absorption loss of the cladding material, especially PES, is detrimental to the usefulness and effectiveness of the proposed scheme of using quasiperiodic cladding structure to stretch the transmission range. In Fig. 9(b), we replace PES with SiO2, and calculate the confinement loss of TE01 mode for the quasiperiodic Bragg fiber with cladding materials of SiO2 and As2Se3 shown in Fig. 9(b). The material dispersion and the material absorption loss of SiO2 are included in the calculation. We can observe an evident broadening of guiding range for structures with pm p1 varying from 1.0 (denoted by solid star line) to 1.2 (denoted by solid square). The upper transmission band edge wavelength increases from 1.25 µ m for pm p1 = 1.0 , to 1.7 µ m for pm p1 = 1.2 without any ‘spikes’ in the confinement loss curve. The materials that are used to form the multilayer cladding require careful selections. In Fig. 9(b) we used SiO2 and As2Se3, which have low absorption loss, with the purpose of demonstrating the capabilities of the hetero-structured cladding in broadening the guiding range. (a)

8


pm/p1=1.0

0.5

pm/p1=1.2

0 1352

6

10

10

6

10

1354 1356 λ (nm)

4

5

10

4

10

Transmission

3

10

2

10

0.6

0.8

1

(b)

8

1


7

10

Transmission

10

1.2

10

SiO2/As2Se3,pm/p1=1.0 SiO2/As2Se3,pm/p1=1.2

2

10

0

10

-2

10

1 -4

10

0.5 0

1.4 λ (µm)

1458

1460 1.6λ (nm)1.8

1462 2

-6

10

0.6

0.8

1

1.2

1.4

1.6

1.8

2

λ (µm)

Fig. 9. Loss of TE01 mode of the periodic (solid curve) and quasiperiodic Bragg fiber (solid curve with circles). (a) The Bragg cladding has the following parameters: nL, nH are from [13], m = 5, N = 6, the periodic Bragg fiber has 1D PC, p1 = p2 = ⋅⋅⋅ = pm = 287 nm. The quasiperiodic Bragg fiber has hetero-structured cladding, p1 = 287 nm, pm p1 = 1.2. (b) The cladding materials are SiO2/As2Se3, the fiber structural parameters are the same as those used in Fig. 9(a) with cladding materials of PES/As2Se3. The Loss of TE01 mode of the quasiperiodic Bragg fiber shows an evident broadening with pm p1 varying from 1.0 (solid star line) to 1.2 (solid square line).

For 1D hetero-structured Bragg blocks of non-absorptive materials, putting more unit cells in the Bragg blocks or having more Bragg blocks in the structure can result in better bandrejection transmission properties [8]. The quasiperiodic Bragg fibers with such non-absorptive hetero-structured claddings will obtain lower modal confinement loss, or leakage loss. For absorptive materials PES and As2Se3 which are considered in the present work, it has been #122629 - $15.00 USD

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reported that the loss level does not vary with the total number of layers in the cladding. This is because for the Bragg fiber with periodic 1D PC cladding, the absorption loss of the material dominates the total fiber loss when the number of layers exceeds 40 [14]. Here we study the effect of increasing the number of unit cells in each Bragg block, N, and the number of Bragg blocks, m, in the hetero-structured cladding on the fiber modal loss. In Fig. 10(a), the number of unit cells of PES/ As2Se3 bilayer in each Bragg block is varied, i.e. N = 6, 12, other fiber parameters are the same as those used in the previous figure. In Fig. 10(b), the number of Bragg block is varied, i.e. m = 5, 10, other fiber parameters are the same as those used in the previous figure. We can observe that the larger N and larger m do not reduce modal loss. This is conforming to the results reported in [14]. In addition, the modal loss exhibit more ‘spikes’ at long wavelengths beyond 1.35 µ m . The ‘spikes’ are caused by larger portion of power residing in the lossy cladding. For instance, at 1.37 µ m eclipsed in Fig. 9(b), the modal power distributed in the hollow core is 99.982% for m = 5, and 99.573% for m = 10. The corresponding azimuthal electric field profiles of TE01 mode are plotted in the inset figures. (a)

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Fig. 10. Loss of TE01 mode of the quasiperiodic Bragg fiber with hetero-structured cladding. The fiber parameters are: p1 = 287 nm, pm p1 = 1.2, nL, nH are from [13]. Other fiber parameters are: (a) m = 5. The loss curve of the fiber with N = 6 is plotted in solid curve with circles, N = 12 is plotted in solid curve with stars. (b) N = 6. The loss curve of the fiber with m = 5 is plotted in solid curve with circles, and m = 10 is plotted in solid curve with stars.

It should be noted that the hollow-core Bragg fiber consisting of PES and As2Se3 in the cladding can be fabricated by preform preparation and fiber drawing technique [1]. Recently, a hollow-core Bragg fiber with all-solid multilayer composite meso-structure having as many as 35 periods, i.e. 70 layers is reported [15]. Therefore, we believe that the proposed hollowcore quasiperiodic Bragg fiber with 60 layers in the cladding is within the capability of current fabrication technology. 4. Conclusions

We present a robust scheme of achieving broadband low loss transmission in hollow-core quasiperiodic Bragg fibers. The cladding of the fiber is made of a series of Bragg blocks with different periods, consisting of two materials with different refractive indices. The effective bandgap edge wavelengths of the cladding, corresponding to the transmission band edge wavelengths of the fiber mode, are determined by the first and the last Bragg block. The common ratio of the periods in two adjacent Bragg blocks has to satisfy certain condition in order to obtain a stretched bandgap. The bandgap properties in the hetero-structured cladding are studied. The loss characteristics of the quasiperiodic Bragg fiber are investigated. It should be noted that due to large absorption loss of the cladding material, increasing the number of unit cells in each Bragg block, and increasing the number of Bragg blocks in the cladding do

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not lead to modal loss reduction in this work. However, if the cladding materials are less lossy, ideally non-absorptive so that the fiber loss is solely from radiation/leakage loss, such as silica, the proposed design scheme can effectively broadens the low loss transmission range. Furthermore, due to scalability of the photonic bandgap of the multilayered cladding, the proposed scheme can be applied to any other desired frequency range with suitable fiber materials. Acknowledgement

The authors would like to thank Dr. Min Yan from Royal Institute of Technology (KTH), Sweden for helpful discussions on fiber modal analysis by transfer matrix method.

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