Porous photonic-crystal fiber with near-zero ultra-flattened dispersion and high birefringence for polarization-maintaining terahertz transmission

Journal Pre-proof Porous photonic-crystal fiber with near-zero ultra-flattened dispersion and high birefringence for polarization-maintaining terahertz transmission Yani Zhang, Lu Xue, Dun Qiao, Zhe Guang

PII:

S0030-4026(19)31715-2

DOI:

https://doi.org/10.1016/j.ijleo.2019.163817

Reference:

IJLEO 163817

To appear in:

Optik

Received Date:

20 September 2019

Revised Date:

11 November 2019

Accepted Date:

19 November 2019

Please cite this article as: Zhang Y, Xue L, Qiao D, Guang Z, Porous photonic-crystal fiber with near-zero ultra-flattened dispersion and high birefringence for polarization-maintaining terahertz transmission, Optik (2019), doi: https://doi.org/10.1016/j.ijleo.2019.163817

This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. 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. © 2019 Published by Elsevier.

Porous photonic-crystal fiber with near-zero ultra-flattened dispersion and high birefringence for polarization-maintaining terahertz transmission

YANI ZHANG1,2,6,*,LU XUE2,6, DUN QIAO3, ZHE GUANG4,5,#

1

Department of Physics, Shaanxi University of Science& Technology, Xi’an 710021, China

2

School of Physics and Optoelectronics Technology, Baoji University of Arts & Science, Baoji, Shaanxi 721016,

China and Optoelectronics Research and Innovation Centre, Faculty of Computing, Engineering and Science,

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3 Wireless

University of South Wales, Pontypridd CF37 1DL, UK 4 School 5

of Physics, Georgia Institute of Technology, 837 State Street, Atlanta, GA 30332, USA

School of Computer Science, Georgia Institute of Technology, 266 Ferst Drive, Atlanta, GA 30332, USA

6 Baoji

Engineering Technology Research Centre on Ultrafast Laser and New Materials, Baoji, Shaanxi 721016,

author: [email protected]

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*Corresponding

author: [email protected]

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#Co-corresponding

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China

Abstract: One unique kind of porous-core photonic crystal fiber (PCF) is proposed with elliptical air-holes in core, and round-corner hexagonal air-holes in cladding, for efficiently transmitting polarization-maintaining terahertz waves. Dispersion, birefringence, effective material loss (EML) and

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confinement loss are investigated by using the full-vector finite element method (FV-FEM) with anisotropic perfectly matched layers. The numerical results indicate that its near-zero ultra-flattened dispersion (-0.01±0.06 ps/THz/cm) and birefringence higher than 7.0×10−2 (highest birefringence of 7.4×10−2 at 1.2 THz) can be achieved simultaneously over almost the same frequency range of 0.75-1.6 THz, with a low EML (0.08 cm-1 at 1.2 THz) and a low confinement loss (3.2×10-13 cm-1 at 1.2THz). Important modal properties, such as power fraction and effective mode area, are discussed, and the

tolerance in fabrication is analyzed to indicate the feasibility of the proposed terahertz waveguides

with PCF structure in manufacturing. Keywords: Polymer optical fiber (POF); Far infrared or terahertz; Birefringence; Dispersion; Fiber optical communication; Fiber design

1. Introduction Terahertz (THz) radiation, which lies between optical and microwave frequencies, can be used extensively for many applications, including security, sensing, spectroscopy and medical imaging [1-5]. Despite all benefits, transmission of THz waves has remained a challenge, with main difficulties in

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selecting low-loss waveguide materials and designing low-dispersion waveguide structures [6]. Recently, various types of THz waveguides have been proposed, such as metallic waveguides [7, 8], hollow-core waveguides [9-11], solid-core waveguides [12], and porous-core waveguides [13-23]. Among them, porous-core waveguides based on polymers have the advantages of relatively low

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absorption loss based on polymers, and the flexibility in waveguide design. For instance, asymmetrical sub-wavelength air-holes were introduced in the fiber core to achieve birefringence of 2.6×10−2 and

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dispersion value ≳0.85 ps/THz/cm [13], and porous fibers with rotated elliptical air-holes were demonstrated with birefringence of 4.45×10−2 [14] and 1.19×10−2 [15]. More recently, Jakeya et al.

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proposed an elliptical-core photonic crystal fiber (PCF) with a birefringence of 8.6×10−2 and a dispersion of 0.53±0.07 ps/THz/cm within 0.5-1.48 THz [22]. Also, with an array of elliptical air-holes

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in core and a rectangularly slotted cladding structure, Islam M. S. et al. proposed a high birefringence and ultra-flattened dispersion PCF based on Zeonex, which had birefringence of 6.3×10−2 and dispersion variation of ±0.02 ps/THz/cm within 1.05-1.5 THz [23]. However, given the remarkable properties of the proposed waveguides, it was found that either THz transmission efficiency can be

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limited by high loss, or the proposed complex patterns in the waveguide can be impractical to fabricate, which would limit the use of the waveguides in polarization-maintained transmission applications. In this paper, a novel and relatively simple-patterned porous-core PCF THz waveguide is proposed

with elliptical air-holes in core and rounded-corner hexagonal air-holes in cladding with a symmetrical lattice structure. We optimize the waveguide structure, based on a full-vector finite element method (FV-FEM) with anisotropic perfectly matched layers, to achieve high birefringence and low-flattened dispersion simultaneously. Also, we calculate important waveguide properties, such as effective material loss, confinement loss, power fraction, and effective mode area, under variant structural

parameters of the core diameter Dcore and the core air filling ration 2a/Λx. The numerical results indicate that the proposed PCF structure has high birefringence, near-zero ultra-flattened dispersion (with three zero dispersion points for y-polarized mode), strong guided-mode confinement, low confinement loss, and low effective material loss. In view of the tolerance of the proposed structure, the PCF should maintain its optimal performance within a variation of ±2% which comes from a standard fabrication process [24]. Based on PCF structure design, the proposed THz waveguide with high birefringence and ultra-flattened dispersion will have potential applications in the fields of ultrafast fiber lasers and THz polarization-maintaining transmission [25-27].

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2. Porous Fiber Structure and Design Principles The cross-section of the proposed PCF THz waveguide is shown in Fig. 1, along with an enlarged view of the porous-core region. In order to facilitate manufacturing, a simple cladding is chosen with

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hexagonal air-hole structures, consisting of 6 ring layers of rounded-corner air-holes. The pitch Λ is the distance between two adjacent air holes, d is the diameter of air holes and dc is the diameter of

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curvature for the air-holes in cladding. In our simulation, dc/Λ is selected to be 0.6 and the ratio of d/Λ is kept at a large value of 0.95 based on the fact that a larger d parameter tends to the better confine the

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light in core [24]. In order to increase the birefringence, elliptical air holes are used in the core region to introduce asymmetry in the structure. Here, the major and minor axes of the ellipse are denoted by 2b and 2a, and the pitch Λx and Λy are the distance between two adjacent elliptical air holes along

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horizontal and vertical directions, respectively. The diameter of the core area along horizontal direction is denoted by 𝐷core .

Cyclic olefin copolymer (trade name, Topas) is selected as the background material for the THz waveguide, which has a nearly constant refractive index of 1.53 between 0.1-2 THz with a low material

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dispersion [20]. It is well known that the bulk material loss of Topas is 0.06 cm−1 at 0.4 THz and increases at a rate of 0.36 cm−1 /THz [15]. Topas has also other important properties such as insensitivity to humidity, decency for bio-sensing, and the flexibility in fabrication to achieve high glass transition temperatures [20]. Here, numerical simulations are performed to calculate the effective indices of the electromagnetic modes in the PCF THz waveguide based on FV-FEM. A perfectly matched layer at depth of 10% radius from the outer boundary is used to reduce the effect of surrounding environment and materials to the

confinement loss. In general, the mechanism of light guidance through the waveguide is total internal reflection (TIR) [1, 24], and the dispersion originates from the waveguide structure and the material itself. Here, we consider only the waveguide dispersion because Topas has near zero material dispersion between 0.1–2 THz. The waveguide dispersion is determined by the geometrical structure, which can be expressed by [19]: 2 𝑑𝑛eff 𝑑𝜔

𝛽2 = 𝑐

+

𝜔 𝑑 2 𝑛eff 𝑐 𝑑𝜔2

(1)

vacuum, and 𝑛eff is the effective refractive index of the fiber.

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where 𝛽2 is the dispersion parameter, 𝜔 is the angular frequency, c is the speed of light in the

To achieve an effective polarization-maintaining PCF THz waveguide, the level of birefringence should be significant, which is [17]:

𝐵 = |𝑛x − 𝑛y |

(2)

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where B stands for birefringence, 𝑛x and 𝑛y are the effective refractive indices of the x and y polarized modes, respectively.

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Effective material loss (𝛼eff ), as an important parameter in THz waveguide, can be quantified as [18]:

(𝜀0 ⁄𝜇0 )1⁄2 ∫𝐴

𝑛𝛼𝑚𝑎𝑡 |𝐸|2 𝑑𝐴

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𝛼eff =

𝑚𝑎𝑡

2 ∫𝐴𝑙𝑙 𝑆𝑧 𝑑𝐴

(3)

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where 𝜀0 and 𝜇0 are the permittivity and permeability of the vacuum, respectively. n is the refractive index of Topas, 𝛼𝑚𝑎𝑡 is the bulk material absorption loss, E is the electric field component, and 𝑆𝑧 is the z-component of the Poynting vector (𝑆𝑧 =1⁄2 (𝐸 × 𝐻) ∙ 𝑧̂ ). The integral in the numerator is performed over the material regions of Topas, and the denominator is over all regions. Confinement loss is inevitable in THz waveguide due to finite extent of the periodic cladding,

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which can be calculated from following the equation [22]: 2𝜋𝑓 )𝐼𝑚(𝑛eff ) 𝑐

𝛼CL = 8.686(

(4)

where f is the operating frequency, and 𝐼𝑚(𝑛eff ) represents imaginary part of the refractive index. Mode power fraction is another important parameter that describes the amount of power propagating through different regions. Our goal is to achieve a high power fraction in the core, which can be expressed by the following [17]:

Power fraction =

∫𝑋 𝑆𝑍 𝑑𝐴

(5)

∫𝐴𝑙𝑙 𝑆𝑍 𝑑𝐴

where in the numerator the z-component of Poynting vector is integrated over the area of interest (denoted by X), and in the denominator the integral is over the total area. Finally, we characterize the effective mode area (𝐴eff ) of the proposed fiber [1]: [∫ 𝐼(𝑟)𝑟𝑑𝑟]2 ∫ 𝐼2 (𝑟)𝑑𝑟]

𝐴eff = [

(6)

where 𝐼(𝑟) = |𝐸𝑡 |2 is the transverse field intensity in the cross section of the fiber. In the following, we investigate the dependence of waveguide dispersion, birefringence, effective

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material loss, and confinement loss of the proposed THz waveguide on its structural parameters in detail, with the goal of optimizing its dispersion profile and birefringence in THz range.

3. Simulation Results and Discussion 3.1 Power flow distribution

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Firstly, the computed power flow distributions are shown in Fig. 2 for various core diameters 𝐷core . It can be observed that the mode power is tightly confined around the asymmetric porous-core, with the

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y-polarized modes better confined than x-polarized modes at Dcore=470 μm, which is essential for low material dispersion as well as low loss in the transmission of THz waves.

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3.2 Dispersion vs. Frequency

Secondly, we explore the dependence of dispersion (𝛽2 ) on core diameter 𝐷core and core air

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filling-ration 2a/𝛬x . Here, 𝛽2 as function of frequency is depicted in Fig. 3(a) for different 𝐷core at 2a/𝛬𝑥 =0.45. It can be observed that a near-zero flattened dispersion profile of the proposed THz waveguide could be achieved for a constant 𝐷core , and the corresponding region of flattened frequency becomes broader when 𝐷core is increased. Moreover, the y-polarized modes show lower dispersion

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than x-polarized modes at the flattened dispersion region, because the mode power is more tightly confined in the core for y-polarized modes than x-polarized modes. Note that when 𝐷core = 470 μm, the flattened dispersion nearly covered the frequency range of interest (0.8-1.4 THz), and the dispersion is 0.22±0.06 ps/THz/cm for y-polarized and 0.65±0.3 ps/THz/cm for x-polarized of the fundamental mode. The dependence of 𝛽2 on 2a/𝛬x is shown in Fig. 3(b) with a fixed 𝐷core = 470 μm. It is evident that 𝛽2 decrease with reducing values of 2a/𝛬x , due to the fact that more mode power is confined in

the core for lower core air-filling ratios. Furthermore, the y-polarized modes offer lower-flattened dispersion than the x-polarized modes over the frequency range (0.75-1.6 THz). Note that when 2a/𝛬x =0.35, near-zero ultra-flattened dispersion is obtained (-0.01±0.06 ps/THz/cm and 0.3±0.1 ps/THz/cm for y- and x-polarized modes, respectively) over the frequency range (0.75-1.4 THz). Also, three zero dispersion points (0.77 THz, 1.04 THz and 1.59 THz) are found for the proposed THz waveguide. Here, we also select 𝐷core = 470 μm and 2a/𝛬x = 0.35 as our optimal parameters. 3.3 Birefringence vs. Frequency Thirdly, the dependence of birefringence (B) on the core diameter 𝐷core and the core air filling-ration

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2a/𝛬x is shown in Fig. 4, with high birefringence on the order of 10−2. Here, birefringence B as

function of frequency for different 𝐷core at 2a/𝛬𝑥 =0.45 is depicted in Fig. 4(a). It can be observed that the frequency for maximum birefringence red-shifted with increasing of 𝐷core . The values of

birefringence exceeds 6.0×10−2 over the frequency range of interest (0.8-1.6 THz) when the 𝐷core is

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470 μm, and birefringence as high as 6.6×10−2 is obtained at f =1.2 THz.

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On the other hand, with fixed 𝐷core =470 μm, birefringence B as function of frequency for different 2a/𝛬x is shown in Fig. 4(b). It is evident that the birefringence increased when 2a/𝛬x

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decreased. When 2a/𝛬x is 0.35, birefringence higher than 7.0×10−2 is obtained over the frequency range (0.8-1.6 THz), with the highest birefringence of 7.4×10−2 at 1.2 THz. In brief, the increasing core diameter 𝐷core and decreasing core-air filling-ratio 2a/𝛬x would enhance the asymmetry in the fiber

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core. As a result, high birefringence can be obtained, where near-zero ultra-flattened dispersion also presents due to the air-core structure. 3.4 Other modal properties

Finally, with fixed 𝐷core =470 μm, important modal properties of the PCF THz waveguide, such as

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effective material loss, confinement loss, power fraction, effective mode area are also investigated for different 2a/𝛬x as shown in Fig. 5. Here, there is the same legend for all the figures in Figs. 5 as shown in Fig. 5(d) only. Figure 5(a) shows the effective material loss (EML) variation as a function of frequency. EML increases steeply with the increase of frequency, and the values of the y-polarized modes are higher than in the x-polarized modes, which could be explained by that the majority of the light propagates through the porous core area in the x-polarized modes with power flow distributions shown previously in Fig. 2. Also, at a fixed frequency, EML decreases with the increasing of 2a/𝛬x ,

with the reason being more light would propagate through the air-holes in the core than the material. Note that, for the optimal parameters (𝐷core = 470 μm and 2a/𝛬x = 0.35), EML is found as 0.08 cm−1 and 0.12 cm−1 for x- and y-polarized mode at 1.2 THz, respectively. Confinement loss (𝛼CL ) of the proposed THz waveguide as a function of frequency is shown in Fig. 5(b), with the trend of 𝛼CL decreasing with the increasing of frequency. When the 2a/𝛬x increases, 𝛼CL increases with smaller values for y-polarized modes than x-polarized modes. The reason is that the difference in effective refractive index between core and cladding for the y-polarized mode is larger than that of x-polarized mode. As a result, light is better confined in y-polarized mode,

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with the mode field distribution of x- and y-polarized modes shown in Fig. 6 for the selected optimal case. Here, 𝛼CL is found to be 3.6×10-10 cm−1 and 3.2×10-13 cm−1 , respectively, for x- and y-polarized modes at 1.2 THz.

The mode power fraction is also an important parameter to describe the propagating power through

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different regions. The fraction of mode power in the core air-holes as a function of 2a/𝛬x is shown in

Fig. 5(c), and it can be observed that when 2a/𝛬x increases the power fraction generally increases. The

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fraction of mode power in y-polarized modes is smaller than x-polarized modes, due to the effects of

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EML. It is found the power fraction in core air-holes was 57% and 33% of the total power for x- and y-polarized modes, respectively, at 1.2 THz for the optimal parameters. Effective mode area (𝐴eff ) is another important parameter to represent the beam quality in

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propagation. 𝐴eff as a function of 2a/𝛬x is shown in Fig. 5(d), where it can be observed that the 𝐴eff decreases with the increasing of frequency, and that 𝐴eff increases with the increasing of 2a/𝛬x . Therefore, the quality of the light beam could be improved by decreasing 2a/𝛬x , as modes would be more confined in the porous-core region. Here, 𝐴eff are 1.0×105 μm2 and 1.2×105 μm2 for x- and

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y-polarized mode, respectively, at 1.2 THz for the optimal parameters. 3.5 Feasibility of fabrication It is well known that the feasibility of fabrication is crucial for THz waveguide applications. Typically, a ±2% variation would arise from a standard fabrication process [24] for the global parameters. Here, the birefringence and dispersion of the proposed THz waveguide are calculated for up to ±2% variation of the optimal parameters, with results as shown in Fig. 7. It can be observed that the variations in 𝐷core and 2a/𝛬x do not significantly affect the high birefringence and near-zero

ultra-flattened dispersion over the frequency range of interest (0.75-1.4THz). Moreover, the proposed THz waveguide is based on simple structures, consisting of elliptical air-holes and hexagonal rounded-corner air-holes, which are commonly used and easy to fabricate. As for the fabrication, we have the experiences to fabricate the PCF with elliptical core by using in situ polymerization of polymer monomers [28] and the porous core POF in the THz frequency range via the extrusion stretching method [29]. The drilling and drawing technique to fabricate a honeycomb bandgap THz fiber has been proposed in our research group [30]. Especially, the technology of 3D printing was demonstrated recently to fabricate different complex-shaped asymmetrical air-holes [31]. Therefore, the

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fabrication of the proposed THz waveguide structure can be further achieved with better quality and accuracy by using 3D printing, drawing and extrusion technique or in situ polymerization.

4. Conclusion

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In conclusion, one type of porous-core waveguide with near-zero ultra-flattened dispersion and high birefringence is proposed for THz transmission applications. It is composed of a porous core of

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elliptical air-holes to induce high birefringence and a cladding with rounded-corner hexagonal air-holes to enhance the guided-mode confinement. Optimal structure parameters are selected for the proposed

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THz waveguide based on a systematic investigation, by the FV-FEM method, over a series of waveguide properties including dispersion, birefringence, effective modal loss, confinement loss, power fraction, and effective mode area. In a broad frequency range of 0.75-1.6 THz, a birefringence of

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higher than 7.0×10-2 and an ultra-flattened dispersion of -0.01±0.06 ps/THz/cm could be achieved simultaneously, with highest birefringence of 7.4×10−2 at 1.2 THz and three zero dispersion frequencies 0.77 THz, 1.04 THz and 1.59 THz. And, the proposed PCF THz waveguide has the good mode confinement due to low EML (<0.08 cm-1) and small confinement loss (~3.2×10-13 cm-1). In addition,

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with common geometric structure and tolerance for fabrication variations, the proposed waveguide is easy to fabricate by using 3D printing and extrusion technique or in situ polymerization. Therefore, the proposed waveguide can potentially serve as a polarization-maintained, lower losses, and ultra-low and ultra-flattened dispersion waveguide for broadband THz transmission applications and transition state of Q-switching/mode-locking.

Acknowledgement

This work was supported by the National Nature Science Foundation of China (No. 11647008), the International Science & Technology Cooperation and Exchanges Project of Shaanxi (No. 2018KW-016 ）, the Key Sciences and Technology Project of Baoji City (No. 2015CXNL-1-3), the Open Research Fund of State Key Laboratory of Transient Optics and Photonics (No. SKLST201802)

and the

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science and technology project of Xianyang City (No. 2018K02-60).

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[30] T. Ma, Andrey Markov, L. L. Wang, Maksim Skorobogatiy, Graded index porous optical fibers – dispersion management in terahertz range, Opt. Express 23 (6) (2015) 7856–7869.

[31] H. Ebendorff-Heidepriem, J. Schuppich, A. Dowler, L. Lima-Marques, T. M. Monro, 3D-printed extrusion dies: a versatile

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approach to optical material processing, Opt. Mater. Express 4 (8) (2014) 1494-1504.

Figure captions: Fig. 1. Cross-section of the PCF THz waveguide, and an enlarged view of the porous-core area. Fig. 2. Power flow distribution of the proposed fiber for (a) x-polarized modes and (b) y-polarized modes for various 𝐷core at the operating frequency of 1.0 THz when 2a/𝛬𝑥 =0.45. Fig.3. Dispersion versus frequency of the proposed fiber (a) for different 𝐷core at 2a/𝛬𝑥 =0.45, (b) for different 2a/𝛬𝑥 at 𝐷core = 470 μm. Fig.4. B versus frequency of the proposed fiber (a) for different 𝐷core at 2a/𝛬x =0.45, (b) for different 2a/𝛬x at 𝐷core =470 μm. Fig.5 (a) Effective material loss, (b) confinement loss, (c) power fraction in core air-holes, (d) effective mode area, over frequency for the proposed THz waveguide at different 2a/𝛬x and fixed 𝐷core =470 μm. Fig. 6. Mode field distribution of the proposed THz waveguide (a) x-polarized mode and (b) y-polarized mode with selected optimal parameters 2a/𝛬𝑥 =0.35 and 𝐷core = 470 μm at frequency of 1.2 THz. Fig.7. The tolerance of the proposed THz waveguide for up to ±2% variation with frequency (a) Birefringence (b)

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Figure captions:

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Fig. 1 by Yani Zhang et al.

Fig.1. Cross-section of the PCF THz waveguide, and an enlarged view of the porous-core

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Fig. 2 by Yani Zhang et al.

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Fig.2. Power flow distribution of the proposed fiber for (a) x-polarized modes and (b) y-polarized modes for various 𝐷core at the operating frequency of 1.0 THz when 2a/𝛬𝑥 =0.45.

Fig. 3 by Yani Zhang et al. Dcore=370m, y-pol

2.5

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2a/x =0.35, x-pol

2a/x =0.4, y-pol

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Fig. 4 by Yani Zhang et al.

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Fig.4. B versus frequency of the proposed fiber (a) for different 𝐷core at 2a/𝛬x =0.45, (b) for different 2a/𝛬x at 𝐷core=470 μm.

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Fig.5 (a) Effective material loss, (b) confinement loss, (c) power fraction in core air-holes, (d) effective mode area, over frequency for the proposed THz waveguide at different 2a/𝛬x and fixed 𝐷core =470 μm.

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Fig. 6 by Yani Zhang et al.

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Fig.6. Mode field distribution of the proposed THz waveguide (a) x-polarized mode and (b) y-polarized mode with selected optimal parameters 2a/𝛬𝑥 =0.35 and 𝐷core = 470 μm at frequency of 1.2 THz.

Fig. 7 by Yani Zhang et al. 0.075

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Fig.7. The tolerance of the proposed THz waveguide for up to ±2% variation with frequency (a) Birefringence (b) Dispersion

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