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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 18 — Sep. 1, 2008
  • pp: 13593–13598

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Low-loss air-core polarization maintaining terahertz fiber

Guobin Ren, Yandong Gong, Ping Shum, Xia Yu, JuanJuan Hu, Guanghui Wang, Michael Ong Ling Chuen, and Varghese Paulose  »View Author Affiliations


Optics Express, Vol. 16, Issue 18, pp. 13593-13598 (2008)
http://dx.doi.org/10.1364/OE.16.013593


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Abstract

We propose a low-loss air-core polarization maintaining polymer fiber for terahertz (THz) wave guiding. The periodic arrangement of square holes with round corners in the cladding offers a bandgap effect for mode guiding. Numerical simulations show that the bandgap effect repels the modal power from the absorbent background polymers, resulting in a significant suppression of absorption loss of the polymers by a factor of more than 25. The phase-index birefringence of the proposed THz fiber is in the order of 10-3.

© 2008 Optical Society of America

1. Introduction

With wavelengths covering the range of 30µm–3mm, terahertz (THz) radiation has an increasing variety of applications in biology and medical science, imaging, spectroscopy and communication technology [1

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1, 97–105 (2007). [CrossRef]

]. At present, most THz systems are large and difficult to use because they rely on free-space to guide and manipulate the THz pulses. This requires users to be well experienced with optical-alignment techniques and also to have direct line-of-sight access to the area or sample of interest. The principal difficulty has been the lack of materials well suited for guided propagation at THz frequencies. Materials such as glasses and polymers that work well at optical frequencies exhibit unacceptably high absorption losses at THz frequencies. Several kinds of THz waveguides have been proposed and demonstrated, such as metallic waveguides [2

J. Harrington, R. George, P. Pedersen, and E. Mueller, “Hollow polycarbonate waveguides with inner Cu coatings for delivery of terahertz radiation,” Opt. Express 12, 5263–5268 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-21-5263. [CrossRef] [PubMed]

], metallic wires [3

K. Wang and D. M. Mittleman, “Metal wires for terahertz wave guiding,” Nature , 432, 376–379 (2004). [CrossRef] [PubMed]

], plastic ribbons [4

R. Mendis and D. Grischkowsky, “Plastic ribbon THz waveguides,” J. Appl. Phys. 88, 4449–4451 (2000). [CrossRef]

], dielectric fibers [5

S. P. Jamison, R. W. McCowan, and D. Grischkowsky, “Single-mode waveguide propagation and reshaping of sub-ps terahertz pulses in sapphire fiber,” Appl. Phys. Lett. 76, 1987–1989 (2000). [CrossRef]

], and photonic crystal fibers [6

H. Han, H. Park, M. Cho, and J. Kim, “THz pulse propagation in plastic photonic crystal fiber,” Appl. Phys. Lett. 80, 2634–2636 (2002). [CrossRef]

, 7

M. Goto, A. Quema, H. Takahashi, S. Ono, and N. Sarukura, “Teflon photonic crystal fiber as terahertz waveguide,” Jpn. J. Appl. Lett. 43, L317–L319 (2004). [CrossRef]

]. The loss coefficients of these THz waveguides are still relatively high. The lowest loss 0.00898 cm-1 was obtained from a 3-mm core diameter fiber at 158.51µm wavelength for hollow polycarbonate waveguides with inner Cu coatings [3

K. Wang and D. M. Mittleman, “Metal wires for terahertz wave guiding,” Nature , 432, 376–379 (2004). [CrossRef] [PubMed]

]. Lu et al. [8

J.-Y. Lu, C.-P. Yu, H.-C. Chang, H.-W. Chen, Y.-T. Li, C.-L. Pan, and C.-K. Sun, “Terahertz air-core microstructure fiber,” Appl. Phys. Lett. 92, art. no. 064105, (2008). [CrossRef]

] demonstrated a low loss (less than 0.01 cm-1) terahertz air-core microstructure fiber. Polarization maintaining THz waveguides are essential for some polarization sensitive applications, such as the measurement of biomaterials in THz frequency band [9

T. W. Crowe, T. Globus, D. L. Woolard, and J. L. Hesler, “Terahertz Sources and Detectors and Their Application to Biological Sensing,” Phil. Trans. R. Soc. Lond. A 362, 365–377 (2004). [CrossRef]

]. Recently, Cho et al. [10

M. Cho, J. Kim, H. Park, Y. Han, K. Moon, E. Jung, and H. Han, “Highly birefringent terahertz polarization maintaining plastic photonic crystal fibers,” Opt. Express 16, 7–12 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-1-7. [CrossRef] [PubMed]

] demonstrated a highly birefringent plastic solid core photonic crystal fiber, which exhibits a large birefringence of ~2.1×10-2. But the attenuation of the fiber is limited by the material absorption, due to the high power fraction in the solid core.

Air-core photonic bandgap fibers (PBGFs) have engendered growing interest over the past few years since they have the potential to provide very low-loss transmission in air, along with delivery of high power and low nonlinearity [11

P. St. J. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003). [CrossRef] [PubMed]

]. To overcome the limitation of highly absorbent materials in the terahertz region, an efficient waveguide design should maximize the guided power fraction in the air, so the bandgap fiber is an instinctive choice for THz guiding.

In this paper, we propose an efficient air-core polarization maintaining THz fiber with high birefringence and low modal absorption loss. The fiber cladding consists with square-lattice subwavelength air holes, which effectively confines the guided modes in a rectangular air-core. Since most of the power fraction of the guided modes is confined in the hollow region, the modal absorption loss is then significantly reduced.

2. Transmission bandwidth

Fig. 1. Bandgap map of square lattice photonic crystal in fiber cladding with parameters: D/Λ=0.96, d/D=0.6 for PTFE polymer. The blank region represents the bandgap, in which the fiber modes are guided. The inset shows the unit cell, in which the four corners of a square hole are rounded with circles.

So far, most PBGFs have been fabricated from silica due to their applications in the optical domain. However, the material loss of silica is prohibitively high at THz frequencies. Thus, for THz applications, low loss plastics need to be used. In this paper, three polymers including Polytetraflouroethylene (PTFE), Polypropylene (PP) and high density Polyethylene (HDPE) will be considered as the background materials of the air-core THz fiber. The nearly constant refractive indices and the small absorption coefficients of PTFE (n=1.445), PP (n=1.51) and HDPE (n=1.534) in the terahertz region enable them to be good candidates for THz waveguide materials [12

Y. S. Jin, G. L. Kim, and S. G. Jeon, “Terahertz dielectric properties of polymers,” J. Korean Phys. Soc. 49, 513–517 (2006).

].

Since the square lattice offers a wider bandgap than that achievable with triangular lattice for bandgap fibers [13

F. Poletti and D. J. Richardson, “Hollow-core photonic bandgap fibers based on a square lattice cladding,” Opt. Lett. 32, 2282–2284 (2007). [CrossRef] [PubMed]

], the periodic arrangement of subwavelength square holes with round corners is chosen to form the cladding of the THz fiber. The inset of Fig. 1 shows the unit cell of the fiber cladding. The unit cells are regularly arranged in a square lattice to form the fiber cladding. The solid line denotes the boundary between the background material and the air. The lattice spacing is Λ; the size of the hollow part is characterized by a square hole with side D, the four corners of the hole are rounded with circles of diameter d (dash lines). A full-vectorial plane wave method is used to calculate the photonic bandgap of the perfect square lattice. We show in Fig. 1 the bandgap map of the fiber cladding, the blank region represents the bandgap, in which the fiber modes are guided. The background material is set as PTFE with refractive index nb =1.445. The air-line neff =1 represents the boundary between states that are propagating or evanescent in the cladding.

The bandgap shown in Fig. 1 introduces a forbidden area where modes cannot propagate along the fiber axis. Introducing a defect core will allow a finite number of modes to propagate in the defect core. The bandgap is characterized by several parameters: fU , where the high frequency band edge (nU ) crosses the air-line (also the highest bandgap frequency); f0 , where the low frequency band edge (nL ) crosses the air-line; fL , the lowest bandgap frequency. Since the guided modes are confined in the bandgap region, the maximum transmission bandwidth is then defined as the frequency interval: Δf=fU - fL . The relative bandwidth is Δf/fc , where the central frequency of the transmission band is fc =(fU +fL )/2.

Fig. 2. (a) Relative bandwidth supported by the square lattice photonic crystal cladding versus d/D for different D/Λ. The background material is PTFE. (b) Relative bandwidth evolution with parameter D/Λ=0.96 for three background material PTFE, PP and HDPE.

Since the THz waves cover a broad frequency region (0.1–10 THz), the THz waveguides are expected to have a matching transmission bandwidth. But due to the guiding mechanism being the bandgap effect, the transmission bandwidth of THz bandgap fibers is limited, and it is determined by the bandgap of the photonic crystal cladding. Fig. 2(a) shows how the relative bandwidths supported by the square lattice photonic crystal vary as a function of corner curvature d/D for different hole size D/Λ. We see that as D/Λ increases, the relative bandwidth increases monotonically. For each D/Λ, the relative bandwidth increases, reaches its maximum and then decreases as d/D increases from 0.1 to 1. But as D/Λ increases from 0.92 to 0.99, the maximum position d/D or the optimized position of d/D, with maximum bandwidth, drops from 0.6 (0.7) to 0.3 (0.4). It is also shown the circular hole (corresponding to d/D=1) supports a relatively narrow bandwidth; the change of the hole’s shape increases the bandwidth effectively. When D/Λ=0.99, the relative bandwidth is ~0.3 for circular hole. While it reaches ~0.64 for d/D=0.3, this corresponds to an efficiently widened bandwidth. In Fig. 2(b), we show the effect of the background material. For a given D/Λ (0.96), the relative bandwidths as a function of d/D are shown for three polymers PTFE, PP and HDPE. As the index of the background material increases, the bandwidth decreases. The PTFE offers the widest bandwidth compared with the other two polymers.

3. Phase-index birefringence

Fig. 3. Bandgap maps and mode dispersion curves within bandgaps. Mode profiles and transverse electric field distributions of fundamental x-polarized fundamental mode (inset (a)) and y-polarized fundamental mode (inset (b)) at frequency f=1 THz. The background index profile is also demonstrated. Fiber parameters: D/Λ=0.96, d/D=0.6 for PTFE.

For a bandgap fiber, an asymmetric air core is generally used to remove the degeneracy of the two orthogonal polarization modes and achieve high birefringence. At the center of the square lattice, a rectangular core is formed by removing 3×5 cells for the proposed THz fiber, as shown in insets of Fig. 3. The surface mode of the bandgap fiber will impair transmission bandwidth and dispersion properties. To eliminate surface modes within the bandgap, according to Ref. [14

M. Digonnet, H. Kim, J. Shin, S. Fan, and G. Kino, “Simple geometric criterion to predict the existence of surface modes in air-core photonic-bandgap fibers,” Opt. Express 12, 1864–1872 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-9-1864. [CrossRef] [PubMed]

], without changing the boundary thickness, the rectangular core is formed by cutting the veins around the air-core, as the index profile shown in insets of Fig. 3. The lattice spacing Λ is chosen as 375 µm to ensure that the central frequency of the transmission bandwidth is 1 THz. A finite-difference method with perfectly matched layer (PML) boundary was employed to calculate the guided modes of the THz fiber. The mode profiles and transverse electric field distributions of the x-polarized fundamental mode and y-polarized fundamental mode at frequency f=1 THz are shown in the insets of Fig. 3. The mode dispersion curves of x and y-polarized modes and the bandgap provided by the cladding are also shown. The geometry parameter d/D=0.6, background material is PTFE polymer. It should be noticed that although the transmission bandwidth is defined as Δf=fU - fL (as shown in Fig. 1), the actual transmission bandwidth is generally narrower, and it is determined by the core size. The phase-index birefringence is defined as the difference of effective refractive indices for x and y-polarized modes. The phase-index birefringence is 1.26×10-3 at f=1 THz. We notice that the modal profile for x and y-polarized mode is slightly different, the effective area is 1 mm2 for x-polarized fundamental mode and 0.94 mm2 for y-polarized fundamental mode. Although the fibers presented in this paper may have multiple modes, the confinement losses of the high order modes are generally much higher than that of the fundamental modes; only the properties of the fundamental modes are concerned in this paper.

In Fig. 4(a), the phase-index birefringence of the THz fiber is shown. The phase-index birefringence monotonically increases with increasing frequency. It is found this observation remains invariant for other configurations of fiber structural parameters. Due to the finite extent of the periodic cladding, the guided modes are intrinsically leaky. We show the confinement loss of x and y-polarized modes for the proposed THz fiber in Fig. 4(a). Seven rings of holes are included in the cladding. The minimum loss occurs around the center of the bandgap. The confinement loss increases dramatically at the edge of the bandgap, where the air guided modes couple with the photonic bands supported by the periodic cladding. The loss of the x-polarized mode is little higher than that of the y-polarized modes, which means that the x-polarized mode is more leaky than the y-polarized mode, which is consistent with the observation in Fig. 3. To further reduce the loss, we can simply increase the number of the cell rings in the cladding. A sufficient number of rings of air-holes will reduce the confinement loss to a negligible level.

Fig. 4. (a) Phase-index birefringence B within the bandgap and confinement loss of the x and y-polarized modes with 7 rings of holes included in the cladding. (b) Central frequency of the transmission band and phase-index birefringence at central frequency as a function of d/D for three different polymers.

To demonstrate the influence of the fiber parameters especially d/D, and the background material on the phase-index birefringence, we show the birefringence at the central frequency fc of transmission band as a function of parameter d/D in Fig. 4(b) for three different polymers. The variations of normalized central frequency for PTFE, PP and HDPE are also shown in figure. The parameter D/Λ is fixed as 0.96. As the index of the background material increases, the central frequency of the transmission band shifts to low frequency. It is seen the birefringence at central frequency decreases, reaches the minimum and then increases with increasing d/D. Moreover, with the highest refractive index, the HDPE shows the highest birefringence of three polymers.

4. Modal absorption loss

Since almost all materials are highly absorbent in the terahertz region, the low attenuation waveguide design must maximize the guided power fraction in the air. The absorption loss of the guided mode due to material absorption in fiber can be qualified by [15

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, New York, 1983).

]:

αab αm= ∫background nb ∣E∣2ds ∫all ∣E∣2ds
(1)

where αab is the modal absorption loss due to material absorption, while αm is the bulk absorption loss of fiber material. “background” and “all” represent the integrals over the background material region versus the entire fiber cross section. nb is the refractive index of the background material.

For the proposed THz fiber, we show the modal power fractions in air-core and the normalized modal absorption loss αab /αm for x and y-polarized fundamental modes in Fig. 5(a). It is noticed that most of the modal power is confined in air-core within 0.87–1.08 THz region. Furthermore, the power fraction of y-polarized mode is generally higher than that of x-polarized mode, which is consistent with the confinement loss shown in Fig. 4(a). The photonic bandgap effect expels the mode power from the background material, resulting in an evidently reduced modal absorption loss. The normalized modal absorption loss is less than 0.04 within the transmission band, which means the modal absorption loss is significantly reduced by a factor of more than 25, compared with bulk absorption loss. The bulk absorption coefficients of the polymers depend on the amount of impurities incorporated during the manufacturing processes. If the bulk absorption coefficient is assumed to be ~0.2cm-1, the modal absorption loss will be less than 0.008 cm-1 within the transmission band.

Fig. 5. (a) Modal power fraction in air-core and normalized modal absorption loss for x and y-polarized modes. (b) Normalized modal absorption loss at central frequency of transmission band of y-polarized mode as a function of d/D for three different background materials.

We quantified the normalized modal absorption losses of various fiber parameters i.e. d/D and the background materials. The absorption losses of y-polarized fundamental mode at the central frequency of the transmission band are shown in Fig. 5(b). We see that the normalized modal absorption loss decreases, until reaches a minimum at 0.3–0.4 and then increases with increasing d/D. The modal absorption loss increases with increasing refractive index of the background material. Compared with Fig. 4(b), the modal absorption loss presents a similar evolution behavior with varied d/D or background materials. The higher background index offers higher birefringence but with a higher absorption loss. This observation indicates that a compromise should be taken between the birefringence and the modal absorption loss for specific practical application.

5. Conclusion

A low-loss air-core polarization maintaining polymer fiber for THz wave guiding is proposed in this paper. An investigation of its transmission bandwidth, phase-index birefringence and modal absorption loss is presented. To demonstrate the influence of the polymer materials, PTFE, PP and HDPE are considered for the THz fibers. The bandgap effect repels the modal power from the absorbent background polymers, resulting in a significant suppression of absorption loss of polymers by a factor of higher than 25. The phase-index birefringence of the proposed THz fiber is in the order of 10-3. It is shown that the HDPE offers the highest birefringence but with a largest absorption loss. This observation indicates that a compromise is required between the birefringence and the modal absorption loss for a specific practical application. As a single material waveguide, such fiber can be fabricated by a stack and drawing (fusing) process, similar to that in conventional PBGF fabrication. The proposed polarization maintaining air-core THz fibers may find their applications in polarizationsensitive devices at THz frequencies such as polarization controllers and filters.

Acknowledgment

This work is supported by grant SBIC RP C-014/2007, Singapore Bioimaging Consortium, Astar, Singapore.

References and Links

1.

M. Tonouchi, “Cutting-edge terahertz technology,” Nat. Photonics 1, 97–105 (2007). [CrossRef]

2.

J. Harrington, R. George, P. Pedersen, and E. Mueller, “Hollow polycarbonate waveguides with inner Cu coatings for delivery of terahertz radiation,” Opt. Express 12, 5263–5268 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-21-5263. [CrossRef] [PubMed]

3.

K. Wang and D. M. Mittleman, “Metal wires for terahertz wave guiding,” Nature , 432, 376–379 (2004). [CrossRef] [PubMed]

4.

R. Mendis and D. Grischkowsky, “Plastic ribbon THz waveguides,” J. Appl. Phys. 88, 4449–4451 (2000). [CrossRef]

5.

S. P. Jamison, R. W. McCowan, and D. Grischkowsky, “Single-mode waveguide propagation and reshaping of sub-ps terahertz pulses in sapphire fiber,” Appl. Phys. Lett. 76, 1987–1989 (2000). [CrossRef]

6.

H. Han, H. Park, M. Cho, and J. Kim, “THz pulse propagation in plastic photonic crystal fiber,” Appl. Phys. Lett. 80, 2634–2636 (2002). [CrossRef]

7.

M. Goto, A. Quema, H. Takahashi, S. Ono, and N. Sarukura, “Teflon photonic crystal fiber as terahertz waveguide,” Jpn. J. Appl. Lett. 43, L317–L319 (2004). [CrossRef]

8.

J.-Y. Lu, C.-P. Yu, H.-C. Chang, H.-W. Chen, Y.-T. Li, C.-L. Pan, and C.-K. Sun, “Terahertz air-core microstructure fiber,” Appl. Phys. Lett. 92, art. no. 064105, (2008). [CrossRef]

9.

T. W. Crowe, T. Globus, D. L. Woolard, and J. L. Hesler, “Terahertz Sources and Detectors and Their Application to Biological Sensing,” Phil. Trans. R. Soc. Lond. A 362, 365–377 (2004). [CrossRef]

10.

M. Cho, J. Kim, H. Park, Y. Han, K. Moon, E. Jung, and H. Han, “Highly birefringent terahertz polarization maintaining plastic photonic crystal fibers,” Opt. Express 16, 7–12 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-1-7. [CrossRef] [PubMed]

11.

P. St. J. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003). [CrossRef] [PubMed]

12.

Y. S. Jin, G. L. Kim, and S. G. Jeon, “Terahertz dielectric properties of polymers,” J. Korean Phys. Soc. 49, 513–517 (2006).

13.

F. Poletti and D. J. Richardson, “Hollow-core photonic bandgap fibers based on a square lattice cladding,” Opt. Lett. 32, 2282–2284 (2007). [CrossRef] [PubMed]

14.

M. Digonnet, H. Kim, J. Shin, S. Fan, and G. Kino, “Simple geometric criterion to predict the existence of surface modes in air-core photonic-bandgap fibers,” Opt. Express 12, 1864–1872 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-9-1864. [CrossRef] [PubMed]

15.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, New York, 1983).

OCIS Codes
(060.2420) Fiber optics and optical communications : Fibers, polarization-maintaining
(160.5470) Materials : Polymers
(060.4005) Fiber optics and optical communications : Microstructured fibers
(110.6795) Imaging systems : Terahertz imaging

ToC Category:
Photonic Crystal Fibers

History
Original Manuscript: April 4, 2008
Revised Manuscript: June 5, 2008
Manuscript Accepted: June 5, 2008
Published: August 20, 2008

Citation
Guobin Ren, Yandong Gong, Ping Shum, Xia Yu, JuanJuan Hu, Guanghui Wang, Michael Ong Ling Chuen, and Varghese Paulose, "Low-loss air-core polarization maintaining terahertz fiber," Opt. Express 16, 13593-13598 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-18-13593


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References

  1. M. Tonouchi, "Cutting-edge terahertz technology," Nat. Photonics 1, 97-105 (2007). [CrossRef]
  2. J. Harrington, R. George, P. Pedersen, and E. Mueller, "Hollow polycarbonate waveguides with inner Cu coatings for delivery of terahertz radiation," Opt. Express 12, 5263-5268 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-21-5263. [CrossRef] [PubMed]
  3. K. Wang and D. M. Mittleman, "Metal wires for terahertz wave guiding," Nature 432, 376-379 (2004). [CrossRef] [PubMed]
  4. R. Mendis and D. Grischkowsky, "Plastic ribbon THz waveguides," J. Appl. Phys. 88, 4449-4451 (2000). [CrossRef]
  5. S. P. Jamison, R. W. McCowan, and D. Grischkowsky, "Single-mode waveguide propagation and reshaping of sub-ps terahertz pulses in sapphire fiber," Appl. Phys. Lett. 76, 1987-1989 (2000). [CrossRef]
  6. H. Han, H. Park, M. Cho, and J. Kim, "THz pulse propagation in plastic photonic crystal fiber," Appl. Phys. Lett. 80, 2634-2636 (2002). [CrossRef]
  7. M. Goto, A. Quema, H. Takahashi, S. Ono, and N. Sarukura, "Teflon photonic crystal fiber as terahertz waveguide," Jpn. J. Appl. Lett. 43, L317-L319 (2004). [CrossRef]
  8. J.-Y. Lu, C.-P. Yu, H.-C. Chang, H.-W. Chen, Y.-T. Li, C.-L. Pan, and C.-K. Sun, "Terahertz air-core microstructure fiber," Appl. Phys. Lett. 92, 064105 (2008). [CrossRef]
  9. T. W. Crowe, T. Globus, D. L. Woolard, and J. L. Hesler, "Terahertz Sources and Detectors and Their Application to Biological Sensing," Phil. Trans. R. Soc. Lond. A 362, 365-377 (2004). [CrossRef]
  10. M. Cho, J. Kim, H. Park, Y. Han, K. Moon, E. Jung, and H. Han, "Highly birefringent terahertz polarization maintaining plastic photonic crystal fibers," Opt. Express 16, 7-12 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-1-7. [CrossRef] [PubMed]
  11. P. St. J. Russell, "Photonic crystal fibers," Science 299, 358-362 (2003). [CrossRef] [PubMed]
  12. Y. S. Jin, G. L. Kim, and S. G. Jeon, "Terahertz dielectric properties of polymers," J. Korean Phys. Soc. 49,513-517 (2006).
  13. F. Poletti and D. J. Richardson, "Hollow-core photonic bandgap fibers based on a square lattice cladding," Opt. Lett. 32, 2282-2284 (2007). [CrossRef] [PubMed]
  14. M. Digonnet, H. Kim, J. Shin, S. Fan, and G. Kino, "Simple geometric criterion to predict the existence of surface modes in air-core photonic-bandgap fibers," Opt. Express 12, 1864-1872 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-9-1864. [CrossRef] [PubMed]
  15. A. W. Snyder and J. D. LoveOptical Waveguide Theory (Chapman and Hall, New York, 1983).

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