1. Introduction

The ability to design and fabricate photonic crystal fibers (PCFs) for the realization of vastly different optical properties made possible with the seemingly unlimited cladding microstructural features is arguably one of the most significant recent advances in fiber optics. Long-period gratings (LPGs) introduced in PCFs further expand the realm of applications. LPG, periodic index perturbations typically hundreds of microns in periodicity, enables the coupling of the fundamental core mode (LP₀₁) to co-propagating cladding modes (LP₀₂ and higher orders) in PCF at well-defined resonance wavelengths [1]. The coupling takes place when the phase-matching condition, ${\lambda }_{\text{resonance}}=\left(n_{core}^{eff}–n_{clad}^{eff}\right)\Lambda$, is satisfied [2], where λ_resonance is the resonance wavelength, $n_{core}^{eff}$ and $n_{clad}^{eff}$ are the respective effective indices of the core and cladding modes, and Λ is the grating period. The high attenuation induced by mode coupling at the resonance wavelength makes LPG robust filters [3]. The dependence of λ_resonance on $n_{core}^{eff}$ and $n_{clad}^{eff}$ enables it a sensitive index transduction platform for multi-parameter sensing [4–8].

Periodic index perturbation is essential for the realization of LPG, which can be accomplished by laser techniques such as UV [9] (with photosensitive dopants and photomask) or CO₂ laser irradiation [10] as well as by non-laser approaches such as arc discharge [11], acoustic wave coupling [12], and periodic mechanical stress loading [13]. CO₂ laser irradiation is by far the most versatile, relatively straightforward, and inexpensive LPG fabrication method especially for a vast majority of PCF based on dopant-free SiO₂. Localized heating by CO₂ laser induces relaxation of the residual stress in PCF, leading to the change in the index of refraction in the laser irradiated regions of period Λ without structural deformation [14]. Excessive heating and high tensile load during laser irradiation can lead to periodic structural deformation (e.g., necking and elongation) [15] or frozen-in viscoelasticity [16] in the irradiated regions, resulting in index perturbation and mode coupling.

Unidirectional irradiation in fiber transverse direction has been the prevailing method for CO₂ laser inscription of LPG in both PCFs and SMFs due to simplicity in instrument configuration and operation. The strong absorbance of SiO₂ at 10.6 m wavelength in short pulses limits the localized heating region to the order of the laser wavelength beneath the fiber surface in the cladding. This approach hence inherently leads to azimuthally asymmetric distribution of the change in the index of refraction, resulting in coupling between the LP₀₁ core mode to azimuthally asymmetric LP_1n cladding modes, as opposed to the symmetric LP_0n cladding modes. Experimental studies involving SMF-LPG have shown that the asymmetric cladding modes are the main origin of the complex resonance characteristics with multiple band features, strong dependence on polarization, as well as orientation dependence of the light coupling efficiency in SMF-LPG [17–19]. By using a heavily doped SMF, the coupling of the asymmetric cladding modes can be suppressed under a small dose of CO₂ laser asymmetric irradiation [18]. A few attempts have been made to reconfigure the CO₂ laser irradiation symmetrically around SMF during LPG inscription to promote the symmetric LP_0n cladding modes while suppressing the asymmetric LP_1n cladding modes [14, 19, 20]. One example includes the fabrication of high-quality SMF-LPG by Grubsky and Feinberg with the aid of a 120° reflector for three-beam equivalent, angularly even spaced CO₂ laser irradiation [19]. The resultant SMF-LPG exhibited negligible insertion loss and well-behaved resonance bands characteristic of coupling to symmetric cladding modes. Another example is the success of Oh et al. [20] in obtaining sharp transmission spectrum in the symmetrically inscribed SMF-LPG (Λ: 600 µm) as well as reducing the polarization-dependent loss by either rotating SMF for each exposure or focusing the CO₂ laser beam into a ring shape around the fiber.

The advantage of symmetric CO₂ laser irradiation in LPG fabrication in SMF has been demonstrated [17, 19]. The need for such approach to create azimuthally symmetric change in the index of refraction in PCF is even more compelling for two reasons. First, coupling to the symmetric cladding modes will enable the full and most efficient contribution of the entire air cladding structure in the LPG structure for sensing applications in particular. Second, symmetric laser irradiation of PCF, especially of azimuthally anisotropic cladding microstructure, such as the hexagonal geometry, will make LPG inscription more reproducible. Yet there is a dearth of knowledge on the correlation of the symmetry of index perturbation with the mode coupling behavior and reproducibility in PCF-LPG, while also recognizing the fact that related studies are also relatively few with SMF-LPG. We report the first numerical and experimental study, to the best of our knowledge, of endlessly single mode (ESM) PCF-LPG with symmetric as well as asymmetric perturbation in the change of index of refraction in the grating region. Parallel investigation of similarly inscribed SMF-LPG is carried out for comparison. Both theoretical and experimental results indicate that symmetric index perturbation induced by laser irradiation using a 120° gold-coated mirror results in LP_0n symmetric mode coupling and asymmetric index perturbation without the mirror leads to LP_1n asymmetric mode coupling in both ESM PCF-LPG and SMF-LPG. We further demonstrate that symmetric inscription yields ESM PCF-LPGs of reproducible transmission characteristics, overcoming the challenge often encountered during unidirectional LPG inscription in ESM PCF. In contrast, LPG inscription in SMF is highly reproducible, independent of the symmetry of laser irradiation.

2. Theoretical simulation and analysis

We first attempted theoretical simulation to study the mode orders with respect to the symmetry of perturbation in refractive index. The contribution to the mode effective index is given by the integral of index perturbation weighted by a value of mode field over fiber cross section. The transmission of the core mode is determined by cos(κL), where L is the grating length and κ is the coupling efficiency defined as [21]:

To evaluate the correlation of the coupled mode orders with the symmetry, a mode solver (COMSOL Multiphysics, version 3.5.a extended with the RF module) based on the Finite Element Method (FEM) was employed to calculate the phase matching condition and the field profiles. It has been found that using the linear, quadratic and exponential refractive-index profiles in numerical simulations resulted only in a quantitative difference in obtained results [22]. Thus, not knowing precisely the refractive index change across the fiber cross-section, simple linear variation of refractive index was used. Two types of index modulations with gradient dn [RIU/µm²)] in the laser direction were considered: $n=n_0+\left(x-R\right)d{n}_1$ for unidirectional (asymmetric) irradiation, where R is the radius of the fiber, $d{n}_1={10}^{-6}$ for SMF or $2\times {10}^{-6}$ for ESM PCF, and $n=n_0+{\displaystyle \sum _{i=1}^{i=3}\left[\left(x\cos {\phi }_i+y\sin {\phi }_i\right)-R\right]}d{n}_2$, where ${\phi }_i=\raisebox{1ex}{$2\pi $}\!\left/ \!\raisebox{-1ex}{$3$}\right.(i-1)$ and $d{n}_2=\raisebox{1ex}{$d{n}_1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$for the tri-directional (symmetric) irradiation. The index of refraction change decreases from its maximum negative value at the fiber surface under direct laser exposure to zero at the opposite fiber surface.

Shown in Fig. 1 are simulated transmission spectra for SMF-LPG illustrating the coupled resonances with respect to the symmetry of index perturbation during SMF-LPG fabrication. For symmetric perturbation, between the lowest 78 modes, only the symmetric LP_0n are coupled. Asymmetry in perturbation induces coupling to the corresponding asymmetric LP_1n modes. Depicted in Fig. 2 are the mode profiles calculated for SMF-LPG that correspond to the cladding mode distributions of symmetric LP₀₄ and its neighboring asymmetric LP₁₃ at the respective resonance wavelengths of 1582 and 1560 nm. For symmetric index perturbation, the mode profile is symmetric and the mode coupling is between LP₀₁ and LP₀₄. For asymmetric index perturbation, cladding mode field is distorted and asymmetric LP₁₃ has its field component in fiber core with the resonance resulting from coupling between LP₀₁ and LP₁₃.

 

Fig. 1 Simulated transmission spectra of SMF-LPG for (a) azimuthally symmetric index perturbation and (b) azimuthally asymmetric index perturbation.

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Fig. 2 Mode profiles of SMF-LPG for cladding modes of (a) LP₀₄ at resonance wavelength of 1582 nm and (b) LP₁₃ at resonance wavelength of 1560 nm.

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Depicted in Fig. 3 are the calculated mode profiles of the ESM PCF-LPG that illustrate the distribution of symmetric LP₀₃ and the neighboring asymmetric LP₁₂ cladding modes at the respective resonance wavelengths of 1500 nm and 1550 nm. For ESM PCF-LPG, symmetric index perturbation favors symmetric mode coupling while asymmetric index perturbation causes asymmetric mode coupling. The results for the index modulation of the zero-gradient and the 3-directional gradient are almost identical for the ESM PCF-LPG. As can be seen from the simulated asymmetric mode profiles, the index modulation in the outer cladding also affects the mode field distribution in the core region of ESM PCF, as such, the coupling coefficients are affected also.

 

Fig. 3 Mode profiles of ESM PCF-LPG for cladding modes of (a) LP₀₃ at resonance wavelength of 1500 nm and (b) LP₁₂ at resonance wavelength of 1550 nm.

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Illustrated in Fig. 4 are calculated phase matching curves (PMC) for LP_0n and LP_1n modes in both SMF-LPG and ESM PCFLPG. They show the relationship between the grating period and the resonance wavelengths for LP₀₁ to LP_0n/LP_1n mode coupling governed by the phase matching condition. With LP₀₁ coupled to a given cladding mode in the wavelength region considered, the resonance wavelength increases as the grating period increases for SMF-LPG, whereas it decreases with the increase of the grating period for ESM PCF-LPG. For SMF-LPG, the difference in the effective indices of core and cladding, $n_{core}^{eff}-n_{clad}^{eff}$, is almost constant [23]. The resonance wavelength thus increases with the grating period. For ESM PCF-LPG, however, light tends to spread to the air channels at longer wavelength and the effective index of the cladding decreases rapidly as a result [24], leading to a negative slope of its PMC. Due to the opposite signs of slopes in PMCs for SMF-LPG and ESM PCF-LPG, the resonance wavelength of LP_1n shifts to the opposite direction with respect to that of their neighboring LP_0n cladding modes. At a given grating period, the resonance wavelength of LP₁₃ is shorter than that of LP₀₄ for SMF-LPG, whereas the resonance wavelength of LP₁₂ is longer than that of LP₀₃ for ESM PCF-LPG.

 

Fig. 4 Phase matching curves for (a) LP₁₃ and LP₀₄ in SMF-LPG and (b) LP₁₂ and LP₀₃ in ESM PCF-LPG.

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3. Experimental results and discussion

3.1 Symmetric and asymmetric inscription of LPG

ESM-12B (NKT Denmark) and SMF-28 (Corning Inc.) were chosen as the respective ESM PCF and SMF fiber types for symmetric and asymmetric LPG inscription using CO₂ laser. Cross-sectional SEM micrographs of the two fibers are shown in Fig. 5, vividly indicating the azimuthal anisotropy for ESM-12B and isotropy for SMF-28 in their cladding structures. Symmetric inscription was accomplished with the use of a 120° gold-coated silicon mirror (reflectivity > 98%), also illustrated in Fig. 5. The fiber was axially aligned with and positioned about $8/\sqrt{3}R$ above the V groove of the mirror, where R is the radius of the fiber. This mirror assembly converts otherwise a unidirectional laser beam to three angularly even spaced (i.e., 120°) beams of basically identical intensity that are directed towards the fiber, ensuring symmetric laser irradiation. Asymmetric inscription was conducted without the mirror, leading to single-sided exposure to laser irradiation. The energy densities for symmetric and asymmetric irradiations are 0.21 J/mm² and 0.63 J/mm², respectively, for the same overall dosage. A Synrad water-cooled 48-1 CO₂ laser source with stable power output up to 10 W and computer interface was used for LPG inscription. The laser is shaped and projected as a 3000 µm x 200 µm rectangular beam on the fiber with the longer beam profile straddling the fiber in radial direction. The fiber was firmly mounted on a motorized translation stage with or without the mirror under 1 g and 30 g tensile load. Our screening test showed that the 1 g load was sufficient to keep the fiber under nominal tension without causing apparent structural deformation whereas the 30 g load resulted in appreciable structural deformation in the inscribed regions. The LPG was inscribed via the point-by-point method. A Super-K ultra-broadband supercontinuum light source and an Anritsu MS9710 C optical spectrum analyzer (OSA) were used for measurements of the transmission characteristics during and after LPG inscription. The grating periods for SMF-LPG and ESM PCF-LPG were 600 μm and 490 μm, respectively, for coupling resonances in the wavelength range of our OSA (1150~1750 nm), using the aforementioned numerical analysis as a guide. SMF-LPG and ESM PCF-LPG samples for subsequent measurements were fabricated under a combination of symmetric and asymmetric irradiation at 1 g and 30 g applied load, with a total of eight distinct sample sets.

 

Fig. 5 Cross-sectional SEM image of (a) ESM-12B PCF with a hexagonal array of cladding air channels and (b) SMF-28 SMF with the center core revealed via gentle etch in buffered HF solution; and schematic illustration of (c) mirror-aided symmetric and (d) mirror-free asymmetric CO₂ laser inscription of LPG in ESM PCF and SMF.

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3.2 Transmission and mode coupling in SMF-LPG

Shown in Fig. 6 are the transmission spectra of four SMF-LPG samples: two symmetrically and asymmetrically inscribed under 1 g load, labeled respectively as a and b; two symmetrically and asymmetrically inscribed under 30 g load, labeled respectively as c and d. The respective LPG lengths of samples a, b, c, and d are 55.8, 66.0, 21.6 and 20.4 mm. Provided as insets in Fig. 6 are the near-field images taken with an IR camera at the corresponding resonance wavelengths of 1580, 1565, 1570 and 1505 nm.

 

Fig. 6 Transmission spectra of symmetrically and asymmetrically inscribed SMF-LPG under 1 g and 30 g tensile load. The insets are the corresponding near-field images taken at the respective resonance wavelengths of 1580, 1565, 1570 and 1505 nm.

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For sample a, the symmetric mode-field distribution at resonance wavelength of 1580 nm as indicated by the near-field image in Fig. 6, is a strong indication of coupling to azimuthally symmetric cladding mode. For sample b, on the other hand, the loss of symmetry in the near-field image, also seen in Fig. 6, is a clear sign of coupling to asymmetric cladding mode at resonance wavelength of 1565 nm. According to our numerical results shown in Fig. 4(a), for SMF-LPG with a grating period of 600 μm, the resonance wavelength at 1582 nm (1580 nm as measured for sample a) corresponds to LP₀₄ symmetric mode whereas the resonance wavelength at 1560 nm (1565 nm as measured for sample b) can be attributed to LP₁₃ asymmetric mode. As the tensile load was increased to 30 g, LPG inscription was accompanied with considerable structural deformation in SMF, which also contributed to index perturbation in the affected regions, besides the commonly observed stress relaxation. Mode coupling in the symmetrically inscribed sample c, seen from the near-field image of mode field distribution (inset in Fig. 6), closely resembles that for sample a, suggesting azimuthally uniform structural deformation and preservation of symmetric index perturbation. For sample d, the resonance wavelengths were not as predicted by our numerical analysis (Fig. 4(a)). The near-field image (inset in Fig. 6) indicates a highly asymmetric and distorted mode field distribution in the resultant SMF-LPG. The resonance wavelengths of both LP₀₄ and LP₁₃ modes for SMF-LPG fabricated under 30 g underwent a blue shift compared to those of the coupled modes under 1 g, respectively. As concluded in [25], fiber deformation also plays a role in the LPG mode coupling and thus has an influence on its resonance wavelength. While the dimensional decrease of the fiber cladding leads to the reduction in $n_{clad}^{eff}$ and subsequent increase in the resonance wavelength, the deformation also results in a dimensional decrease in the fiber core, which tends to decrease the $n_{core}^{eff}$ and the concurrent decrease in the resonance wavelength. The two counteracting factors as a whole have an effect of decreasing the resonance wavelength, resulting in a blue shift for the SMF-LPG fabricated under 30 g tensile load.

3.3 Transmission and mode coupling in ESM PCF-LPG

Exhibited in Fig. 7 are the transmission spectra of four ESM PCF-LPG samples: two symmetrically and asymmetrically inscribed under 1 g load, labeled respectively as a and b; two symmetrically and asymmetrically inscribed under 30 g load, labeled respectively as c and d. Their corresponding grating lengths are 58.8, 54.9, 23.0 and 22.5 mm. Insets in the figure are the near-field images of the samples at respective resonance wavelengths of 1501, 1555, 1510 and 1570 nm.

 

Fig. 7 Transmission spectra of symmetrically and asymmetrically inscribed ESM PCF-LPG under 1 g and 30 g tensile load. The insets are the corresponding near-field images taken at the respective resonance wavelengths of 1501, 1555, 1510 and 1570 nm.

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Although the near-field images in all cases suggest mode coupling with mode field extension throughout the cladding region, none exhibits specific patterns that would allow definitive determination of the symmetry of mode distribution. It is generally recognized that near-field image of cladding modes in ESM PCF is very difficult to experimentally measure due to their strong attenuation and high sensitivity to even the slightest variations in the cladding structure along the optical path, compared to the SMF counterpart. Nevertheless, based on our numerical results shown in Fig. 4(b), for ESM PCF-LPG with a grating period of 490 μm, the resonance wavelengths at 1501 nm and 1510 nm (versus the predicted 1506 nm) for samples a and c correspond to LP₀₃ symmetric mode whereas the resonance wavelengths at 1555 nm and 1570 nm (predicted at 1570 nm) for sample b and d stem from LP₁₂ asymmetric mode. We note that upon release of the ESM PCF-LPG from the inscription stage, the symmetrically inscribed samples retained their straightness. In contrast, the asymmetrically inscribed samples became instantly curved, a clear indication of the asymmetric stress, hence, refractive index distribution in the transverse direction of the fiber. As a result, the mode coupling and the resonance wavelength of asymmetrically inscribed ESM PCF-LPG can be very sensitive even to routine sample handling and positioning.

To compare and contrast the symmetric and asymmetric methods in their ability to reproducibly inscribe LPG structures, we fabricated multiple samples under the same conditions as described before, i.e., symmetric and asymmetric irradiation at 1 g and 30 g tensile load for SMF and ESM PCF. Shown in Figs. 8(a)-8(d) are the transmission spectra of four sets of three SMF-LPG samples, while Figs. 8(e)-8(h) are the transmission spectra of four sets of three ESM PCF-LPG samples, each set under identical conditions. Figures 8(a)-8(d) indicate excellent reproducibility for all SMF-LPG samples, regardless of the symmetry of laser irradiation. These results are not unexpected given the highly isotropic nature of the all-solid cladding in SMF-LPG. The consistency as demonstrated in our LPG inscription in SMF is also a confirmation of the high degree of control in our laser inscription technique that serve to affirm the reliability of our experimental results involving the structurally more complex ESM PCF. Compared to SMF, the reproducibility in LPG fabrication in ESM PCF showed a strong dependence on the symmetry of laser irradiation, as seen clearly in Figs. 8(e)-8(h). Symmetric inscription generally leads to ESM PCF-LPG samples of similar transmission characteristics and nearly identical resonance wavelength under both light and heavy tensile load. Symmetric irradiation makes it far less sensitive to the azimuthal anisotropy of the hexagonally arrayed cladding air channels, due to the averaging effect of the overall energy distribution in the cladding structure. In contrast, the ESM PCF-LPG samples asymmetrically inscribed exhibited considerable inconsistency in the transmission spectral features, a strong indication of the challenge in quality fabrication of ESM PCF-LPG of azimuthal anisotropy in the cladding microstructure.

 

Fig. 8 Transmission spectra of eight sets of three LPG samples with each set inscribed under the same condition: (a) symmetric at 1 g, (b) asymmetric at 1 g, (c) symmetric at 30 g, and (d) asymmetric at 30 g for SMF-LPG; and (e) symmetric at 1 g, (f) asymmetric at 1 g, (g) symmetric at 30 g, and (h) asymmetric at 30 g for ESM PCF-LPG.

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4. Conclusion

We have carried out a comprehensive numerical and experimental study of core mode to cladding mode coupling in both SMF-LPG and ESM PCF-LPG with symmetric and asymmetric perturbation in the change of index of refraction in the grating regions. We have shown that symmetric index perturbation afforded by a 120° gold-coated reflective mirror during laser irradiation promotes the coupling into symmetric cladding mode LP_0n and suppresses coupling into asymmetric cladding mode LP_1n. Asymmetric perturbation by unidirectional laser irradiation without the aid of the mirror leads to the coupling into asymmetric cladding mode LP_1n. We have demonstrated that symmetric inscription yields ESM PCF-LPG of highly reproducible transmission and resonance characteristics whereas asymmetric inscription results in ESM PCF-LPG of inconsistent properties. In contrast, both symmetric and asymmetric inscriptions lead to reproducible SMF-LPGs. Given the easy adoption and low cost of the 120° reflective mirror assembly for symmetric laser irradiation, it is strongly recommended that it be the standard method of choice for ESM PCF-LPG research that utilizes CO₂ laser as the inscription technique for sensing and optical device applications where symmetric mode coupling is desired.