## Abstract

We present the design of a solid-core microstructured optical fiber with steering-wheel pattern of large holes in cladding as platform for evanescent-field sensing. Both geometry and optical properties of the fiber are numerical computed and analyzed in consideration of manufacturability using sol-gel casting technique as well as by evaluating a triangular lattice of holes with three rings in the design structure so that effective parameters can be established using effective step-index model. We predict less than 0.7 dB/m confinement loss at 850 nm, 29 %, 13.7 %, and 7.2 % of light intensity overlap in air holes at 1500 nm, 1000 nm, and 850 nm wavelength, respectively, in such fiber. With the low loss and high mode-field overlap, the steering-wheel structured fiber is well suited for evanescent-field sensing and detection of chemical and biological species.

©2006 Optical Society of America

## 1. Introduction

Microstructured optical fibers (MOFs), also termed photonic crystal fibers (PCFs) or holey fibers [1, 2], consist of regularly spaced air holes in the cladding and/or core along the fiber length. Such fibers have gained increasing attention for potential applications in chemical and biological sensing [3–8] due to permissible gas or liquid transport in the air holes as well as long light-analyte interaction path. Despite the popularity of evanescent-field sensing modality for fiber optic sensors, it continues to face the challenge of insufficient mode-field overlap with measurands and thus limited sensitivity. The successful emergence of several MOF-fabrication techniques such as stack-and-draw [9], sol-gel casting [10], drilling [11], and extrusion [12], have enabled considerable design flexibility, making it possible to obtain unique and intricate cladding microstructures in MOFs with structural and optical properties tailored for sensing applications. This paper describes the design of steering-wheel microstructured optical fiber (SW-MOF) for evanescent-field based sensing. Optimal evanescent-field overlap with the air holes is theoretically realized. The advantages of the design structure include (1) large power percentage localized in holes resulting in high sensitivity; (2) low non-linearity without sacrificing light confinement; and (3) macro-bending insensitive due to relatively large air holes in the cladding. Furthermore, the air-hole arrangement in the cladding is directly correlated to that in fiber preform using sol-gel casting technique, making high performance and low-cost SW-MOFs commercially feasible.

## 2. Design of SW-MOF

The design strategy is based on an effective step-index model derived from classical fiber-optics theory so that the traditional numerical approaches can be used for the description of our solid-core SW-MOF configuration. Depicted in Fig. 1(a) are the steps leading to the eventual design structure starting with three rings of triangularly arranged air holes in inner-cladding. The triangular air-hole array can be characterized by hole-to-hole pitch (*ʌ*) and hole diameter (*d*). The air hole array will collapse to form three-large hexagonal air holes when *d*/*ʌ* approaches unity and with the inclusion of three radically spaced thin webs, leading to the steering-wheel pattern shown in Fig. 1(b). Such design can achieve an air-filling fraction of about 93.7%. The resultant SW-MOF is considered as a dual-clad fiber with a depressed-index inner cladding so that the relation with effective parameter can be established without compromising evaluation accuracy as measured by *V*
^{2}(*n*
_{sikica}, *n*
_{FSM}) = *U*
^{2}(*n*
_{silica}, *n*
_{eff}) + *W*
^{2}(*n*
_{eff}, *n*
_{FSM}). In this expression, *V* is effective normalized frequency constant; *U* is effective normalized transverse phase constant; *W* is effective normalized attenuation constant; and *n*
_{silica}, *n*
_{eff}, and *n*
_{FSM} are refractive indices of silica, effective and fundamental space-filling mode, respectively.

In general, all types of MOFs can be fabricated by the various established techniques mentioned above. Our SW-MOF design approach is based on feasibility and implementation of sol-gel casting technique. In order to simplify the theoretical calculation in the situation when the ratio of hole diameter and pitch (*d*/*ʌ*) approaches one, it is appropriate to model MOF with 3-ring of air holes instead of steering-wheel pattern. All optical characteristics of SW-MOF could be accurately estimated following this evaluation method.

#### 2.1 Mode field diameter and numerical aperture

Power distribution in SW-MOF is determined by mode field diameter (MFD) which constrains most of light propagating through the solid core. Originally, MFD is quantified by the effective area (*A*
_{eff}) in conventional optical fiber. Relation *A*
_{eff} = *π*
${w}_{\mathit{\text{eff}}}^{2}$, where *w*
_{eff} is the effective modal spot size (one half of MFD), can be used for the evaluation of MFD in MOF by full-vector finite element method [13]. The effective area is weakly dependant on the number of air-hole rings. MFD of SW-MOF can therefore be calculated without loss of accuracy. Furthermore, the numerical aperture (NA) is related to the effective area with NA = sin*θ*≃(1 + *π*
*A*
_{eff}/*λ*
^{2})^{-1/2} [14]. This relation is used for the evaluation of NA of SW-MOF shown in Fig. 2. According to this figure, MFD is independent of wavelength whereas NA is proportional to wavelength.

#### 2.2 Near-field image and confinement loss

The near-field image of SW-MOF is shown in Fig. 3(a). It is modeled using beam propagation method that allows analysis of the fundamental mode in core and its optical properties. The pattern of power distribution is dependent on mode density with triangular rotational symmetry. The power distribution somewhat extended into the three thin webs but is nevertheless mainly restricted in the core region. SW-MOF allows a guided mode to overlap with adjacent air holes more efficiently because sufficient surface area of core is exposed to the evanescent field.

Confinement loss is numerically calculated using power leakage formula given by [15]

where *a* is effective core radius; *b* is effective radius of whole-hoe area; *λ* is wavelength of free space; and *K*
_{1} is the first-order modified coefficient of Bessel function. The formula is based on classical dual-clad normal fiber approximation and transformation of hexagonal airhole pattern into quasi-steering-wheel structure shown as inset of Fig. 3(b). The effective ratio of *d*/*ʌ* in SW-MOF is larger than 0.8, corresponding closely with MOF containing 3-ring of hole with same *d*/*ʌ*. For this reason, the confinement loss of SW-MOF is less than 0.7 dB/m at wavelength of 0.85 μm.

#### 2.3 Power distribution

For index-guidance MOF, intensity of evanescent-wave can be expressed as follows:

where *I*
_{0} is intensity of incident light; *z* is a point within penetration depth (*d*) that is given by

where *n*
_{1} and *n*
_{2} are refractive indices of core and cladding, respectively; *θ*
_{ι} is incident angle; and *θ*
_{c} (sin *θ*
_{c} = *n*
_{2}/*n*
_{1}) is critical angle for total internal reflection (*n*
_{1} > *n*
_{2}). The calculated intensity distribution of guided mode in SW-MOF is presented in Fig. 4(a) with left inset being far field pattern and the right inset being 3-D power profile of fundamental mode. When *V* <*π*, a significant portion of power is extended to the air-hole region. Single mode
propagation is realized here as *V* is less than 2.405, Single mode or few modes in MOF are highly desirable for evanescent-field sensing applications. Since web thickness is much larger than operating wavelength, most light power will be located within silica, resulting in sharply reduced mode overlap with air hole and increased confinement loss. By reducing the web thickness, analogous to shortening pitch in step index model, the overlap would be significantly strengthened. Fig. 4(b) shows the power percentage in air holes as a function of wavelength with web thickness.

#### 2.4 Macro-bending loss

High macro-bending loss of MOF hinders sensor performance and system miniaturization. For prediction of macro-bending loss induced from the irradiation modes in SW-MOF, the ratio of the amplitude coefficient of the mode field in the cladding (*A*
_{e}) and the power transmitted by the fundamental mode (*P*) should be considered. The Gaussian-beam approximation of guided mode is derived to give rise to a relation of ${A}_{\mathrm{e}}^{2}$/*P* = 1/*A*
_{eff} for the calculation at different curvatures [16]. The results are shown in Fig. 5 with inset of macro-bending radius at different cut-off wavelength. Apparently, there is even macro-bending loss edge in short wavelength region, but it does not appear till the wavelength of 2 μm, indicating that SW-MOF can be used for sensing in the near infrared range. With this feature, the critical bending radius is about 2 cm at wavelength of 800 nm. Under this condition, macro-bending loss of SW-MOF is negligeble.

## 3. Closing Remarks

Ideally, a silica rod suspended in air without any support is the best evanescent-field sensing platform. Its implementation is not practical, however. If the core of optical fiber is sustained through silica bridges, without doubt, confinement loss will be present no matter how big the air hole is in the cladding. On the other hand, smaller core diameter allows tight mode confinement in a MOF yet MOF with small core exhibits significant attenuation detrimental to evanescent-wave propagation. There is therefore a trade-off between core size and confinement loss. The web thickness is another factor that affects the light intensity distribution in fiber. A very thin web can benefit greatly light confinement in the core area even if the core size is relatively large. Very small core diameter with thin web can give rise to high light intensity in the core, leading to non-linear effect and thus substantial spectral broadening. Non-linearity is not desirable for spectroscopy. In addition, surface modes formed by capillary wave frozen in thin silica web at nanometer scale are another source to induce fiber loss.

In a continuous sensing scheme, gas or liquid phase must be transported through the air holes of a MOF. Provided below is Poiseulle’s formula [17],

where *Q* is volume flow rate, *r* is radius of air hole, *p*
_{1} is pressure at entrance, *p*
_{2} is pressure at exit, *l* is fiber length, *η* is viscosity of gas or liquid, and *p*
_{0} is reference pressure for volume. According to the equation, the flow rate follows a power law of four with respect to the air hole radius. Clearly, the larger the air holes, the higher the flow rate under given pressure differential. Conversely, for the same flow rate, the larger the air holes, the smaller the required pressure differential. A SW-MOF with high mode-field overlap, low confinement loss, and large air hole radius is an ideal platform for evanescent-field sensing.

In conclusion, we have designed a solid-core MOF with steering-wheel patterned air holes in the cladding. The fiber can be fabricated cost-effectively by sol-gel casting method. We have theoretically predicted that optimal evanescent-field sensing can be achieved by the adjusting silica web thickness, core diameter, and air-hole size. There is also a complex interlay between limiting confinement and bending losses and achieving power intensity overlap in the air holes. In the end, preference should be given to high power percentage in fiber air holes for high-sensitivity sensing applications based on techniques such as absorption, fluorescence, or Raman scattering spectroscopy.

## Acknowledgments

We thank Prof. Svetlana Sukhishvili, Prof. Rainer Martini, Dr. Denis Pristinski, and Dr. Alexander Raspopin for many stimulating discussions. This work was supported by the National Science Foundation under grant number ECS-0404002 and by US Army TACOM/ARDEC under contract W15QKN-05-D-0011.

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