## 2 Coupling Model of the GRIN Fiber Probe and HC-PCF  
The GRIN fiber probe is an all-fiber optical lens which is fused by a single-mode fiber, no-core fiber, and GRIN fiber in turn. The coupling model of the HC-PCF and GRIN fiber probe is shown in Figure 1.  
Symbols are set as follows: \(\lambda\) is the light of the wavelength and \(\omega_{0}\) is the beam radius of the Gaussian beam, the refraction index of the NCF is \(n_{0}\), the length of the NCF is \(L_{0}\), the GRIN fiber lens has the refractive index in the center \(n_{1}\) and the gradient constant \(g\) with length \(L\), the refractive index of the transmission medium in the application environment is \(n_{2}\), and \(z\) is the coupling distance or gap width between the GRIN fiber probe and HC-PCF. If light has no energy loss in the optical fibers, the working distance \(z_{\omega}\) and focusing spot size \(2\omega_{f}\) of the GRIN fiber probe are expressed as [21]  
\[z_{\omega}=\frac{S_{1}\cos(2 gL)+S_{2}\sin(2 gL)}{S_{0}-S_{3}\cos(2 gL)-S_{4}\sin (2 gL)} \tag{1}\]  
where \(S_{0}=n_{1}^{2}g^{2}+n_{1}^{2}S_{2}^{1}n_{0}^{2}a^{2}+n_{0}^{2}a^{2}\), \(S_{1}=-2n_{0}n_{2}L_{0}a^{2}\), \(S_{2}=n_{1}n_{2}g+n_{1}n_{2}gL_{0}^{2}a^{2}-n_{2}n_{0}^{2}a^{2}/(n_{1}g)\), \(S_{3}=n_{1}^{2}g^{2}+n_{1}^{2}g^{2}L_{0}^{2}a^{2}-n_{0}^{2}a^{2}\), \(S_{4}=2n_{0}n_{1}gL_{0}a^{2}\), \(a=\lambda/(n_{0}\pi\omega_{0}^{2})\);  
Figure 1: Coupling model of the GRIN fiber probe and HC-PCF. \[2\omega_{f}=2\omega_{0}\sqrt{P_{0}\cos^{2}(gL)+P_{1}sin^{2}(gL)-P_{2}\sin(2 gL)} \tag{2}\]  
where \(P_{0}=1+a^{2}(L_{0}+n_{0}z_{\omega}/n_{2})^{2}\), \(P_{1}=z_{\omega}^{2}n_{1}^{2}g^{2}/n_{2}^{2}+a^{2}(n_{0}/(n_{1}g)-n_{1}z_{\omega} L_{0}g/n_{2})^{2}\), \(P_{2}=z_{\omega}n_{1}g/n_{2}-a^{2}(L_{0}+n_{0}z_{\omega}/n_{2})(n_{0}/(n_{1}g)-n _{1}z_{\omega}L_{0}g/n_{2})\). \(\omega(z)\) is set as the radius of the beam from the GRIN fiber probe at distance \(z\) and can be given as  
\[\omega(z)=\omega_{f}\sqrt{1+[\frac{\lambda(z-z_{\omega})}{\pi\omega_{f}}^{2} ]} \tag{3}\]  
The power of emitted light from the GRIN fiber probe is set to \(p_{0}\). To avoid a large number of complex calculations for coupling efficiency using an overlap integral [22], a simplified model that only takes distributions of the input beam into consideration is adopted. The emitted light from the GRIN fiber can be approximated as a Gaussian beam, so the power received at the receiving end can be expressed as  
\[p_{t}(z)=\int_{0}^{\frac{d}{2}}\frac{2p_{0}}{\pi\omega^{2}(z)}\exp(-\frac{2r^ {2}}{\omega^{2}(z)})2\pi rdr \tag{4}\]  
where \(d\) is the mode field diameter (MFD) of HC-PCF. From Equations (3) and (4), the coupling efficiency of the GRIN fiber probe coupled with HC-PCF can be expressed as  
\[\eta=\frac{p_{t}}{p_{0}}=1-\exp(-\frac{d^{2}/2}{\omega_{f}^{2}+[\frac{\lambda (z-z_{\omega})}{\pi\omega_{f}}]^{2}}) \tag{5}\]  
As a comparison, the spot size of the Gaussian beam after it is emitted from SMF [12] is  
\[\omega(z)=\omega_{0}\sqrt{1+[\frac{\lambda z}{\pi\omega_{0}}^{2}]^{2}} \tag{6}\]  
and the SMF coupling efficiency is expressed as  
\[\eta^{\prime}=\frac{p_{t}}{p_{0}}=1-\exp(-\frac{d^{2}/2}{\omega_{0}{}^{2}+[ \frac{\lambda z}{\pi\omega_{0}}]^{2}}) \tag{7}\]  
Comparing Equations (5) and (7), the coupling efficiency of the GRIN fiber probe coupled with HC-PCF in Equation (5) achieves the maximum value when the coupling distance \(z\) is equal to the working distance \(z_{\omega}\) of the probe, while the coupling efficiency of SMF coupled with HC-PCF in Equation (7) achieves the maximum value when \(z\) is equal to zero; that is, a GRIN fiber probe can focus an emitted spot at the working distance of the probe when the maximum coupling efficiency is achieved, while SMF achieves the minimum spot and maximum coupling efficiency at zero distance. As a result, the GRIN fiber probe achieves a higher coupling efficiency at a longer distance, which demonstrates the superiority of the GRIN fiber probe over SMF when coupled with HC-PCF.