In general, AR-HCFs features irregular silica cladding and an air core, which induces a special light propagation path. Here, AR-HCFs obey the light-guiding mechanisms of the anti-resonant reflecting optical waveguide (ARROW) [
11], rather than the principle of total internal reflection of SMF. ARROW exploits coherent reflections at the air–silica interface to effectively guide the forward propagation light into the central hollow-core. To intuitively reflect the light propagation in the AR-HCFs, the hollow-core capillary with regular silica cladding is selected as the representative to describe the anti-resonant theory. As shown in a, hollow-core capillary with a length
L is sandwiched by the two segments of SMFs. The hollow-core capillary with regular ring cladding can be regarded as the FP cavity along the radial direction. The light beams,
I1 and
I2, represent the first reflection of their incident light at the air–silica and silica-air interface, respectively. Here, the incident light of
I2 is the refract light generated by the initial incident light. Then, the two beams
I1 and
I2 will form interference in the air core on the condition that the initial incident angle
θ meets the grazing incidence (
θ~90°). When the incident light of
I2 reaches the phase matching condition, it will leak out of the cladding. The leaked light is called resonant light, similarly, the reflected part is AR light. In short, AR-HCFs exploit coherent reflections from cladding to confine the guided light and propagate in the central air core. Thus, these light transmission mechanisms reveal the basic principle of the AR effect in HCFs. In addition, it is worth noting that the generation of the AR effect is limited by the length of the HCF. Obviously, if
L is short enough, the Fabry–Perot (FP) cavity formed by the two fusion splicing interfaces dominates the whole transmission spectrum, as exhibited in b. With an increasing
L, however, the FP effect will gradually be weak or disappear caused by the increasing space loss. At this moment, nearly all of the light follows the ARROW transmission mechanisms. It can be seen that a critical length exists between FP and AR effect. The critical length
Lc corresponds to the axial transmission length of beam
I2, it can be expressed as follows [
12]:
where
n0,
n1, and
n2 represent the refractive index of air, fiber core of SMF, and cladding, respectively.
r and
d denote the radius of the air core and the thickness of the ring cladding, respectively. If the capillary length is longer than
Lc, the AR effect will be excited in the whole process. Otherwise, the sandwich structure only induces the FP effect. b shows that several resonant wavelengths with a periodic distribution that is located in the AR effect-based transmission spectrum. According to the phase matching condition, the resonant wavelength
λr can be given as follows [
13]:
where
m is the resonance order. All of the aforementioned statements about AR-HCFs are based on the regular shape of the silica cladding, while most of the application scenes rely on an irregular structure to enhance the sensitivity. In view of diversified sensing applications, the next part will accordingly introduce the various structure of AR-HCFs.