3.1. Evanescent wave
When a light beam is totally internally reflected on a dielectric interface separating an optically high-index medium and optically lower-index medium, the reflected light beam is slightly shifted compared to the one in the classical approach using geometrical optics. This phenomenon is known as the Goos–Hänche effect . The Goos–Hänche shift (GHS) is a lateral shift of totally reflected beams along the optical interface. That is, the points of incidence and reflection do not coincide. This is attributed to the phase changes due to the evanescent wave that propagates into the lower-index medium near the interface . In this case, the beam appears to travel as an evanescent wave over a short distance through the optically low-index medium. Thus, in each reflection along the circular path of WGMs, the light seeps into the surroundings as an evanescent wave [2-4, 9, 54, 82-87]. A larger GHS corresponds to a larger evanescent wave as it propagates into the lower-index medium near the interface. The characteristics and theoretical simulation of GHS and evanescent wave are still actively investigated [88-91] although they are sometimes ignored because the exponentially small magnitude is far below that of the refracted field. However, the evanescent wave plays an important role in optical microcavities with a tubular geometry and thin walls (several micrometers or subwavelength thick). Most of optical properties in microcavities show a strong relationship with the evanescent wave and it will be discussed in the following sections. To demonstrate the relationship between the optical properties of tubular microcavities and evanescent wave, the influence of polarization state and wavelength of incident light wave on the evanescent wave is first discussed.
The GHS can be different if the polarization state of the incident light wave is changed. The optical-beam displacements in the incident plane are not the same for TE-polarized wave (the electric vector parallel to the incident plane) and TM-polarized wave (the electric vector perpendicular to the plane of incidence). The difference between the GHS reaches a maximum between TE- and TM-polarized waves with the same wavelength . The polarization-dependent GHS phenomena are confirmed by experiments and theoretical simulation  and the penetration depth is larger in the TE mode (s polarization) than the TM mode (p polarization) [80, 81, 92]. It should be noted that the definition of TE and TM mode (polarization) is under debate. The details will be discussed in Section 'Polarization of resonant light' with Figs. 4 and 10. To compare the experimental data with theory, the GHS versus the incident angle relationship is derived numerically using Artmann's formulas :
where dTE (TM) represents the GHS for the TE (TM) eigenstate, i is the incident angle on the plane dielectric surface, and n is the refractive index.
The difference between the TE and TM longitudinal GHS versus the angle of incidence at λ = 0.67 µm (n = 1.511, circle and solid line) and λ = 1.083 µm (n = 1.506, cubic and break line) are shown in Fig. 6. The relative sign of the two signals is to ensure that the TE polarization state is characterized by a higher displacement than the TM state. The TE mode thus suffers a larger optical loss than the TM mode at the surface of the optical microcavities, where the evanescent wave is possibly coupled with the radiation mode or absorbed slightly by the surrounding low-index media (Fig. 6). Figure 6 indicates that the agreement between the experimental measurements and theoretical calculation is good . Furthermore, the GHS is proportional to the incident light wavelength . In the short-wavelength limit λ 0, the GHS disappears, leading to the standard ray dynamics of geometric optics and the displacement of TM (or TE) polarization increases with incident wavelength (Fig. 6). Consequently, the evanescent wave increases with increasing incident wavelengths.
Figure 6. Difference between the TE and TM longitudinal GHS versus the angle of incidence at λ = 0.67 µm (n = 1.511, circle and solid line) and λ = 1.083 µm (n = 1.506, cubic and break line), respectively. Reproduced with permission from ref.  (the polarization states are redefined in this review).
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This difference in the optical loss caused by the evanescent wave is thought to affect the polarization characteristics, Q-factor, and sensitivity of tubular optical microcavities . The light wavelengths in WGMs microresonators are confined by continuous total internal reflection. In each reflection along the circular path of WGMs, the light seeps into the surroundings as an evanescent wave [2-4, 9, 54, 82-87]. Based on the discussion on the GHS and evanescent wave, two points are noted. First, the TE mode shows a larger light loss in each reflection along the circular path of the tubular WGMs microresonators than the TM mode. And secondly, the light loss in the tubular WGMs microresonator increases with the light wavelength.
The evanescent waves affect the properties of optical microcavities such as the Q-factor and polarization and the application of microcavities is also greatly influenced by the evanescent waves. At the same time, the evanescent waves are used to characterize the optical properties of optical microcavities and fabricate new optical devices. These issues will be discussed in detail later in this review.
3.2. Q-factor dependence on tubular wall characteristics
Among the various important characteristics of optical microresonators, the Q-factor is one of most basic parameters. It is a measure of the energy loss and defined by the time-averaged energy in the cavity divided by the energy loss per cycle :
The Q-factor in a resonant mode is an estimation of the finite photon lifetime in practice and can be determined by the ratio of the resonant frequency and the full width at half-maximum (FWHM) bandwidth of the resonance peak. In order to characterize the property of a resonator by keeping its energy inside for a decay time τ, Q-factor can also be expressed as [12, 33]:
where ∆ω is the linewidth (FWHM) of the Lorentzian peak associated with the considered resonance. The Q-factor of the WGM is determined by other mechanisms that cause light losses in WGMs. Thus, it is convenient to introduce the partial Q-factors related to each type of light losses in the microcavities and to describe Q by the well-known expression [12, 94, 95].
Normally, in an isolated (not coupled) WGM optical microcavity, the overall Q-factor (the intrinsic Q-factor of the resonators) is determined by the individual loss terms according to :
The first term Qrad denotes radiative (curvature) losses. vanishes exponentially with increasing size and for resonator diameters larger than 10 µm radiative losses are negligible [12, 96]. Qmat is associated with absorption and bulk Rayleigh scattering in the materials constituting the microcavities. Qs.s denotes scattering losses due to residual surface inhomogeneity. There are many different expressions for Qs.s [95, 97-100]. Qcont denotes the losses introduced by surface contaminates during the fabrication process. The limitation of the ultimate Q-factor caused by adsorption of atmospheric water on the surface of fused-silica microresonators is demonstrated in 1996  and Qs.s and Qcont are both defined as Qs.s (scattering light losses due to surface inhomogeneity).
In WGM microresonators, the most important loss terms are bulk absorption and scattering at surface inhomogeneities :
Similarly, in an isolated (not coupled) WGM optical microcavity with a tubular geometry, the overall Q-factor can be expressed as :
where Qrad, Qwall, and Qs.s are the Q-factors that are reasonably determined by the radiation loss, loss in the wall medium, and loss resulting from the surface scattering, respectively . Based on a detailed analysis on the Q-factors of the WGMs, it is noted that all WGMs (up to the 15th order) have Qrad ≫ 1011 . Thus, Qrad is negligible. If the WGM optical microcavities with a tubular geometry is filled with liquid, the Q-factor Qliq is determined by :
In WGM microresonators with a tubular geometry, the most important light loss terms are light loss in the wall medium (Qwall), scattering at surface inhomogeneity (Qs.s), and light loss caused by the liquid in the tubes core (Qsol), and η1 is the fraction of evanescent wave light outside the tube wall. The Q-factor increases as η1 decreases. The influence of wall thickness and materials index, surface roughness, and liquid medium on the Q-factor of WGM optical microcavities with the tubular geometry cannot be overlooked.
Based on Mie scattering theory simulation, the Q-factor of tubular optical microcavities can be increased by increasing the wall thickness and effective index constant  and reduced by increasing the wavelength  because of the properties of the evanescent waves. The evanescent waves impact the Q-factor significantly.
Figure 7 indicates that light can be confined in the wall of the fiber-drawing glass capillary  and self-rolled optical microcavities . Meanwhile, the modes generate outer and inner evanescent wave at the regions r < a (r is the radial position; a is the inner radius) and r > b (b is the outer radius) defined as the inner evanescent wave and outer evanescent wave, respectively. Compared to traditional capillary optical microcavities (Fig. 7a), the evanescent wave fraction of self-rolled optical microcavities with ultrathin wall thickness is higher (Fig. 7b). The Q-factor decreases with larger fraction of evanescent wave. The Q-factor indicates light confinement and the evanescent wave is generally related to light loss. Therefore, the Q-factor of the self-rolled tubular optical microcavities (<5000) is smaller than that of other types of optical microresonators with a tubular geometry (>104). However, the inner evanescent wave and outer evanescent wave inside and outside the tube are different in tubular optical microcavities. This ensures efficient coupling of the external emitters or analyte with the optical modes and opens up, for example, the possibility of evanescent field-coupled lasing  of external emitters in microtube resonators or lab-on-a-chip sensors to detect analytes such as DNA, cells, molecules, proteins, and viruses . Some of these interesting applications will be discussed in Section 'Applications of tubular optical microcavities'. Three parameters: the wall thickness and morphology, index contrast, and materials in the tube walls influence the Q-factors of tubular optical microcavities and the influence of the wall thickness, materials index, and liquid medium on the Q-factor of WGM optical microcavities with a tubular geometry will be discussed in detail in the following section. The influence of the evanescent wave will also be briefly described.
The wall of the tubular optical microcavities supports the WGMs. Owing to the Goos–Hänche effect, in each reflection along the circular path of WGMs, the light seeps into the surroundings as an evanescent wave [2-4, 9, 54, 82-87]. Meanwhile, based on the Mie scattering theory, the Q-factor bears a strong relationship with the wall thickness . In tubular microcavities with the same index constant, the wall thickness and morphology are crucial for achieving (ultra-)high Q-factor. Hollow structures show higher Q-factors than solid structures for spherical microcavities by a factor of about 10 . Figure 7a shows the calculated angle-averaged radial distributions of the two modes (TE165.1 and TE137.3) for the hollow and solid spheres with the same radius. In the solid case, the radial distributions extends far below the inner boundary (r = a = 0.926b) compared to the hollow case. The light is strongly confined within a < r < b in the hollow case . The mode distributes deeper from the outside boundary and the corresponding cavity Q-factor decreases, especially for the higher mode order (l > 1). The second (radial) mode order (l) indicates the number of maxima in the radial distribution of the internal electric field .
Figure 7. (a) Calculated angle-averaged radial distribution of TE165.1 and TE137.3 in a homogeneous sphere and a shell for a = 0.926b (inner radius a ∼11.9 µm and outer radius b ∼12.8 µm) . (b) Radial field distribution calculated for a ring with the diameter of the self-rolled tube and a wall thickness of 120 nm. The refractive index of the tube wall (gray shaded region) is assumed to be 3.3. Outside the tube we set n = 1. For the red (black) curve a refractive index inside the tube of ni = 1.49 for toluene (ni = 1 for vacuum) is used .
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Based on Eq. (22), the Q-factor increases when η1 (the fraction of evanescent wave light outside the tube wall) decreases. Figure 7 shows that the evanescent wave percentage of the glass capillary with thick wall (∼1 µm, mode order l = 1) is smaller than that of self-rolled microtubes with an ultrathin wall (∼150 nm thick). In other words, the light loss in a self-rolled tubular optical microresonator is dramatically higher than that in the glass capillary resonator. Hence, compared to glass capillary microcavities with a thick wall (> 0.5 µm), the silica glass self-rolled microtubes with an ultrathin wall (subwavelength, <150 nm) have a smaller Q-factor (less than 103). For one given azimuthal mode (m) of tubular optical microcavities with thick walls (wall thickness/outer diameter >0.05), the Q-factor increases slightly with wall thicknesses , whereas for one given azimuthal mode (m) of tubular optical microcavities with (ultra-)thin walls (wall thickness/outer diameter <0.05), the Q-factor increases dramatically with increasing wall thickness [104, 105].
Compared to (ultra-)high Q-factor (higher than 104) optical microresonators resembling microspheres, microspheroids, and toroids, the Q-factor of tubular microcavities without coupling is quite small (less than 5000). In some optical microresonators, self-interference can produce a resonant mode that is strongly localized along the axial direction to produce 3D optical confinement, but the WGMs in a uniform long tubular microresonator without taper coupling are delocalized. More details about 3D optical confinement are described in Section '3D optical confinement in a geometry-defined tube'.
Both theoretical simulation and experimental data indicate that high-index materials are good for (ultra-)high Q-factor microcavities. Based on Artmann's formulas (see Section 'Evanescent wave'), the evanescent wave increases with decreasing index contrast between the wall of the microcavity and low-index medium. Hence, the tubular microcavities with a high index contrast wall support low-light-loss WGMs [106, 107], meaning that the high index contrast microcavities possess (ultra-)high Q-factors compared to the low index contrast ones. Both the ultrathin wall (subwavelength) and effective index constant of the self-rolled microresonator are smaller than those of the glass capillary. The self-rolled optical microcavities with (ultra-)thin walls show small light confinement in the microcavities. Therefore, it is reasonable that the Q-factor (less than 5000) of a self-rolled microtube with ultrathin wall is still smaller than the Q-factor (larger than 104) of the fiber-drawing glass capillary. Meanwhile, the experimental results and Mie scattering show that the self-rolled microcavities with high effective index possess a high Q-factor compared to those with a low effective index . Furthermore, the main limit of the WGM Q-factor in Eq. (22) is related to the contribution Qwall associated with absorption and bulk Rayleigh scattering in the materials constituting the tubular microcavities. Qwall can be approximated as [12, 108]:
where λ is the light wavelength in vacuum, α is the absorption coefficient, and n is the refractive index. This approximation shows that the Q-factor increases as the refractive index (n) increases. As the index constant of the semiconductor (n > 3) is higher than that of silica (n ∼1.5), the Q-factor of InGaAs/GaAs rolled-up microtube laser (∼3500) [58, 109] is higher than that of SiO/SiO2 bilayer self-rolled tubular microcavities without a coating (Q-factor less than 1000) .
High index contrast microcavities support low-loss WGMs . In order to optimize the optical properties of tubular microcavities, two processes can be adopted, (1) choosing high index constant materials: Y2O3/ZrO2 self-rolled microcavities without surface modification with larger Q-factors (>1500) than SiO/SiO2  and (2) coating the tube walls with high index constant materials. When the self-rolled optical microcavities are coated with the same thickness materials with different indexes, the high index coating will effectively enhance the Q-factor. For example, Fig. 8 indicates that HfO2 effectively improves light confinement compared to the Al2O3 coating since HfO2 has a larger refractive index .
Figure 8. (a) 3D schematic diagram of a rolled-up nanomembrane in liquid. The bottom-right inset illustrates the multilayered structure of the wall, the top-left inset shows an optical microscope image of an ordered array of rolled-up nanomembranes, and the bottom-left inset shows a bird's-eye view SEM image of the opening of a rolled-up nanomembrane. (b) Photoluminescence (PL) spectra obtained in water from rolled-up nanomembranes with HfO2 and Al2O3 coating layers (30 nm thick in both cases) taken under excitation of the 442-nm line of a He-Cd laser .
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Besides the diameter of optical microcavities (see Section '3D optical confinement in a geometry-defined tube'), the same effect can be achieved by creating an “index bottle” without involving any diameter change along the axial direction . These prolate “index bottle” WGM microcavities possess ultralarge Q-factors of 2 × 105 .
The main limit of the WGM Q-factor in Eq. (22) is related to the contribution Qsol associated with light loss caused by the liquid in the tube core . The wall of the tubular optical microcavities supports WGMs and the evanescent wave interacts with the inside and outside surrounding medium when the evanescent wave light seeps into the surrounding medium [9, 85, 94, 111-113]. Based on Eq. (21), the Q-factor increases as η1 (the fraction of evanescent wave light outside the tube wall) decreases. The Q-factor is influenced by the inside and outside liquid media and here, both experimental and theoretical results are discussed.
The liquid index influences the self-rolled-up optical microcavities with subwavelength wall thickness as observed by Huang et al.  and Moon et al. . Huang et al. noticed that as the refractive indices of the surrounding media increased, light loss for WGMs modes increased and the Q-factor of the WGMs in the tubular optical microcavities decreased. However, light loss in the TE modes was much more prominent than in the TM modes, rendering the TE modes undetectable in liquids . The Q-factors of big microcavities (the diameter is around 9 µm) in air and liquid are ∼480 and ∼220, respectively, for the mode at ∼2 eV, and those of the small microcavity (the diameter is around 7 µm) are ∼660 and ∼250, respectively . Moon et al.'s results indicate the Q-factor of glass capillary microcavities increases as the refractive index of the inner region increases . As the liquid index increases from 1.0 to 1.1 and 1.15, the Q-factor of TE137.3 increases from 4.9 × 103 to 9.0 × 103 and 1.5 × 104, respectively . The experimental results are consistent with theoretical simulation .
Based on Mie scattering theory, a theoretical demonstration about the influence of the liquid medium on the rolled-up tubular optical microcavities was reported by Zhao et al. . Figure 9a shows the cross-sectional schematic view of a microtube. The microtube is placed in four different surroundings: air (Fig. 9d), with a liquid inside and air outside (Fig. 9e), with a liquid outside and air inside (Fig. 9f), and in a liquid (Fig. 9g). The liquid has a refractive index of nL. The Q-factors of the ideal microtubes depend on the index n3 of the outer medium (Fig. 9b). For the ideal microtubes with a given diameter, the Q-factor increases with wall thicknesses (Δ/h < 0.05). When Δ/h is smaller than 0.05, the increase in the Q-factor is negligible. When the microtube is surrounded by liquid (nL > 1) on the outside, the index contrast between the microtube wall and liquid is smaller than the index contrast between the microtube wall and air. Under these conditions (Figs. 9f and g), the optical loss caused by the evanescent wave is larger than those under the other two conditions. Thus, for the ideal tubular optical microcavities with (ultra-)thin wall, the Q-factors depend on the index n3 of the liquid medium. The liquid media inside and outside the microtube result in light loss.
Figure 9. (a) Cross-sectional schematic views of a microtube. The microtube has a diameter of h, wall thickness of Δ and refractive index of n2 in the wall. The refractive indices are n1 and n3 for media inside and outside the microtube, respectively. The incident light propagates in the x–y plane and has E field along the z direction. (b) Q-factors for the m = 40 resonant modes of the microtubes in (c)–(f) with n2 = 2 in the tube walls. (c) Q-factors for the m = 40 resonant modes of the microtubes in (c)–(f) with n2 = 2 + 0.004i in tube walls. (d)–(g) The microtube is placed in four kinds of environment: (d) in air (green line in b and c), (e) with liquid inside and air outside (black line in b and c), (f) with liquid outside and air inside (red line in b and c), and (g) in liquid (blue line in b and c). The liquid has a refractive index of nL .
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The above studies are performed on ideal microtubes with Q-factors of Qi. Actually, the fabrication process also introduces surface imperfections. In rolled-up microtubes, the Q-factor , where Qs.s is related to the loss from surface imperfection and cone effects (Q < Qi, Qs.s and Qs.s < 5000 in rolled-up microtubes). The diameters of rolled-up tubes vary linearly along the tube axis, which is called the cone effect. As shown in Fig. 9c, we simulate the imperfect microtubes by using a complex refractive index n2 = 2 + 0.004i in the tube walls. For the tubular optical microcavities with thick walls (Δ/h > 0.05), under all four conditions, the surface imperfections (Qs.s) are the main limit of the WGM Q-factors with values smaller than 2000 (Fig. 9c). In the tubular optical microcavities with (ultra-)thin walls (Δ/h < 0.05), the liquid medium influences the surface imperfections more (Fig. 9c). Compared to the tube in air (Fig. 9c, green line), the inside and outside liquid media (nL > 1) lead to light loss and dramatically decrease the Q-factors (black, red, and blue lines in Fig. 9c).
The experimental and theoretical demonstration inside and outside the liquid medium indicate that the Q-factor is reduced by the liquid medium surrounding the tube microcavities with (ultra-)thin walls (Δ/h < 0.05). In optical microcavties with thick walls (Δ/h > 0.05), the influence of surface imperfections is more substantial than that of the elements.
The interaction between the evanescent wave (WGMs) and surrounding media can be used to modify the optical microcavities. Dye-doped optical–gain media have been chosen in microlaser design since 1970s [39, 114, 115]. Various polymeric materials such as PMMA [84, 116], polystyrene (PS) , polyurethane (PU) , poly(p-phenylene-vinylene) (PPV) [25, 117, 118], and poly(1-vinyl-2-pyrrolidone) (PVP) can be used and the main dyes include Rh 6G , and pyrromethene .
The tubular geometry is suitable to contain dye solutions liquid as the gain medium. Compared to the solid gain medium, liquid dye solutions lead to more applications as they can be used as carriers of dyes [40, 86, 87, 92] and luminescent quantum dots [31, 70, 85, 119-123] to introduce special optical properties.
3.3. Polarization of resonant light
Generally, the spectrum of tubular optical microcavities is split into two well-known polarization modes, the transverse-electric (TE) mode and transverse-magnetic (TM) mode defined by the orientation of electric and magnetic fields. However, the definitions of TM polarized and TE polarized vary. In some references, the TM-polarized wave has the electric vector parallel to the incident plane (namely ), whereas a TE-polarized wave has the electric vector perpendicular to the plane of incidence (namely ) [91, 92, 95]. In other references, the orientation of the electric field of a TE-polarized wave is parallel to the incident plane but that of the magnetic field of a TM-polarized wave is parallel to the incident plane . In the TE mode, the electric field is tangential to the microcavity surface whereas in the TM mode, its electric field is normal to the microcavity surface [9, 72, 104, 125]. In this review, the following polarization definitions are adopted: TE modes with the magnetic field vector parallel to the tube axis and TM modes with the electric field vector parallel to the tube axis (Fig. 10) .
The Q-factors bear a strong relationship to the polarization states. Based on Mie scattering theory, compared to the TE modes, the TM modes show larger decay in air . Hence, TM modes are used in experiments and sensors because they have larger Q-factors than the TE modes normally [47, 54, 121, 126]. Another possible reason for the polarization dependence is the mode-field expansion, i.e. the penetration depth of the light wave in the total internal reflection process . The penetration depth or the GHS (Section 'Evanescent wave'), which is induced by the phase shift on reflection, is larger in the TE mode (p polarization) than TM mode (s polarization) . Hence, the TE mode suffers a larger optical loss at the interface, where the evanescent wave is possibly coupled with the radiation mode or absorbed slightly by the surrounding medium . The difference in the optical loss is thought to affect the polarization characteristics.
The detection sensitivity of optical microcavities changes according to the polarization states. The TE modes exhibit higher sensitivity when the surrounding medium is changed compared to the TM modes . If the Q-factor is not very important, it is better to use the TE modes because the peak shifts are larger than those in the TM modes for the same change in the liquid index constants . Under the same Q-factor condition, the TE modes show higher sensitivities for refractive index detection of the surrounding medium, as verified experimentally . The TM modes of WGMs resonances of capillaries with submicrometer wall thickness show a smaller wavelength shift as a function of the refractive index of the medium that fills the interior . The sensitivity of the TM modes (50–70 nm/RIU) is lower than those for the TE modes (130–170 nm/RIU) . Tubes with thin walls are required in order to produce high-sensitivity sensors  and the self-rolled optical microresonators have superior performance in this respect [35, 37, 85, 127-129]. The possible reason for this polarization dependence is the evanescent wave. The TE mode has a larger GHS than the TM mode at the interface, where the evanescent wave is possibly coupled with and absorbed slightly by the surrounding medium . Consequently, the TE modes have stronger interaction with the surrounding medium.
The TM modes and TE modes are important to the application of self-rolled optical microcavities [37, 47, 63, 128, 130]. Using the single-scatter-induced coupling mechanism of a pair of counterpropagating high-Q-factor WGMs, the toroidal microcavity can be used to investigate single nonspherical nanoparticles with high sensitivity. The nonspherical particles may produce distinct frequency splitting and additional damping for TE and TM WGMs. This polarization-dependent effect allows the study of the orientation of single biomolecules, molecule–molecule interaction on the microcavity surface, and distinguishing different inner configurations of similar biomolecules [125, 131].