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Surface nanoscale axial photonics at a capillary fiber

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Abstract

We present the theory and first experimental demonstration, to the best of our knowledge, of a sensing platform based on surface nanoscale axial photonics (SNAP) at a capillary fiber. The platform explores optical whispering gallery modes, which circulate inside the wall of a capillary and slowly propagate along its axis. Due to the small thickness of the capillary wall, these modes are sensitive to spatial and temporal variations of the refractive index of the media adjacent to the internal capillary surface. In particular, the developed theory allows us to determine the internal effective radius variation of the capillary from the measured mode spectra. Experimentally, a SNAP resonator is created by local annealing of the capillary with a focused CO2 laser followed by internal etching with hydrofluoric acid. The comparison of the spectra of this resonator in the cases when it is empty and filled with water allows us to determine the internal surface nonuniformity introduced by etching. The results obtained pave the way for a novel advanced approach in sensing of media adjacent to the internal capillary surface and, in particular, in microfluidic sensing.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Corrections

T. Hamidfar, A. Dmitriev, B. Mangan, P. Bianucci, and M. Sumetsky, "Surface nanoscale axial photonics at a capillary fiber: publisher’s note," Opt. Lett. 42, 4828-4828 (2017)
https://opg.optica.org/ol/abstract.cfm?uri=ol-42-23-4828

26 October 2017: A typographical correction was made to the author listing.

The sensing platform developed in this Letter extends the surface nanoscale axial photonics (SNAP) platform [13] to the case of thin-walled capillary fibers, including those filled with gas, liquid, or solid media. Following the idea of Ref. [4], we explore whispering gallery modes (WGMs), which circulate inside the capillary wall and slowly propagate along the capillary axis for sensing the media adjacent to the internal capillary surface and, in particular, the nonuniformity of the internal surface itself. The wavelength of WGMs considered is very close to their cutoff wavelength. For this reason, the propagation constant of these modes is small and their speed along the capillary axis is slow [3]. It has been shown that the axial distribution of slow WGMs is sensitive to extremely small nanoscale variations of the optical fiber radius [1]. In the case of a capillary fiber, it is of great interest to investigate how the spatial and temporal variations of the media adjacent to the internal capillary surface affect the spectrum of the SNAP resonator created at the capillary wall. In particular, it is important to find out if it is possible to determine the internal effective radius variation of the capillary from the measured spectra, as was done for the outer radius in SNAP [13].

Our method illustrated in Fig. 1 bridges and extends the ideas of the liquid core resonance sensing method [5] and SNAP [1]. White, Oveys, and Fan [5] proposed and demonstrated a fused silica capillary with a few-micron-thick wall acting as a WGM resonator sensor of the refractive index of liquid carried by the capillary. In their first demonstration and followup publications (see reviews [6,7]), an axially uniform silica capillary coupled to a transverse microfiber or planar waveguide was explored. Sensing of fluid inside the capillary was performed locally at the position of the transverse waveguide and was based on the measurement of variation of a single resonance. A SNAP microresonator introduced along the capillary surface (Fig. 1) can significantly advance this approach. In fact, it enables the detection of changes, which happen away from the waveguide position along the length of the resonator by controlling the variation of resonant spectra of this resonator [4].

 figure: Fig. 1.

Fig. 1. Illustration of a capillary fiber coupled to an input–output microfiber. The capillary was processed with a CO2 laser beam and, internally, with hydrofluoric acid. Inset: magnified cross section of the capillary wall (not to scale).

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The WGMs of our concern are adjacent to the external capillary surface. For a capillary with a thick wall, these modes have negligible overlap with the internal surface and, therefore, are independent of the optical properties of media situated at or in a small vicinity of this surface. For example, in the illustration of Fig. 1, the WGM is much less sensitive to the internal surface variations near the internal bump. Away from the bump, the wall becomes thinner and the WGMs spectra start reacting to the changes at the internal surface of the capillary.

In application to the capillary fibers, the SNAP theory is developed as follows. The dependence of external and internal radii of the capillary, rint(z)=rint(0)+Δrint(z) and rext(z)=rext(0)+Δrext(z), are assumed to be adiabatically slow and the refractive index is defined as

n(ρ,z)={nint,ncap,next,0<ρrint(z),rint(z)<ρrext(z),rext(z)<ρ.

The WGMs considered are numerated by the azimuthal quantum number m, radial quantum number p, and axial quantum number q (the latter is introduced if the WGM is localized along the capillary axis) and can be expressed in the cylindrical coordinates as Empq(z,ρ,ϕ)=exp(imϕ)Qmp(ρ,z)Ψmpq(z). The slowness of WGMs is manifested in the small value of their propagation constant, β(z), or, equivalently, in the proximity of their wavelength λ to the cutoff wavelength λmp(cut)(z) [8]. Functions Qmp(ρ,z) and Ψmpq(z) are determined by the adiabatic separation of variables in the wave equation, which leads to the well-known equation for the radial distribution of modes in an optical fiber [9]:

d2Qmpdρ2+1ρdQmpdρ+((2πn(ρ,z)λ)2m2ρ2β2(z))Qmp=0,
where the dependencies on the axial coordinate z are parametric. Consequently, the dependence of Ψmpq(z) on z is determined by the one-dimensional wave equation
d2Ψmpqdz2+β2(z)Ψmpq=0.

Here we are interested in the situation when the variation of the cutoff wavelength, λmp(cut)(z)λmp(0), and the deviation of the radiation wavelength from the cutoff, λλmp(cut)(z), are small. This condition is satisfied for small variations of radii, Δrint(z) and Δrext(z), and for the evanescent values of Qmp(ρ,z) near the internal wall surface. Under these assumptions, we set the propagation constant equal to

β(z)=2πncapλmp(cut)2(λλmp(cut)(z))λmp(cut)
and ignore the terms that are of higher order in λλmp(cut)(z). Then Eq. (2) is simplified to the following equation independent of wavelength λ:
d2Qmpdρ2+1ρdQmpdρ+((2πn(ρ,z)λmp(cut))2m2ρ2)Qmp=0.
We are looking for the solution of this equation in the form
Qmp(ρ,z)={AJm(kmpnintρ),BJm(kmpncapρ)+CYm(kmpncapρ),DJm(kmpnextρ),0<ρrint(z),rint(z)<ρrext(z),rext(z)<ρ.
Here kmp=2π/λmp(cut), functions Jm(x) and Ym(x) are the Bessel functions, and parameters A,B,C, and D are determined from the condition of continuity of Qmp(ρ,z) and its derivative over ρ at the interfaces r1(z) and r2(z) [10]. The solution of Eq. (6) with the refractive index profile determined by Eq. (1) allows us to express the cutoff wavelength, λmp(cut)(z), through the external and internal radii, rext(z) and rint(z). Next, to determine the WGMs and their spectrum, we solve Eq. (3) with the propagation constant determined by Eq. (4).

The device fabricated and investigated in this Letter consists of a silica fiber capillary with an external radius r0=21μm and an initial wall thickness of 6 μm. WGMs are excited in the capillary by a biconical fiber taper having a microfiber waist, which was fabricated of a conventional single-mode fiber. To measure the resonant spectrum of WGMs, the taper is connected to the optical spectrum analyzer.

A SNAP microresonator was introduced at the capillary surface by local annealing with a focused CO2 laser beam. The WGM spectra of this resonator were measured at microfiber positions spaced by 20 μm along the capillary axis [Fig. 2(a)]. The length of the resonator was 300μm. From Fig. 2(a), the spectral width of the resonator is Δλ00.15nm at a radiation wavelength of λ01.57μm. The height of the introduced external effective radius variation (ERV) is estimated as Δr0Δλ0r0/λ02nm. The resonant spectra shown in Fig. 2(a) correspond to the vicinity of a cutoff wavelength having the azimuthal and radial quantum numbers m2πncapr0/λ0 and p=0, respectively. For a silica capillary with ncap1.5, we have m100.

 figure: Fig. 2.

Fig. 2. Surface plots of spectra of the fabricated SNAP resonator measured with 20 μm resolution along the capillary axis. (a) Before etching, p=0. (b) After etching, empty capillary, p=0 (bottom) and p=1 (top). (c) After etching, capillary filled with water, p=0 (bottom) and p=1 (top). (d) Parabolic approximation of the cutoff wavelength for the empty (black curve) and water-filled (blue curve) capillary. Dashed red curve is the difference of these curves. (f) The restored internal ERV. The solid, dashed, and dotted curves correspond to λwλe equal to 0.05 nm, 0.04 nm, and 0.06 nm, respectively.

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In order to make the WGM spectrum of the SNAP microresonator sensitive to the presence of fluid inside the capillary, its wall thickness was reduced by internal etching with the hydrofluoric acid [5]. The etching process was controlled by the simultaneous measurement of the resonant spectrum and was stopped when the shift of resonances was observed.

Figures 2(b) and 2(c) show the SNAP resonator spectra after etching for the empty and the water-filled capillary, respectively. It is seen that the axial WGM resonances adjacent to the cutoff wavelength with the fundamental radial number series (p=0, q=0,1,2,, bottom) experience relatively small distortion. However, the axial resonances of WGMs with a lager radial quantum number (hypothetically the p=1 series, top) change dramatically. The p=0 axial resonances are narrower, i.e., less lossy than the p=1 resonances, due to the fact that they have less overlap with the internal surface of the capillary. The dramatic shrinking of the p=1 WGMs compared to the p=0 WGMs along the axial direction can be explained by the axial nonuniformity introduced by hydrofluoric etching. We suspect that, in addition to the nanoscale variation of the wall thickness, the CO2 laser annealing deforms the capillary wall as a whole. This deformation disturbs the hydrofluoric acid flow and leads to the creation of a bump of the capillary wall illustrated in the inset of Fig. 1. While the p=0 WGM series have small overlap with the internal wall surface and, therefore, are not noticeably sensitive to the appearance of this bump, it causes the additional localization of the p=1 WGMs evident from Figs. 1(b) and 1(c).

Comparison of Figs. 2(b) and 2(c) shows that filling the capillary with water led to the reduction of separation between the axial series of resonances with axial quantum numbers q=0,1,2, clearly seen for the p=1 series. In our experiment, adding water introduced the shift of the SNAP microresonator spectrum by 0.13 nm as a whole. This shift is presumably due to mechanical deformation of the capillary pressurized by water. Below, in the theoretical analysis of experimental data, we assumed that, besides this global shift, the effect of water on the p=0,q=0 resonances is negligible. This fact allowed us to calibrate the relative positions of surface plots in Figs. 2(b) and 2(c) by equalizing the positions of the p=0, q=0 resonances.

We used the developed theory to estimate the internal ERV, rint(z)=rint(0)+Δrint(z), from the measured spectra of the SNAP resonator. To this end, we numerically analyzed and compared the p=1 resonances for the empty and water-filled capillaries shown in Figs. 2(b) and 2(c), respectively. The major contribution to the appearance of these resonances was caused by variation of the internal radius rint(z). Therefore, in our calculations, we neglected the variation of the external radius. Figure 3(a) compares the dependencies of the cutoff wavelength on the internal radius for rint(z)r0=21.12μm, ncap=1.46, two azimuthal quantum numbers, m=113 and 105, and three radial quantum numbers, p=0,1,2. We suggest that our experimental situation can be approximated by the behavior of the m=113, p=0 and m=105, p=1 series. Figure 3(b) compares the deviations of dependencies shown in Fig. 3(a) from their value λmpq(0) for a thick capillary (rint<16μm) magnified to the nanometer wavelength scale of our interest. The curves in Fig. 3(b) look remarkably similar and coincide with good accuracy after the horizontal translation into the darker rectangle in this figure. Thus, all of them can be defined by a common function Δλ(cut)(rint+s) with the appropriate choice of shift s. Consequently, the internal ERV can be found as λmpq(cut)(rint)=Δλ(cut)(rint+s)+λmpq(0). While the actual value of s is not important for the determination of variation of the internal radius, the direct application of this result to our experiment is complicated because the accurate measurement of λmpq(0) is not possible. To solve this problem, we proceed as follows. Figures 3(c) and 3(d) compare the behavior of cutoff wavelength with m=113 and q=0,1,2 for the empty capillary and the capillary filled with water. These dependencies, again, coincide with a good accuracy after horizontal translation [they are compared in the darken rectangles of Figs. 3(c) and 3(d)]. In particular, the shift of the cutoff wavelength caused by filling the capillary with water is defined by a common function:

Δλ˜(cut)(rint+s)=λmpq(cut)(rint+s)|waterλmpq(cut)(rint+s)|empty,
which does not depend on λmpq(0). Thus, this function does not depend on the actual values of the azimuthal and radial quantum numbers m and p, respectively, which were not precisely determined from the experiment.

 figure: Fig. 3.

Fig. 3. (a) Cutoff wavelength as a function of internal ERV for the quantum numbers indicated on the plot. (b) Dependencies shown in (a) magnified and shifted along the vertical axis. (c) Cutoff wavelength as a function of internal ERV for the empty and water-filled capillaries for the quantum numbers indicated on the plot. (d) Dependencies shown in (c) magnified and shifted along the vertical axis. Curves shown in (b) and (d) are compared by horizontal translation into the darker rectangles.

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In both cases of empty and water-filled capillaries, we estimate the cutoff wavelength dependency corresponding to p=1 spectral series in Figs. 2(b) and 2(c) by parabolas, λwater(cut)(z)=λwγwz2 and λempty(cut)(z)=λeγez2, respectively. Here parameters γw,e are expressed through the separation of resonances along the axial quantum number, δλw,e, as γw,e=2(πncapδλw,e)2λ03 [11]. From Figs. 2(b) and 2(c), setting the top of the p=0 resonance series as a reference (which, as noted above, has a negligible effect of water), we estimate δλe0.1nm, δλw0.08nm, and λwλe0.05nm. Consequently, γw=1.1·107μm1 and γe=6.9·108μm1. Figure 2(d) shows the dependencies λempty(cut)(z) and λwater(cut)(z) (black and blue curves, respectively), which are translated into surface plots on Figs. 2(b) and 2(c) and show good agreement with the experiment. Finally, the black solid curve in Fig. 2(f) shows the internal ERV Δrint(z) restored from the difference λwater(cut)(z)λempty(cut)(z) [red dashed curve in Fig. 2(d)] following Eq. (7). This variation appears to be quite different from parabola and has the micrometer scale. In contrast, as it follows from the SNAP theory of a regular fiber [13], the same cutoff wavelength variation can be introduced by the external ERV Δrext(z)=(λempty(cut)(z)λmpq(0))r0/λ0, which has nanometer rather than micrometer scale.

In summary, we have developed a theory of a capillary SNAP platform and experimentally demonstrated a SNAP microresonator at the surface of a capillary that is sensitive to the presence of fluid inside the capillary. The resonator was created by local annealing of the capillary with a focused CO2 laser beam and internal etching with hydrofluoric acid. We investigated the variation of the spectra of fabricated microresonator resulted from thinning of the capillary wall and the presence of water inside the capillary. Using the developed theory, we determined the internal effective radius variation of the capillary from the spectra of the SNAP resonator measured experimentally. We believe that the future development of this theory will allow for the simultaneous determination of the internal and external effective radius variation of the capillary. In addition, we suggest that the developed approach will allow for the determination of the complex structure of microfluidic components adjacent to the internal capillary surface, e.g., the resonant structure of microparticles [12]. Generally, this demonstration provides the groundwork for advanced microfluidic sensing with SNAP microresonators.

Funding

Royal Society (WM130110); Horizon 2020 Framework Programme (H2020) (H2020-EU.1.3.3, 691011); Engineering and Physical Sciences Research Council (EPSRC) (EP/P006183/1).

Acknowledgment

MS acknowledges the Royal Society Wolfson Research Merit Award.

REFERENCES AND NOTES

1. M. Sumetsky, D. J. DiGiovanni, Y. Dulashko, J. M. Fini, X. Liu, E. M. Monberg, and T. F. Taunay, Opt. Lett. 36, 4824 (2011). [CrossRef]  

2. M. Sumetsky, Nanophotonics 2, 393 (2013). [CrossRef]  

3. M. Sumetsky, Phys. Rev. Lett. 111, 163901 (2013). [CrossRef]  

4. M. Sumetsky, Opt. Lett. 39, 5578 (2014). [CrossRef]  

5. I. M. White, H. Oveys, and X. Fan, Opt. Lett. 31, 1319 (2006). [CrossRef]  

6. X. Fan and I. M. White, Nat. Photonics 5, 591 (2011). [CrossRef]  

7. T. Reynolds, N. Riesen, A. Meldrum, X. Fan, J. M. M. Hall, T. M. Monro, and A. Francois, Laser Photon. Rev. 11, 1600265 (2017). [CrossRef]  

8. The fact that the cutoff wavelengths of the WGMs with very large azimuthal quantum numbers m correspond to the zero propagation constant, β=0, does not contradict to the well-known relation kmpnext<β<kmpncap for small m (see, e.g., [9]).

9. A. W. Snyder and J. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

10. For the case kmpncapr0m1 and kmpnextr0<m considered, the Bessel function Jm(kmpnextρ) exponentially decays with the growth of ρ for 0<kmpnext(ρr0)m.

11. M. Sumetsky and J. M. Fini, Opt. Express 19, 26470 (2011). [CrossRef]  

12. Y. Li, F. Abolmaali, K. W. Allen, N. I. Limberopoulos, A. Urbas, Y. Rakovich, A. V. Maslov, and V. N. Astratov, Laser Photon. Rev. 11, 1600278 (2017). [CrossRef]  

References

  • View by:

  1. M. Sumetsky, D. J. DiGiovanni, Y. Dulashko, J. M. Fini, X. Liu, E. M. Monberg, and T. F. Taunay, Opt. Lett. 36, 4824 (2011).
    [Crossref]
  2. M. Sumetsky, Nanophotonics 2, 393 (2013).
    [Crossref]
  3. M. Sumetsky, Phys. Rev. Lett. 111, 163901 (2013).
    [Crossref]
  4. M. Sumetsky, Opt. Lett. 39, 5578 (2014).
    [Crossref]
  5. I. M. White, H. Oveys, and X. Fan, Opt. Lett. 31, 1319 (2006).
    [Crossref]
  6. X. Fan and I. M. White, Nat. Photonics 5, 591 (2011).
    [Crossref]
  7. T. Reynolds, N. Riesen, A. Meldrum, X. Fan, J. M. M. Hall, T. M. Monro, and A. Francois, Laser Photon. Rev. 11, 1600265 (2017).
    [Crossref]
  8. The fact that the cutoff wavelengths of the WGMs with very large azimuthal quantum numbers m correspond to the zero propagation constant, β=0, does not contradict to the well-known relation kmpnext<β<kmpncap for small m (see, e.g., [9]).
  9. A. W. Snyder and J. Love, Optical Waveguide Theory (Chapman and Hall, 1983).
  10. For the case kmpncapr0≈m≫1 and kmpnextr0<m considered, the Bessel function Jm(kmpnextρ) exponentially decays with the growth of ρ for 0<kmpnext(ρ−r0)≪m.
  11. M. Sumetsky and J. M. Fini, Opt. Express 19, 26470 (2011).
    [Crossref]
  12. Y. Li, F. Abolmaali, K. W. Allen, N. I. Limberopoulos, A. Urbas, Y. Rakovich, A. V. Maslov, and V. N. Astratov, Laser Photon. Rev. 11, 1600278 (2017).
    [Crossref]

2017 (2)

T. Reynolds, N. Riesen, A. Meldrum, X. Fan, J. M. M. Hall, T. M. Monro, and A. Francois, Laser Photon. Rev. 11, 1600265 (2017).
[Crossref]

Y. Li, F. Abolmaali, K. W. Allen, N. I. Limberopoulos, A. Urbas, Y. Rakovich, A. V. Maslov, and V. N. Astratov, Laser Photon. Rev. 11, 1600278 (2017).
[Crossref]

2014 (1)

2013 (2)

M. Sumetsky, Nanophotonics 2, 393 (2013).
[Crossref]

M. Sumetsky, Phys. Rev. Lett. 111, 163901 (2013).
[Crossref]

2011 (3)

2006 (1)

Abolmaali, F.

Y. Li, F. Abolmaali, K. W. Allen, N. I. Limberopoulos, A. Urbas, Y. Rakovich, A. V. Maslov, and V. N. Astratov, Laser Photon. Rev. 11, 1600278 (2017).
[Crossref]

Allen, K. W.

Y. Li, F. Abolmaali, K. W. Allen, N. I. Limberopoulos, A. Urbas, Y. Rakovich, A. V. Maslov, and V. N. Astratov, Laser Photon. Rev. 11, 1600278 (2017).
[Crossref]

Astratov, V. N.

Y. Li, F. Abolmaali, K. W. Allen, N. I. Limberopoulos, A. Urbas, Y. Rakovich, A. V. Maslov, and V. N. Astratov, Laser Photon. Rev. 11, 1600278 (2017).
[Crossref]

DiGiovanni, D. J.

Dulashko, Y.

Fan, X.

T. Reynolds, N. Riesen, A. Meldrum, X. Fan, J. M. M. Hall, T. M. Monro, and A. Francois, Laser Photon. Rev. 11, 1600265 (2017).
[Crossref]

X. Fan and I. M. White, Nat. Photonics 5, 591 (2011).
[Crossref]

I. M. White, H. Oveys, and X. Fan, Opt. Lett. 31, 1319 (2006).
[Crossref]

Fini, J. M.

Francois, A.

T. Reynolds, N. Riesen, A. Meldrum, X. Fan, J. M. M. Hall, T. M. Monro, and A. Francois, Laser Photon. Rev. 11, 1600265 (2017).
[Crossref]

Hall, J. M. M.

T. Reynolds, N. Riesen, A. Meldrum, X. Fan, J. M. M. Hall, T. M. Monro, and A. Francois, Laser Photon. Rev. 11, 1600265 (2017).
[Crossref]

Li, Y.

Y. Li, F. Abolmaali, K. W. Allen, N. I. Limberopoulos, A. Urbas, Y. Rakovich, A. V. Maslov, and V. N. Astratov, Laser Photon. Rev. 11, 1600278 (2017).
[Crossref]

Limberopoulos, N. I.

Y. Li, F. Abolmaali, K. W. Allen, N. I. Limberopoulos, A. Urbas, Y. Rakovich, A. V. Maslov, and V. N. Astratov, Laser Photon. Rev. 11, 1600278 (2017).
[Crossref]

Liu, X.

Love, J.

A. W. Snyder and J. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Maslov, A. V.

Y. Li, F. Abolmaali, K. W. Allen, N. I. Limberopoulos, A. Urbas, Y. Rakovich, A. V. Maslov, and V. N. Astratov, Laser Photon. Rev. 11, 1600278 (2017).
[Crossref]

Meldrum, A.

T. Reynolds, N. Riesen, A. Meldrum, X. Fan, J. M. M. Hall, T. M. Monro, and A. Francois, Laser Photon. Rev. 11, 1600265 (2017).
[Crossref]

Monberg, E. M.

Monro, T. M.

T. Reynolds, N. Riesen, A. Meldrum, X. Fan, J. M. M. Hall, T. M. Monro, and A. Francois, Laser Photon. Rev. 11, 1600265 (2017).
[Crossref]

Oveys, H.

Rakovich, Y.

Y. Li, F. Abolmaali, K. W. Allen, N. I. Limberopoulos, A. Urbas, Y. Rakovich, A. V. Maslov, and V. N. Astratov, Laser Photon. Rev. 11, 1600278 (2017).
[Crossref]

Reynolds, T.

T. Reynolds, N. Riesen, A. Meldrum, X. Fan, J. M. M. Hall, T. M. Monro, and A. Francois, Laser Photon. Rev. 11, 1600265 (2017).
[Crossref]

Riesen, N.

T. Reynolds, N. Riesen, A. Meldrum, X. Fan, J. M. M. Hall, T. M. Monro, and A. Francois, Laser Photon. Rev. 11, 1600265 (2017).
[Crossref]

Snyder, A. W.

A. W. Snyder and J. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Sumetsky, M.

Taunay, T. F.

Urbas, A.

Y. Li, F. Abolmaali, K. W. Allen, N. I. Limberopoulos, A. Urbas, Y. Rakovich, A. V. Maslov, and V. N. Astratov, Laser Photon. Rev. 11, 1600278 (2017).
[Crossref]

White, I. M.

X. Fan and I. M. White, Nat. Photonics 5, 591 (2011).
[Crossref]

I. M. White, H. Oveys, and X. Fan, Opt. Lett. 31, 1319 (2006).
[Crossref]

Laser Photon. Rev. (2)

T. Reynolds, N. Riesen, A. Meldrum, X. Fan, J. M. M. Hall, T. M. Monro, and A. Francois, Laser Photon. Rev. 11, 1600265 (2017).
[Crossref]

Y. Li, F. Abolmaali, K. W. Allen, N. I. Limberopoulos, A. Urbas, Y. Rakovich, A. V. Maslov, and V. N. Astratov, Laser Photon. Rev. 11, 1600278 (2017).
[Crossref]

Nanophotonics (1)

M. Sumetsky, Nanophotonics 2, 393 (2013).
[Crossref]

Nat. Photonics (1)

X. Fan and I. M. White, Nat. Photonics 5, 591 (2011).
[Crossref]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. Lett. (1)

M. Sumetsky, Phys. Rev. Lett. 111, 163901 (2013).
[Crossref]

Other (3)

The fact that the cutoff wavelengths of the WGMs with very large azimuthal quantum numbers m correspond to the zero propagation constant, β=0, does not contradict to the well-known relation kmpnext<β<kmpncap for small m (see, e.g., [9]).

A. W. Snyder and J. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

For the case kmpncapr0≈m≫1 and kmpnextr0<m considered, the Bessel function Jm(kmpnextρ) exponentially decays with the growth of ρ for 0<kmpnext(ρ−r0)≪m.

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Figures (3)

Fig. 1.
Fig. 1. Illustration of a capillary fiber coupled to an input–output microfiber. The capillary was processed with a CO 2 laser beam and, internally, with hydrofluoric acid. Inset: magnified cross section of the capillary wall (not to scale).
Fig. 2.
Fig. 2. Surface plots of spectra of the fabricated SNAP resonator measured with 20 μm resolution along the capillary axis. (a) Before etching, p = 0 . (b) After etching, empty capillary, p = 0 (bottom) and p = 1 (top). (c) After etching, capillary filled with water, p = 0 (bottom) and p = 1 (top). (d) Parabolic approximation of the cutoff wavelength for the empty (black curve) and water-filled (blue curve) capillary. Dashed red curve is the difference of these curves. (f) The restored internal ERV. The solid, dashed, and dotted curves correspond to λ w λ e equal to 0.05 nm, 0.04 nm, and 0.06 nm, respectively.
Fig. 3.
Fig. 3. (a) Cutoff wavelength as a function of internal ERV for the quantum numbers indicated on the plot. (b) Dependencies shown in (a) magnified and shifted along the vertical axis. (c) Cutoff wavelength as a function of internal ERV for the empty and water-filled capillaries for the quantum numbers indicated on the plot. (d) Dependencies shown in (c) magnified and shifted along the vertical axis. Curves shown in (b) and (d) are compared by horizontal translation into the darker rectangles.

Equations (7)

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n ( ρ , z ) = { n int , n cap , n ext , 0 < ρ r int ( z ) , r int ( z ) < ρ r ext ( z ) , r ext ( z ) < ρ .
d 2 Q m p d ρ 2 + 1 ρ d Q m p d ρ + ( ( 2 π n ( ρ , z ) λ ) 2 m 2 ρ 2 β 2 ( z ) ) Q m p = 0 ,
d 2 Ψ m p q d z 2 + β 2 ( z ) Ψ m p q = 0 .
β ( z ) = 2 π n cap λ m p ( cut ) 2 ( λ λ m p ( cut ) ( z ) ) λ m p ( cut )
d 2 Q m p d ρ 2 + 1 ρ d Q m p d ρ + ( ( 2 π n ( ρ , z ) λ m p ( cut ) ) 2 m 2 ρ 2 ) Q m p = 0 .
Q m p ( ρ , z ) = { A J m ( k m p n int ρ ) , B J m ( k m p n cap ρ ) + C Y m ( k m p n cap ρ ) , D J m ( k m p n ext ρ ) , 0 < ρ r int ( z ) , r int ( z ) < ρ r ext ( z ) , r ext ( z ) < ρ .
Δ λ ˜ ( cut ) ( r int + s ) = λ m p q ( cut ) ( r int + s ) | water λ m p q ( cut ) ( r int + s ) | empty ,

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