Open Access
Add to BibSonomy
Abstract
This paper demonstrates whispering gallery mode (WGM) resonance with the help of an encaved optical nano-probe developed inside an optical fiber tip cavity. The nano-probe generates a tightly focused beam with a spot-size of ∼3 µm. A barium titanate microsphere is placed besides the optical axis inside the cavity. The focused beam remains off-axis of the microresonator and excites the WGM. The off-axis excitation shows unique resonating properties depending on the location of the resonator. A resonant peak with quality factor as high as Q ∼7 × 104 is achieved experimentally. Another design with a shorter cavity length for a bigger resonator is also demonstrated by embedding a bigger microsphere on the cleaved fiber tip surface. The optical probe holds great potential for photonic devices and is ideal for studying morphology-based scattering problems.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical scattering by an object of size comparable to the incident wavelength of light has drawn great interest to a broad range of scientific disciplines. Since the work of Ashkin [1] demonstrating scattered light as a tool to detect surface waves or morphology dependent resonances from a dielectric spherical object, it has been studied extensively both in theory and experiment. The resonances are also known as a class of whispering gallery modes (WGMs) which are excited by the far field off-axis beam and have potential applications in the fields of lasing, sensing, nonlinear photonics, spectroscopy, [2–6]. The focused off-axis beam excites WGM resonances in microspheres more uniformly and efficiently as the distribution of the optical intensity is more on the rim of microsphere avoiding illumination on front and back surfaces [7]. It is a scattering phenomenon, analogous to quantum mechanical resonance [8–10], where photons are treated analogous to electrons and depends upon the potential well introduced by shape of the object. The photons or the electromagnetic energy is trapped in a potential well of dielectric resonator and is tunneled through its forbidden region by surpassing the centrifugal barrier and vice versa. This surface wave can be described by introducing effective potential (${V_{eff}}) $ trapping the electromagnetic energy at the surface of the resonator. The potential function is combination of a potential well and a centrifugal barrier as defined by [8,9,11]:
where, l is the quantized angular momentum, n(r) is the radial refractive index of resonator, k is the wave number.Several state-of-the-art works explain the said WGMs analytically from the view point of scaler wave optics and Generalized Lorentz-Mie theory (GLMT) [10,12–15]. Theoretically it is established that an off-axis focused Gaussian beam interacts with the scattered mode of the microsphere around its surface and excites WGMs efficiently compared to the on-axis excitation [12,13,16]. Analytical expressions are developed with the help of a partial wave expansion of the incident field in spherical coordinates. According to GLMT, the plane-wave partial-wave expansion coefficients ${a_l}$ (${b_{l\; }}$) for TM (TE) resonances are replaced by the partial-wave expansion coefficients ${a_{lm}}($ ${b_{lm }})$ for TM (TE) modes to find the contribution of TE mode as ${b_{lm}}\; = \; {b_l}B_l^m$ . This has been researched in details by Lock et al. [2] by introducing the coupling function $B_l^m$ for angular overlap between the excitation field and the spherical harmonics of the scattered field as
In this work, we have demonstrated a novel experimental configuration to excite WGMs in a fiber tip cavity with an encaved nano-probe inside the cavity. The cavity holds microresonator, maintaining a gap between the optical axis of the nano-probe and the axis of the resonator. This off-axis focused beam excites WGMs, resulting in sharp resonant peaks. The probe works in reflection mode.
2. Experiment
An encaved nano-probe is developed by chemical etching of a photosensitive optical fiber (GF3) in 48% Hydrofluoric acid (HF). The details of the etching technique used to develop such probe can be found in our earlier work [17]. The fiber is made to touch the meniscus of the HF solution kept in an etching container tube. It exploits the advantage of a higher etching rate of the Germanium-doped optical fiber core compared to the silica cladding, resulting in the formation of the nano-probe with a cavity along the core-cladding boundary. However, one can control the cavity length by controlling the etching time.
For this work, we have considered two etching periods of 17 minutes and 1.5 minutes to achieve a cavity length of ∼48 µm with opening ∼43 µm (Fig. 1(a)) and ∼12 µm with opening ∼15 µm (Fig. 1(b)) respectively, with nano-probe seating inside the core of the fiber. A probe with a longer cavity length can hold the smaller resonator inside the cavity, while the shorter one can hold the bigger resonator on the cleaved surface of the fiber tip.
Fig. 1. (a) Microscopic image of optical fiber tip cavity (etched for 17 minutes) with nano-probe as shown inside the marked circle. (b) SEM images of typical enclosed optical nanoprobe used to excite the micro-resonators (c) Experimentally obtained Gaussian-like beam profile from the nanoprobe (represented by the red dot on left) (d) Transverse beam profile showing the variation of beam waist with the distance.
The nanoprobe is located coaxially at the fiber’s center and generates a focused beam at the end of its tip. The experimentally obtained longitudinal beam profile from the nano-probe is shown in Fig. 1(c). The beam profile is obtained using a self-designed near-field beam profiler consisting of a fiber clamp mounted on a 3-axis nano-max translation stage for translation of the probe and the beam. A 40x microscopic objective is used to magnify the beam and a microscopic camera records the profile while the probe is translated stepwise. The images are captured at every step and stacked together to get the complete profile. The measured near-field optics from the nanoprobe show beam waist, ∼ 1 µm, and ∼3 µm, Fig. 1(d), depending upon the distance of ∼ 12 µm and ∼25 µm from the tip end, respectively. The said focused beam is used to excite WGMs by embedding a microresonator inside the cavity and off-axis to the beam.
Figure 2 is the schematic of the WGM coupling experiment of the proposed design. A broadband amplified spontaneous emission (ASE) source of wavelength range 1525 nm −1575 nm is coupled to port 1 of an optical circulator, which is further coupled to the fabricated axicon nano-probe through port 2. The reflected signal from the resonator is collected by Optical Spectrum Analyser (OSA) through port 3 of the circulator. Barium Titanate microsphere resonators of refractive index ∼2 is considered for the proof of the concept experiments.
Fig. 2. Schematic of WGM coupling experiment. (a) SEM image of the typical nanoprobe, (b) microscopic image of the embedded microsphere resonator of diameter D ∼15 µm inside the cavity.
3. Results and discussions
3.1 Excitation of WGM for microsphere kept inside the cavity etched for 17 minutes
As shown in Fig. 2 (Inset: (b)), a barium titanate microsphere with a diameter (D) of ∼15 µm (density ∼ 4250 kg/m3 and refractive index (n) ∼ 2.0) is inserted inside the conical cavity. Figure 1(a) is the microscopic image of the said fiber tip cavity with nanoprobe sitting inside. As seen in the figure, it maintains a gap between the axis of the nano-probe and the resonator which is to facilitate off-axis excitation of the resonator. The obtained reflected spectrum for the resonator located ∼25 µm from the nano-probe is shown in Fig. 3(a). The quality factor Q∼6.6 × 104 is achieved at wavelength 1537 nm for the WGMs. The spectrum consisting dips and peaks are due to the clockwise and counter clockwise WGMs inside the microsphere and it helps to determine the radial mode numbers [18]. The characteristic mode spacing of one mode family is defined as Free spectral range (FSR) which is theoretically computed to be FSR(T)∼25 nm. It closely matches with the experimentally observed FSR(E)∼29 nm, Fig. 3(a). Figure 3(b) represents the reflected spectrum for resonator of diameter ∼20 µm, which results in $Q$∼ 2.3 × 104. The quality factor of the resonator is low due to far-field detection of the reflected signal which may not be able to measure the splitting of the mode due to inevitable asphericity [19].
Fig. 3. (a) Reflected spectrum of resonator of diameter ∼15 µm placed inside the etched cavity off axis (in black) and on axis (in red) to the nanoprobe. (b) Reflected spectrum of resonator of diameter ∼20 µm placed inside the etched cavity off axis to the nanoprobe.
As per Van de Hulst's localization principle, the focal point for maximum excitation of structural resonances, should lie in the proximity of the edge of the resonator [20]. It is observed from the experiments that the resonator placed on the axis of nanoprobe i.e., $\; D/2x\; \to 0$ (Fig. 3(a), red curve) (x is the displacement of the resonator from its centre to the axis of nano-probe) does not show any significant peak. In contrast, as the resonator was moved off axis i.e., $\; D/2x\; \to 1.05$, notable peaks (Fig. 3(a), black curve) of WGMs are appeared. For the off-axis configuration, the beam coming from nanoprobe is scattered less compared to the on-axis configuration. The on-axis symmetry arrangement results in complete blockage of the transversely focused beam by the resonator resulting in maximum scattering [12]. An analytical expression for coupling efficiency to excite WGMs with off-axis focusing is derived approximately as [13]:
The exciting efficiency for the on-axis focus is theoretically obtained as ∼10−22 as calculated from the expression (4) given in Ref. [13].
Figure 4(a) is the simulation result of the Barium titanate resonator placed off axis to the nanoprobe inside the etched cavity. The simulations are carried in COMSOL Multiphysics. It is observed that the focused beam propagating from nanoprobe off-axially excites the WGMs in the microresonator of diameter ∼15 µm at wavelength 1540.2 nm. The simulation is swept over a wavelength region of 1525-1575 nm and a typical resonance peak is presented with fine tuning of the wavelength from 1540 nm to 1540.4 nm.
Fig. 4. (a) Simulation results showing the electric field distribution of resonator of diameter ∼15 µm placed inside the etched cavity off axis (see Visualization 1). (b) Electric field distribution v/s wavelength spectrum of the excited resonator
The corresponding video can be found in the multimedia link (Visualization 1). It may be mentioned that the excitation of WGMs is due to the travelling wave interaction as we have observed experimentally that placing the microresonator on the optical axis does not excite any WGM, Fig. 3(a), which has been simultaneously confirmed by simulation. Figure 4(b) shows the electric field distribution v/s wavelength spectrum of the same resonator. A Fano like resonant peak with Q∼1.7 × 105 is observed at wavelength ∼1540.2 nm similar to our experimental results.
To check the tunability of the typical resonant peak shown within the dotted box, Fig. 3(a), the resonator was translated stepwise towards the nanoprobe by a motorized optical fiber pointed tip with a step size of 500 nm from the point when the peaks start to appear. The cumulative translated distance from the resonator to the nanoprobe is represented by a variable g.
It is observed that the resonance peaks experience a redshift [21–23] and for the particular peak at λ∼1537 nm, a shift of magnitude 0.1 nm is observed when the resonator is translated by the distance ∼2.5 µm, Fig. 5. The translation of microresonator leads to change in coupling condition and perturbation in the scattered field around the resonator due to the presence of the nanoprobe causing the wavelength shift. It is to be mentioned that such perturbations are also encouraged for tunning similar resonating peaks as can be found in literatures both in theory [24] and experiments [21,22,25].
Fig. 5. Reflected spectrum showing tuning of resonating peak (shown within the dotted box, Fig. 3(a)) with the translation of resonator.
The Lorentzian and Fano characters of resonating dips are also observed to change when the resonator is translated. The azimuthal displacement in the position of the resonator with respect to the nanoprobe, as it is translated, accounts for this behavior [26]. We have observed quasi-Lorentzian behavior for $g$= 1 µm and 1.5 µm.
The corresponding spectral lines are fitted using the Fano equation:
The quality factor, Q(Qc,Q0), of the resonating peak is contributed by quality factor due to intrinsic loss, Q0 and coupling loss, Qc [28] resulting in a change in the value of Q as the microsphere position is changed. The nanoprobe introduces the dissipative effects as we change g [23]. We have calculated Q values as ∼6.6 × 104 and ∼ 4.69 × 104 corresponding to the $g$ = 1 µm ($q$ = −2.00) and $g$ = 2.5 µm ($q$=−22.69) respectively showing a change of Q value by ∼2 × 104.
3.2 WGM excitation outside the cavity etched for 1.5 minutes
To consider the bigger resonator, a Barium titanate resonator of diameter ∼ 65 µm is glued on the outside surface of the etched cavity, maintaining the probe to resonator distance ∼ 12 µm as shown in the inset of Fig. 6(a). Figure 1(b) is the SEM image of a typical probe similar to the probe used in this experiment. Figure 6(a) is the reflected spectrum of the said excitation configuration. It achieves Q∼3.0 × 104 at wavelength 1551 nm as shown in Fig. 6(a).
Fig. 6. (a) Reflected spectrum of resonator of diameter 65 µm placed outside the etched cavity. (b) Reflected spectrum of resonator of diameter 50 µm placed outside the etched cavity.
The number of resonant peaks is large for the said resonator compared to the resonators of diameter, D∼15 µm and 18 µm. This can be the fact that the spectral density varies with the diameter of the resonator [29]. The imperfections in the spheres also lead to the formation of multiple intermediate resonances [30].
Figure 6(b) represents the reflected spectrum for resonator of diameter ∼50 µm, which results in $Q$∼ 3.4 × 104. The resonance peaks are fitted by Eq. (5) and the q values obtained are −1 and 2.4, as shown in the inset of Fig. 6. It may be mentioned that the simulation for external resonator also shows WGM characteristics. Experimentally, FSR(E) ∼ 8 nm is obtained for this particular resonator which closely matches with the FSR(T) ∼ 7.6 nm. Figure 6(b) also shows how the FSR is obtained from the experimental results.
4. Conclusion
In conclusion, we have experimentally proposed an optical fiber probe-based coupling method to realize WGMs in micro resonators, placed off axis to the incident focussed gaussian beam. The probe is a chemically etched nanoprobe encaved inside a conical cavity showing a resonant peak in BaTiO3 microresonator of quality factor Q∼6.6 × 104. We have demonstrated WGM excitation for different size of resonators when the conical cavity was etched for 17 minutes and 1.5 minutes making the nanoprobe at 50 µm and 12 µm deep inside the cavity from the cleaved surface of the fiber respectively.
To demonstrate the tuning characteristics of the resonating peak, the resonator is translated 500 nm stepwise. The resonance peaks show redshift and quasi-Lorentzian characteristic on translation. The potential to collect the reflected resonant signal by the same fiber, makes the encaved nanoprobe an excellent fiber-based WGM microprobe. It can be an alternative arrangement to study fundamental scattering optics from micro resonators [9] through controlled experimentation and has potential applications in studying droplet photochemistry and in chemical and biological agent detection. Further studies should be done to enhance quality factor by optimization of the design and the resonator material.
Funding
Science and Engineering Research Board (CRG/2019/001215).
Acknowledgments
This work was also partly supported by the CNRS under the grant International Emerging Actions ORACLe. Authors would like to acknowledge Dr. Umesh Kumar Tiwari for his cooperation during the experiment.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. A. Ashkin and J. M. Dziedzic, “Observation of optical resonances of dielectric spheres by light scattering,” Appl. Opt. 20(10), 1803 (1981). [CrossRef]
2. C. Mathai, R. Jain, V. G. Achanta, S. P. Duttagupta, D. Ghindani, N. R. Joshi, R. Pinto, and S. S. Prabhu, “Sensing at terahertz frequency domain using a sapphire whispering gallery mode resonator,” Opt. Lett. 43(21), 5383 (2018). [CrossRef]
3. C. H. Dong, Y. Yang, Y. L. Shen, C. L. Zou, F. W. Sun, H. Ming, G. C. Guo, and Z. F. Han, “Observation of microlaser with Er-doped phosphate glass coated microsphere pumped by 780 nm,” Opt. Commun. 283(24), 5117–5120 (2010). [CrossRef]
4. H. Li, W. Song, Y. Zhao, Q. Cao, and A. Wen, “Optical trapping, sensing, and imaging by photonic nanojets,” Photonics 8(10), 434 (2021). [CrossRef]
5. A. N. Oraevsky, “Whispering-gallery waves,” Quantum Electron. 32(5), 377–400 (2002). [CrossRef]
6. S. C. Hill and R. E. Benner, “Morphology-dependent resonances associated with stimulated processes in microspheres,” J. Opt. Soc. Am. B 3(11), 1509 (1986). [CrossRef]
7. A. Serpengüzel, S. Arnold, G. Griffel, and J. A. Lock, “Enhanced coupling to microsphere resonances with optical fibers,” J. Opt. Soc. Am. B 14(4), 790 (1997). [CrossRef]
8. B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,” J. Opt. Soc. Am. A 10(2), 343 (1993). [CrossRef]
9. J. T. Marmolejo, A. Canales, D. Hanstorp, and R. Méndez-Fragoso, “Fano Combs in the Directional Mie Scattering of a Water Droplet,” Phys. Rev. Lett. 130(4), 043804 (2023). [CrossRef]
10. G. Gouesbet, J. A. Lock, and G. Gréhan, “Generalized Lorenz-Mie theories and description of electromagnetic arbitrary shaped beams: Localized approximations and localized beam models, a review,” J. Quant. Spectrosc. Radiat. Transf. 112(1), 1–27 (2011). [CrossRef]
11. L. G. Guimarães and H. M. Nussenzveig, “Theory of Mie resonances and ripple fluctuations,” Opt. Commun. 89(5-6), 363–369 (1992). [CrossRef]
12. X. Zambrana-Puyalto, D. D’Ambrosio, and G. Gagliardi, “Excitation Mechanisms of Whispering Gallery Modes with Direct Light Scattering,” Laser Photonics Rev. 15, 2000528 (2021). [CrossRef]
13. J. A. Lock, “Excitation efficiency of a morphology-dependent resonance by a focused Gaussian beam,” J. Opt. Soc. Am. A 15(12), 2986 (1998). [CrossRef]
14. A. Demir, E. Yüce, A. Serpengüzel, and J. A. Lock, “Geometrically enhanced morphology-dependent resonances of a dielectric sphere,” Appl. Opt. 50(36), 6652–6656 (2011). [CrossRef]
15. G. Gouesbet, “Generalized Lorenz-Mie Theory and Applications,” J. Quant. Spectrosc. Radiat. Transfer 110(11), 800–807 (2009). [CrossRef]
16. D. W. Vogt, A. H. Jones, and R. Leonhardt, “Free-space coupling to symmetric high-Q terahertz whispering-gallery mode resonators,” Opt. Lett. 44(9), 2220 (2019). [CrossRef]
17. S. K. Mondal, A. Mitra, N. Singh, S. N. Sarkar, and P. Kapur, “Optical fiber nanoprobe preparation for near- field optical microscopy by chemical etching under surface tension and capillary action,” Opt. Express 17(22), 19470–19475 (2009). [CrossRef]
18. Y. Zhou, F. Luan, B. Gu, and X. Yu, “Controlled excitation of higher radial order whispering gallery modes with metallic diffraction grating,” Opt. Express 23(4), 4991 (2015). [CrossRef]
19. N. Riesen, T. Reynolds, A. François, M. R. Henderson, and T. M. Monro, “Q-factor limits for far-field detection of whispering gallery modes in active microspheres,” Opt. Express 23(22), 28896 (2015). [CrossRef]
20. J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: Focal point positioning effects at resonance,” J. Appl. Phys. 65(8), 2900–2906 (1989). [CrossRef]
21. F. Azeem, L. S. Trainor, P. A. Devane, D. S. Norman, A. Rueda, N. J. Lambert, M. Kumari, M. R. Foreman, and H. G. L. Schwefel, “Dielectric perturbations: anomalous resonance frequency shifts in optical resonators,” Opt. Lett. 46(10), 2477 (2021). [CrossRef]
22. D. W. Vogt, A. H. Jones, H. G. L. Schwefel, and R. Leonhardt, “Anomalous blue-shift of terahertz whispering-gallery modes via dielectric and metallic tuning,” Opt. Lett. 44(6), 1319 (2019). [CrossRef]
23. J. M. Ward, F. Lei, S. Vincent, P. Gupta, S. K. Mondal, J. Fick, and S. Nic Chormaic, “Excitation of whispering gallery modes with a “point-and-play,” fiber-based, optical nano-antenna,” Opt. Lett. 44(13), 3386 (2019). [CrossRef]
24. F. Ruesink, H. M. Doeleman, R. Hendrikx, A. F. Koenderink, and E. Verhagen, “Perturbing Open Cavities: Anomalous Resonance Frequency Shifts in a Hybrid Cavity-Nanoantenna System,” Phys. Rev. Lett. 115(20), 203904 (2015). [CrossRef]
25. M. R. Foreman, F. Sedlmeir, H. G. L. Schwefel, and G. Leuchs, “Dielectric tuning and coupling of whispering gallery modes using an anisotropic prism,” J. Opt. Soc. Am. B 33(11), 2177 (2016). [CrossRef]
26. S. Wan, F.-J. Shu, R. Niu, G.-C. Guo, C.-L. Zou, and C.-H. Dong, “Perpendicular coupler for standing wave excitation and wavelength selection in high-Q silicon microresonators,” Opt. Express 28(11), 15835 (2020). [CrossRef]
27. M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, “Fano resonances in photonics,” Nat. Photonics 11(9), 543–554 (2017). [CrossRef]
28. F. Vanier, C. La Mela, A. Hayat, and Y.-A. Peter, “Intrinsic quality factor determination in whispering gallery mode microcavities using a single Stokes parameters measurement,” Opt. Express 19(23), 23544 (2011). [CrossRef]
29. M. S. Murib, E. Yüce, O. Gürlü, and A. Serpengüzel, “Polarization behavior of elastic scattering from a silicon microsphere coupled to an optical fiber,” Photonics Res. 2(2), 45 (2014). [CrossRef]
30. G. Adamovsky and M. V. Ötügen, “Morphology-dependent resonances and their applications to sensing in aerospace environments,” J. Aerosp. Comput. Inf. Commun. 5(10), 409–424 (2008). [CrossRef]
