Open Access
Add to BibSonomy
Abstract
Measuring the boundary shape of a deformed liquid microjet is of great importance for using it as an optical resonator for various applications. However, there have been technical challenges due to transparency and uncertainty in the refractive index of the liquid. In this study, we have developed a spectroscopic technique that enables simultaneous determination of the boundary shape and the refractive index of a liquid deformed microjet. A detailed procedure of the technique based on imposition of one-to-one correspondence between experimentally observed resonances and numerically calculated ones are presented along with the measurement results including the refractive index of ethanol between a wavelength of 550 nm and 670 nm.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
A liquid deformed microjet is known to be a versatile tool for studying quantum chaos, microcavity lasers and non-Hermitian physics owing to its high quality factor, high output directionality and shape tunability. It has been utilized in studying directional emission [1,2], energy-level dynamics [3], dynamical tunneling [4–6], exceptional points (EPs) [7], and so on. In such applications, the boundary shape as well as the refractive index of the jet should be determined accurately since the cavity resonances and the wave chaotic properties of a dielectric microcavity are sensitive to these parameters [8,9].
However, measuring the boundary shape of a deformed microjet remains challenging in two respects. First, due to the transparency and the smoothness of the surface, CCD imaging or triangulation are not applicable for a liquid volume [10]. A forward shadow diffraction pattern can be used to determine the boundary shape [11], but it requires high concentration of a dye material and thus cannot be performed during actual experiments using low dye concentrations. High dye concentration may also change the properties of the liquid, including its density, that are related to the surface shape of the jet [12,13]. Thus the technique utilizing the forward shadow diffraction is inappropriate when a real-time fine-tuning of the shape is critical such as in studying exceptional points [7,14,15].
Second, nonintrusive optical measurements for a transparent volume are usually related to the refractive index of the material. But in microjet cavities, there may be an uncertainty in the refractive index because of either temperature variation or discrepancies in the published data of refractive indices. Such uncertainty in the refractive index can make ambiguities in the boundary shape measurements.
Rainbow refractometry technique [16,17] with vectorial complex ray model [18–20] may overcome these difficulties, since it can measure the boundary and refractive index of various liquid volumes such as spheroid or oblate droplets and circular jets. However, its application for mapping the surface of a jet with asymmetrical-deformed cross sections has not been demonstrated yet.
In this paper, we report a novel technique that is nonintrusive, spectroscopy-based and allows simultaneous determination of the boundary shape as well as the wavelength dependence of the refractive index (chromatic dispersion) of a columnar or two-dimensional liquid component with high accuracies. These unknowns were determined by matching the cavity-resonance-modified fluorescence spectrum from the microjet with numerically calculated resonances at once for trial parameters. We present the measured boundary shapes of several microjets as well as the chromatic dispersion of ethanol. Because our technique measures the boundary shape and the chromatic dispersion simultaneously, it can be utilized for not only surface profiling, but also refractometry of liquid materials.
2. Backgrounds
The microjet we consider is a liquid stream that is vertically ejected from a non-circular orifice by gaseous pressure [21]. We use ethanol or an ethanol-water mixture as a liquid material, and dope it with dye molecules which provide a gain. The surface of the liquid jet oscillates due to the surface tension, and the oscillation amplitude decays due to the viscous decay as the jet advances. For a fixed ejection pressure, the boundary modulation of the jet column becomes stationary, and its cross-sectional area varies with the height due to the surface oscillation as well as the decrease of the stream speed [22], up to the height where the jet breaks up into droplets due to capillary instability [23]. We use the horizontal cross-sections of the jet where the magnitude of the surface oscillation is locally maximized as two-dimensional deformed microcavities. These microcavities, of which positions are denoted as the critical positions from now on, support whispering-gallery-like optical modes with extremely high-quality (Q) factors owing to total internal reflection.
When a critical position of the jet is excited by a pump laser, the fluorescence is emitted from the jet (Fig. 1). The fluorescence is modified by the cavity resonances: it is enhanced at the resonances of the cavity due to the cavity quantum electrodynamic effect [24]. We determine the boundary profile of the cavity by collective comparison between the experimentally observed resonances with high Q factors and the numerically obtained resonances. The intra-cavity intensity distributions of the high-Q resonances, which are usually whispering-gallery mode (WGM) type, are strongly localized near the boundary [22]. Consequently they exhibit good visibilities in the cavity-modified fluorescence (CMF) spectrum, and the resonant wavelengths are sensitive to the boundary conditions such as the boundary profile and refractive index of the fluid. A small change in such conditions causes small but noticeable inhomogeneous shifts of resonant wavelengths. If the assumed boundary profile and refractive index in the numerical calculation are approximately correct, the resulting resonant wavelengths would coincide with those seen in the experiment as a whole, within the experimental error. Otherwise, while some of the experimental and theoretical resonant wavelengths may coincide accidentally, it is practically impossible that all of them exhibit one-to-one correspondence in sequence.
Fig. 1. Experimental setup and a CMF spectrum. (a) A schematic of the experimental setup for CMF spectroscopy. The incident laser is polarized in the direction parallel to the jet column. (b) An example of CMF spectrum. The spectrum is measured at the fourth lowest critical position from the orifice with $P=1.408$ bar. The room temperature is ${19.5\,}^{\circ }$C. The jet medium is ethanol doped with Rhodamine $590$ dye at a concentration of ${0.05\,}\text {mM/L}$. The coloured short bars indicate the wavelengths of the high-Q resonances. Each resonance group is marked by an individual color.
The boundary profile of the jet at a critical position can be expressed in polar coordinates by a multipolar ($\cos m\theta, m=1, 2, 3, \ldots$) decomposition:
where $a$ is the mean radius, which is estimated to be about 13 ${\mu }$m - 15 ${\mu }$m from CCD images, $F(\theta )$ is the relative boundary modulation (RBM) and ${\eta }_{m}$ is the amplitude of the $m$th multipolar component of the modulation. Because of the shape of the orifices we use and the viscous decay on the jet, it is expected that the RBM is mainly quadrupole (associated with ${\eta }_{2}$) and octapole (associated with $\eta _4$), of which the former is much more dominant [22].The optical pumping is done with a cw 532-nm laser with its electric field polarized along the jet direction (i.e., transverse magnetic or TM). The resolving power of the spectrometer (SPEX 1404 double monochromator) is ${\lambda }/{\Delta }{\lambda }{\sim }{10}^{5}$. The high-Q resonance structure of the CMF spectrum is clearly visible in the range of 550 nm - 670 nm, and these high-Q resonances can be categorized into several groups (typically five groups) by their distinct spectral ranges (FSR’s). Let us denote these groups as G1, G2, G3, G4 and G5, which are numbered in ascending order in their FSR’s. Figure 1(b) shows an example. We have identified 146 resonances of cavity in all throughout the CMF spectrum. The presented figure shows the region with the best visibility, which contains 43 resonances.
3. Principles of the developed technique
3.1 Wave equation for resonances of a microjet cavity
Theoretically [25,26], a resonance seen in a CMF spectrum from a jet microcavity correspond to one of the solutions of Helmholtz equation in the polar coordinates
with both ${\psi }$ and $\partial {{\psi }}/{\partial }r$ continuous at the cavity boundary described by Eq. (1). In Eq. (2), $\psi$ is the vertical ($z$ directional) electric field, $n$ is the real refractive index and $k=k_{\text {r}}+ik_{\text {i}}$ ($k_{\text {r}}>0$ , $k_{\text {i}}<0$ ) is the complex wave number. Furthermore, $k_{\text {r}}=2{\pi }/{\lambda }$ and ${k_{\text {i}}}^{-1}=-2c{\tau }$, where ${\lambda }$ is the resonant vacuum wavelength and $\tau$ is the lifetime of the resonance. With defining a dimensionless parameter ${\xi }{\equiv }r/a$, the surface boundary is and Eq. (2) can be rewritten as where $X=ka$ the size parameter. Inside the cavity, $n=n_\text {cav}$ and the solution of Eq. (4) can be expressed asNote that $n_{\text {cav}}$ of the jet material is a function of the wavelength and the jet temperature $T$. Furthermore, near a reference wavelength, it can be empirically written as
where $C$ is the thermo-optic coefficient of the liquid and $R$ is a function of wavelength. The liquid we use in this study is ethanol doped with Rhodamine 590 at a concentration of $0.05\,\text {mM/L}$. Since the concentration is very low, the chromatic dispersion of the liquid would be nearly the same as that of ethanol [27]. The published value of $C$ of ethanol around ${20\,}^{\circ }$C for a visible vacuum wavelength is approximately $-4{\times }{{10}^{-4}\,}^{\circ }{\text {C}}^{-1}$ from the literature [28–31]. The jet temperature $T$ can be approximated to the room temperature, since the liquid reservoir is in thermal equilibrium with air in the laboratory, and a change in temperature of the jet after its ejection due to vaporization and laser pumping of a few $10$ mWs is very small (estimated to be only ${0.1\,}^{\circ }$C). The reported $R({\lambda })$ functions vary from literature to literature although they exhibit a similar overall trend. Figure 2 shows some of the reported wavelength dispersion of ethanol [32–35].Fig. 2. Refractive index of ethanol at ${20\,}^{\circ }$C. The black, red, blue, and magenta curves are drawn based on previous studies [32–35]. The green curve (“Fitted”) is obtained by using the measurement method described in this study.
In the experiments, we control the room temperature with an accuracy of ${0.1\,}^{\circ }$C in the range of ${16\,}^{\circ }\text {C} - {24\,}^{\circ }$C. Thus the uncertainty in the refractive index due to temperature is order of $10^{-3}\%$. The disagreement between the reported data can be much larger than this, as high as $0.1\%$ at some wavelengths. Therefore, let us temporarily regard the exact form of $R({\lambda })$ as another unknown parameter, and express it in the manner of Cauchy’s equation (up to the third term). Then Eq. (7) becomes
3.2 Numerical calculation of cavity resonances
To find the cavity resonances numerically, we employ the boundary element method (BEM) [25]. Only ${\eta }_{2}$ and ${\eta }_{4}$ are varied in the calculation, although the mean radius $a$ along with $P_0$, $P_2$, and $P_4$ are also unknowns to be determined (other ${\eta }_m$s would be negligible, as mentioned above). We use as small number of varying parameters as we can, to save computing power and time. In this section, it will be discussed what is obtained from the numerical calculation, and how the results can be connected with the experiments.
We calculate the size parameters $X=ka$ of resonances of a $2$-D cavity, whose boundary profile is given and $n$ inside it is a constant. The RBM $F(\theta )$ of the cavity is described by varying parameters ${\eta }{\equiv }{\eta }_{2}$ and $B{\equiv }{\eta }_{4}/{\eta }^{2}$, which are called RBM parameters from now on. All the other ${\eta }_m$’s are zero, and the mean radius is set to unity in the $(\xi,\theta )$ coordinates. For $n$ inside the cavity, $n_{0}=1.361$ (the averaged value of the data shown in Fig. 2 for $\lambda =600$ nm at ${20\,}^{\circ }$C) is used.
For a pair $(\eta,B)$, the resultant calculated resonances can be grouped by their well-defined FSR’s just like the real resonance wavelengths. Among these calculated results, we collect only the groups with $5$ highest Q factors on average in order to emulate the experiments (since typically only $5$ highest-Q groups of resonances are observed). The collection will be denoted as $\{X_{\text {cal}, n_{0}}^{({\eta },{B})}\}$.
The following two properties of the resonances are helpful to describe the relationship between the experimentally observed resonances and a collection $\{X_{\text {cal}, n_{0}}^{({\eta },{B})}\}$. First, when $\eta$ and $B$ of the cavity and $n$ inside it are fixed constants, the set of calculated size parameters is independent of the choice of $a$, i.e., $\{X_{\text {cal}, n}^{({\eta },{B})}\}$ is unique for the given condition. This property can be easily seen in Eqs. (3)–6). Second, for the two sets of calculated size parameters that are obtained with the same RBM parameters but slightly different constants ($n_1$ and $n_2$) for $n$ inside the cavity, each of their elements are mutually correlated as
These properties can be understood from the form of Eq. (5) as long as the effect of the absence of $n_{\text {cav}}$ in Eq. (6) is negligible. As $n_{\text {cav}}$ changes, the loss or the quality factor $Q$ of the cavity, which is determined by the existence of the outside region of the cavity, also changes and it can cause a frequency shift [36]. However, as long as $\Delta n_{\text {cav}}/n_{\text {cav}}\ll 1$, the frequency shift is negligible and thus Eq. (9) holds approximately.Utilizing Eq. (9), we can now deduce the following relation. If $\lambda$ is an observed resonant wavelength from a real microjet cavity at a temperature $T$ with ${\eta }={\eta }_{\text {r}}$ and $B={B}_{\text {r}}$, it can be associated with one element of $\{X_{{\rm cal}, n_{0}}^{(\eta _{\text {r}},B_{\text {r}})}\}$ that satisfies
3.3 Comparison between calculation and experiment
Now, let us consider a set of calculated size parameters of resonance obtained with an arbitrary $({\eta },B)$. If that $({\eta },B)$ is close to $({\eta }_{r},{B}_{r})$, then $\{{\lambda }\}$ and $\{X_{\text {cal},n_0}^{({\eta },B)}\}$ can be approximately connected by $g$. This is the key to find the unknown experimental conditions. In order to facilitate checking whether the correspondence $g$ is valid, we define
4. Procedure of finding the RBM
In the above procedure of determining the unknown experimental conditions, finding of the RBM parameters is the most essential part. In this section, three important steps of the RBM finding will be described.
4.1 Preparations
We have repeatedly performed the numerical calculations in the range of $125{\leq }{X}_{\text {cal},n_0}^{({\eta },B)}{\leq }155$ for various RBM parameters and made a database. The range 125 - 155 in size parameter roughly corresponds to wavelength range of 550 nm - 670 nm, which is the wavelength range in the experiment if $a$ is assumed to be 13.5 ${\mu }$m, the average of the estimated radii of our microjet cavities obtained from CCD images. The parameter $\eta$ is varied from 0.080 to 0.200 by an interval of 0.005, while $B$ is varied by 0.01 from 0.30 to 0.55 in the calculations.
We then construct $S$ functions with the above database and supposed $\tilde {g}$’s. There can be many different $\tilde {g}$’s for the same $\{{\lambda }\}$ and $\{X_{\text {cal},n_0}^{({\eta },B)}\}$ due to translation of wavelengths. Only $\tilde {g}$’s that connect $\{{\lambda }\}$ and $\{X_{\text {cal},n_0}^{({\eta },B)}\}$ in the same way as the actual $g$ would do are worthy to be considered. In other words, $\lambda$’s and $\tilde {g}(\lambda )$’s must be structurally similar to each other: the group (G) orders of $\lambda$’s and relative distances between them should be maintained in $\tilde {g}(\lambda )$’s. Figure 3 shows an example. It is noticeable that $5$ observed resonant wavelengths are close to each other, and they are belong to the resonance groups G$3$-G$5$-G$4$-G$1$-G$2$ (from left to right). Obviously, their associates should be gathered together and belong to the resonance groups in the same order. The connection ${\tilde {g}}_{2}$ fails to satisfy the latter, thus it cannot be an approximation of $g$. On the other hand, ${\tilde {g}}_{1}$ do not exhibit such a mismatch.
Fig. 3. Comparison between experiments and numerical simulation. ${\tilde {g}}_{1}$ and ${\tilde {g}}_{2}$ are two examples of $\tilde {g}$’s. The middle row shows the observed resonant wavelengths, taken at $P=1.584$ bar and $T={16.9\,}^{\circ }$C and at the $5$th lowest critical position, near $595\,\text {nm}$ (an arbitrarily chosen reference), in ascending order from left to right. The top and the bottom rows are two different segments of the calculated size parameters of resonance, obtained with $\eta =1.20$ and $B=0.40$, in descending order from left to right. The connection ${\tilde {g}}_{1}$ (${\tilde {g}}_{2}$) consists of the associations, each of which is represented by a line that connect an element of the middle row and one of the top (bottom) row.
4.2 Ruling out pseudo-matchings
One can find several $({\eta },B,\tilde {g})$’s which make the standard errors (STE’s) of fitting by Eq. (12) considerably small. Moreover, their $({\eta },B)$’s tend to cluster around some centers, for which STEs are the local minima, as shown in Fig. 4(a). Since the size parameters of resonance are slowly varying with $({\eta },B)$, a clustering of $({\eta },B)$’s around a center can be seen as its natural consequence. The real $(\eta _{\text {r}},B_{\text {r}})$, however, should be found around only one of such centers. The other centers are “inappropriate”, and give false boundary profiles as well as false chromatic dispersions. We call these inappropriate centers as pseudo-matchings from now on. A simple way to rule out the pseudo-matchings is to check the dispersion of $S({\lambda })$. Due to the false chromatic dispersion, $S({\lambda })$ for a pseudo-matching exhibits a much stiffer or gentler slope than that is anticipated from the published data shown in Fig. 2.
Fig. 4. An appropriate center and pseudo-matchings in the RBM-finding procedure. (a) The STE surfaces of fitting by Eq. (12) for the experimental observations taken at the same condition as Fig. 3. Each sub-surface (indicated by individual color) corresponds to $({\eta },B,\tilde {g})$s cluster around each of the centers with similar index of refraction curves. (b) $S({\lambda })$ for $\eta =0.120$ and $B=0.40$. Each of the high-Q resonance groups is indicated by individual color. The fitting result is shown as the violet curve. (c) $S({\lambda })$ for $\eta =0.130$, $B=0.45$. (d) $S({\lambda })$ for $\eta =0.110$, $B=0.35$.
For example, in Fig. 4(a), the only appropriate center for the given experimental condition is $({\eta },B)=(0.120,0.40)$ [Fig. 4(b)], while the other two centers - $(0.13,0.45)$ [Fig. 4(c)] and $(0.11,0.35)$ [Fig. 4(d)] are pseudo-matchings. This can be seen from their $S({\lambda })$s. Throughout the visible range of our experiments ($550$ - $670$ nm), $S({\lambda })$ for $(0.120,0.40)$ changes about $0.0035a$, whereas it changes about $0.0021a$ for $(0.11,0.35)$, and $0.0049a$ for $(0.13,0.45)$ respectively. When we use the estimated mean radius in Eq. (13), only $(0.120,0.40)$ agrees well with the previously reported data, since the average change of the index of refractions in Fig. 2 throughout the same wavelength range is $0.0036{\pm }0.0003$.
4.3 Determination of RBM parameters
As a final step, we apply a Gaussian fitting to the STE surface near the appropriate center. We determine the critical point of the Gaussian surface as the approximate RBM parameters. For example, from Fig. 4(a), the determined RBM parameters for the experimental condition that produces the observed resonances are $(\eta =0.1202{\pm }0.0001, B=0.404{\pm }0.002)$. The uncertainties noted here, that may be as small as ${10}^{-4}$, are fitting errors of the Gaussian fit. Because the step sizes in the numerical calculation for constructing the dataset are $0.005$ in $\eta$ and $0.01$ in $B$, the lower limit of the uncertainty of our measurement method would be ${\pm }0.0025$ in $\eta$ and ${\pm }0.005$ in $B$.
The accuracy of our method is limited by the intrinsic error of Eq. (10). The published refractive index data of ethanol indicate that $|n_{\text {cav}}-n_{0}|$ in our experiments may be as large as $0.004$. For $|n_{1}-n_{2}|=0.004$, the averaged magnitude of the deviation between the two sides of Eq. (9) is about $0.005\%$. Therefore, the error of Eq. (10) too, may reach $0.005\%$ in a calculated size parameter of resonance. Since a WGM-type resonance can be corresponded to a periodic ray orbit close to the cavity boundary, roughly the size parameter of such a resonance would be inversely proportional to the cavity perimeter, if $a$ and $n$ are fixed. It implies that a change in RBM that causes a variation of the cavity perimeter less than $0.01\%$ - corresponds to ${\pm }0.00005$ in $\eta$ and ${\pm }0.005$ in $B$ when $\eta {\sim }0.1$ - can not be detected by the present method, in principle.
5. Results
Figure 5 shows the measured surface profile data for $14$ experiments at the $5$th lowest critical positions of the jets under various temperature-pressure conditions. The quadrupolar (${\eta }_{2}=\eta$) and octapolar (${\eta }_{4}$) components in each measured RBM are shown in Fig. 5(a) and (b), respectively. The octapolar component is well fit by a quadratic fit $\eta _4=B\eta ^2$ with $B=0.41{\pm }0.01$, indicated by a dashed violet curve in Fig. 5(b). According to Niels Bohr’s theoretical analysis for elliptical jets, $B=5/12{\simeq }0.41\dot {6}$ at a critical position of the jet [37]. The present shape measurement results agree well with Bohr’s analysis, and it is also consistent with the previous study [22]. This agreement and consistency can be regarded as a supporting evidence for the validity of the present measurement technique.
Fig. 5. The measured surface profiles data. (a) The amplitude of the quadrupolar component of boundary modulation for various temperature-pressure conditions. (b) The amplitude of the octapolar component of boundary modulation. The dashed violet curve represents a quadratic fit ${\eta }_{4}=0.41{{\eta _{2}}}^{2}$. (c) The mean radius. The error bars of the Gaussian fittings are drawn in all three figures.
The mean radius $a$ for each of the above experiments is obtained by using Eq. (13). Here, $S_0$, $S_2$, and $S_4$ are constructed by linear interpolation of $S(\lambda )$’s near the measured RBM parameters for the experiment. Figure 5(c) shows the results. When the temperature is fixed, the mean radius decreases with $\eta$. This can be understood as a manifestation of the continuity of fluid, because a higher $\eta$ tends to be obtained at a higher ejection pressure (hence a faster speed of the jet stream) at the same critical position and temperature. The vertical error bars drawn in Fig. 5(c) come from the fitting errors in $S_0$, $S_2$, and $S_4$. The uncertainty of measurement of the mean radius is ${\pm }0.01$ ${\mu }$m, which is about $0.07{\text { }}\%$ of the radius, due to the step size in numerical calculation and the uncertainty in the reference index of refraction (i.e., $1.3610{\pm }0.0005$ for $\lambda =600$ nm at ${20\,}^{\circ }$C).
The chromatic dispersion of ethanol is determined as follows. We calculate the refractive index at a target temperature of ${20.0\,}^{\circ }$C as a function of $\lambda$ for each of the experiments by using Eq. (8) and Eq. (14) with the mean radius obtained above. Fitting the whole results into Eq. (8) gives ${P}_{0}=1.36009({\pm }0.00007)$, ${P}_{2}=3.45({\pm }0.05){\times }10^{-15}{\text { m}}^{2}$, and ${P}_{4}=-5.1({\pm }0.9){\times }10^{-29}{\text { m}}^{4}$. In Fig. 2, the green curve represents the chromatic dispersion of ethanol at ${20.0\,}^{\circ }$C which is constructed by using these coefficients. The uncertainty of the measured refractive index is ${\pm }2.2{\times }10^{-5}$. It is noticeable that our result agrees well with that reported by Rheims et al. [32], which was measured by an Abbe refractometer.
For the measurements we utilize high-Q WGM-like resonant modes, which are mainly distributed close to the perimeter of the microjet cavity. As a result, we can obtain relatively high accuracy in the shape and refractive index measurements. The modes are sensitive to the boundary conditions of the cavity such as the boundary shape and interior/exterior refractive indices, since the modes correspond to multiple circulations of ray confined along geometrically-well-defined trajectories in the cavity. On the other hand, for the same reason, the measured refractive index in this study is not sensitive to its local change or gradient due to inhomogeneity of temperature or pressure interior the jet. It remains as a future study to check the spatial variance of the refractive index by utilizing various modes mainly distributed in the inner region of the cavity.
6. Conclusion & summary
We have developed a nonintrusive technique that can determine the boundary shape and the chromatic dispersion of columnar liquid volumes such as deformed microjets. The technique is based on both spectroscopic observations of cavity-modified spectrum from a cross-section of the liquid volume and theoretical analysis of the WGM-type resonances supported by such a cross section as a microcavity. The products of the wavelength and the calculated size parameter for resonances can be well fitted to the Cauchy’s equation when the given boundary modulation in calculation is approximately right and the correspondence function between experiment and calculation is appropriately selected. The boundary shapes and the chromatic dispersions of ethanol microjets for various experimental conditions have been simultaneously measured by using this method. The results are in good agreement with the hydrodynamical prediction for the boundary shape and particularly with the previously published refractive index data measured with an Abbe refractometer. The technique can resolve the RBM parameters with accuracies of ${\pm }0.0025$ in $\eta$ and ${\pm }0.005$ in $B$, of which the former would be improved by more intensive numerical calculations. The uncertainties of the measured mean radii and refractive indices are ${\pm }0.01{\text { }}{\mu }$m and ${\pm }2.2{\times }10^{-5}$, respectively. The technique in this study would be applicable to other liquid materials, especially when a fine-tuning of their shape is important such as in liquid lenses [38] and in a fluidic-cavity-based platform for studying EPs.
Funding
Ministry of Science and ICT, South Korea (IITP-2021- 2018-0-01402); Korea Research Foundation (2020R1A2C3009299).
Disclosures
The authors declare no conflicts of interest.
Data availability
The previously reported chromatic dispersion used to draw Fig. 2 are available in Ref. [32–35]. Other 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. J. U. Nöckel, A. D. Stone, G. Chen, H. L. Grossman, and R. K. Chang, “Directional emission from asymmetric resonant cavities,” Opt. Lett. 21(19), 1609–1611 (1996). [CrossRef]
2. S.-B. Lee, J.-B. Shim, S. W. Kim, J. Yang, J.-H. Lee, H.-W. Lee, and K. An, “Universal output directionality of single modes in a deformed microcavity,” Phys. Rev. A 75(1), 011802 (2007). [CrossRef]
3. S.-B. Lee, J. Yang, S. Moon, J.-H. Lee, and K. An, “Chaos-assisted nonresonant optical pumping of quadrupole-deformed microlasers,” Appl. Phys. Lett. 90(4), 041106 (2007). [CrossRef]
4. S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, “Quasieigenstate evolution in open chaotic billiards,” Phys. Rev. A 80(1), 011802 (2009). [CrossRef]
5. J. Yang, S.-B. Lee, S. Moon, S.-Y. Lee, S. W. Kim, T. T. A. Dao, J.-H. Lee, and K. An, “Pump-induced dynamical tunneling in a deformed microcavity laser,” Phys. Rev. Lett. 104(24), 243601 (2010). [CrossRef]
6. H. Kwak, Y. Shin, S. Moon, S.-B. Lee, J. Yang, and K. An, “Nonlinear resonance-assisted tunneling induced by microcavity deformation,” Sci. Rep. 5(1), 9010 (2015). [CrossRef]
7. S.-B. Lee, J. Yang, S. Moon, S.-Y. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, “Observation of an exceptional point in a chaotic optical microcavity,” Phys. Rev. Lett. 103(13), 134101 (2009). [CrossRef]
8. J. D. Eversole, H.-B. Lin, A. L. Huston, A. J. Campillo, P. T. Leung, S. Y. Liu, and K. Young, “High-precision identification in optical processes in microdroplets,” J. Opt. Soc. Am. B 10(10), 1955–1968 (1993). [CrossRef]
9. H. G. L. Schwefel, N. B. Rex, H. E. Tureci, R. K. Chang, A. D. Stone, T. Ben-Messaoud, and J. Zyss, “Dramatic shape sensitivity of directional emission patterns from similarly deformed cylindrical polymer lasers,” J. Opt. Soc. Am. B 21(5), 923–934 (2004). [CrossRef]
10. D. Narita, M. Baba, and K. Ohtani, “Three-dimensional shape measurement of a transparent object using a rangefinding approach,” in Proceedings of the 20th IEEE Instrumentation Technology Conference, vol. 2 (IEEE, 2003), pp. 1022–1027.
11. S. Moon, J. Yang, S.-B. Lee, J.-B. Shim, S. W. Kim, J.-H. Lee, and K. An, “Nondestructive high-resolution soft-boundary profiling based on forward shadow diffraction,” Opt. Express 16(15), 11007–11020 (2008). [CrossRef]
12. Lord Rayleigh, F. R. S., “On the capillary phenomena of jets,” Proc. R. Soc. London 29(196-199), 71–97 (1879). [CrossRef]
13. Sir H. Lamb, Hydrodynamics (Dover Publications, 1945).
14. S.-Y. Lee, J.-W. Ryu, J.-B. Shim, S.-B. Lee, S. W. Kim, and K. An, “Divergent Petermann factor of interacting resonances in a stadium-shaped microcavity,” Phys. Rev. A 78(1), 015805 (2008). [CrossRef]
15. Y. Shin, H. Kwak, S. Moon, S.-B. Lee, J. Yang, and K. An, “Observation of an exceptional point in a two-dimensional ultrasonic cavity of concentric circular shells,” Sci. Rep. 6(1), 38826 (2016). [CrossRef]
16. J. P. A. J. van Beeck, D. Giannoulis, L. Zimmer, and M. L. Riethmuller, “Global rainbow thermometry for droplet-temperature measurement,” Opt. Lett. 24(23), 1696–1698 (1999). [CrossRef]
17. J. Hom and N. Chigier, “Rainbow refractometry: simultaneous measurement of temperature, refractive index, and size of droplets,” Appl. Opt. 41(10), 1899–1907 (2002). [CrossRef]
18. K. F. Ren, F. Onofri, C. Rozé, and T. Girasole, “Vectorial complex model and application to two-dimensional scattering of plane wave by a spheroidal particle,” Opt. Lett. 36(3), 370–372 (2011). [CrossRef]
19. F. R. A. Onofri, K. F. Ren, M. Sentis, Q. Gauber, and C. Pelcé, “Experimental validation of the vectorial complex ray model on the inter-caustics scattering of oblate droplets,” Opt. Express 23(12), 15768–15773 (2015). [CrossRef]
20. Q. Duan, X. Han, S. Idlahcen, and K. F. Ren, “Three-dimensional light scattering by a real liquid jet: VCRM simulation and experimental validation,” J. Quant. Spectrosc. Radiat. Transfer 239, 106677 (2019). [CrossRef]
21. J. Yang, S. Moon, S.-B. Lee, S. W. Kim, J.-B. Shim, H.-W. Lee, J.-H. Lee, and K. An, “Development of a deformation-tunable quadrupolar microcavity,” Rev. Sci. Instrum. 77(8), 083103 (2006). [CrossRef]
22. S. Moon, Y. Shin, H. Kwak, J. Yang, S.-B. Lee, S. Kim, and K. An, “Experimental observation of Bohr’s nonlinear fluidic surface oscillation,” Sci. Rep. 6(1), 19805 (2016). [CrossRef]
23. Lord Rayleigh, F. R. S., “On the instability of jets,” Proc. Lond. Math. Soc. s1-10(1), 4–13 (1878). [CrossRef]
24. P. R. Berman, Cavity Quantum Electrodynamics (Academic, 1994).
25. J. Wiersig, “Boundary element method for resonances in dielectric microcavities,” J. Opt. A: Pure Appl. Opt. 5(1), 53–60 (2003). [CrossRef]
26. P. W. Barber and S. C. Hill, Light scattering by particles : Computational methods (World Scientific, 1990).
27. B. Bilenberg, T. Rasmussen, S. Balslev, and A. Kristensen, “Real-time tunability of chip-based light source enabled by microfluidic mixing,” J. Appl. Phys. 99(2), 023102 (2006). [CrossRef]
28. T. E. Smith and R. F. Bonner, “Refractive index-temperature data for anhydrous ethyl alcohol,” Anal. Chem. 24(3), 517–518 (1952). [CrossRef]
29. D. Solimini, “Loss measurement of organic materials at 6328 Å,” J. Appl. Phys. 37(8), 3314–3315 (1966). [CrossRef]
30. R. C. Kamikawachi, I. Abe, A. S. Paterno, H. J. Kalinowski, M. Muller, J. L. Pinto, and J. L. Fabris, “Determination of thermo-optic coefficient in liquids with fiber Bragg grating refractometer,” Opt. Commun. 281(4), 621–625 (2008). [CrossRef]
31. M. J. Weber, ed. Handbook of optical materials (The CRC, 2002).
32. J. Rheims, J. Köser, and T. Wriedt, “Refractive-index measurements in the near-IR using an Abbe refractometer,” Meas. Sci. Technol. 8(6), 601–605 (1997). [CrossRef]
33. I. Z. Kozma, P. Krok, and E. Riedle, “Direct measurement of the group-velocity mismatch and derivation of the refractive-index dispersion for a variety of solvents in the ultraviolet,” J. Opt. Soc. Am. B 22(7), 1479–1485 (2005). [CrossRef]
34. S. Kedenburg, M. Vieweg, T. Gissibl, and H. Giessen, “Linear refractive index and absorption measurements of nonlinear optical liquids in the visible and near-infrared spectral region,” Opt. Mater. Express 2(11), 1588–1611 (2012). [CrossRef]
35. G. Lan, Y. Gao, and X. Zhang, “Measurement of chromatic dispersion of liquid in a wide spectral range based on liquid-prism surface plasmon resonance sensor,” Sens. Biosensing Res. 16, 32–36 (2017). [CrossRef]
36. H.-J. Moon, Y.-T. Chough, J. B. Kim, K. An, J. Yi, and J. Lee, “Cavity-Q-driven spectral shift in a cylindrical whispering-gallery-mode microcavity laser,” Appl. Phys. Lett. 76(25), 3679–3681 (2000). [CrossRef]
37. N. Bohr, “Determination of the surface-tension of water by the method of jet vibration,” Phil. Trans. R. Soc. Lond. A 209(441-458), 281–317 (1909). [CrossRef]
38. S. K. Y. Tang, C. A. Stan, and G. M. Whitesides, “Dynamically reconfigurable liquid-core liquid-cladding lens in a microfluidic channel,” Lab Chip 8(3), 395–401 (2008). [CrossRef]
