PHYSICAL REVIEW E 96, 022702 (2017)

Electrical and thermal tuning of quality factor and free spectral range of optical resonanceof nematic liquid crystal microdroplets
Junaid Ahmad Sofi, M. A. Mohiddon, N. Dutta, and Surajit Dhara*
School of Physics, University of Hyderabad, Hyderabad 500046, India(Received 22 March 2017; revised manuscript received 16 June 2017; published 4 August 2017)
We experimentally study the effect of temperature and electric field on the quality (Q) factor and free spectral
range (FSR) of whispering-gallery-mode optical resonance of dye-doped nematic liquid crystal microdroplets.
Both the Q factor and the FSR are highly sensitive to the temperature and electric field and are tunable. The
Q factor decreases, whereas the FSR increases substantially, with increasing temperature and electric field.
The variation of the Q factor and FSR is understood based on the change in the effective refractive index andthe dynamic size of the microdroplets.
DOI: 10.1103/PhysRevE.96.022702
II. EXPERIMENTAL

I. INTRODUCTION

Optical microcavities possess a small mode volume andhigh quality (Q) factor. These microresonators find wide
application in laser sources, active filters, all-optical switches[1–4], etc. The high Q-factor value associated with the resonantmodes make them highly sensitive biological and chemicalsensors [5–13]. Their small size and easy tunability by external
electric and magnetic fields make them future contenders as
crucial components in photonic devices [14–17]. In sphericalmicrocavities, the light circulates inside the sphere due to thetotal internal reflection provided the refractive index of thesurrounding medium is lower than that of the microcavity.
The resonance condition is achieved when the circulating
light meets in-phase and this is known as whispering-gallerymode (WGM) resonance. Solid dielectric-based sphericalmicroresonators have a high quality factor and have beenstudied to a great extent [1]. However, they are not easily
tunable by external fields such as electric and magnetic. In
this context, liquid crystal (LC) based microresonators havedrawn significant attention. Humar et al. first reported WGM
resonance in nematic LC microdroplets [14]. Subsequently
they showed three-dimensional lasing and surfactant sensing
ability of the microdropets [15,16]. Wang et al. reported on
the tunable WGM lasing of cholesteric LC microdroplets [17].
Kumar et al. showed that WGM resonance can be a diagnostic
tool for detecting subtle phase transition in liquid crystals[18]. Thermal applications based on the WGMs of cholestericLC microdroplets have been reported by Wang et al. [19].
Very recently we have shown the effect of a magnetic fieldon the structure and WGM lasing of ferromagnetic nematic
LC microdroplets [20]. In this paper, we report experimental
studies on the effect of the temperature and electric field on thequality factor and free spectral range (FSR). We show that thevariation of these two resonance properties can be understoodbased on the effective refractive index of the microdroplets
observed by the circulating light. Their high sensitivity andeasy tunability make them potential candidates for various
applications in photonic devices.

*

sdsp@uohyd.ernet.in

2470-0045/2017/96(2)/022702(5)

We used 4 -pentyl-4-biphenyl-carbonitrile (5CB) liquid
crystal obtained from Sigma-Aldrich. It exhibits the following
phase transitions: Cr 23.65◦ C N 33.5◦ C I. It shows positive
dielectric anisotropy and positive birefringence. The LC microdroplets were suspended in polydimethylsiloxane (PDMS)
using a micropipette. Prior to droplet formation a small
quantity (2 wt%) of fluorescent dye [4-dicyanomethylene2-methyl-6-(p-dimethylaminostyryl)-4H-pyran], commonly
known as DCM, was doped. The LC-polymer mixture isconfined between two indium-tin-oxide coated glass plates.
Microdroplets of the desired size were excited with a focusedlaser light of wavelength 532 nm (Nd:YAG second harmonic).
The absorption peak of the DCM dye coincides with thewavelength of the laser light. The resonance spectra comingfrom the microdroplets were studied with the help of ascanning near-field optical microscope (WiTec Alpha-200)
fitted with a spectrometer. The minimum spectral resolution of
the spectrometer is 0.09 nm. A schematic of the experimental
setup is shown in Fig. 1. To study the effect of an electric fielda sinusoidal voltage was applied across the cell with the helpof a function generator (Tektronix-AFG 3102) and a voltage
amplifier (TEGAM-2350). A temperature controller (Instec
Inc. mk1000) was used to vary the sample temperature.
III. RESULTS AND DISCUSSION

A polarized optical micrograph of PDMS film with 5CB
liquid crystal droplets is shown in Fig 2. A large number ofspherical droplets of various sizes are formed. Figure 3(a)shows an isolated microdroplet under a polarizing optical
microscope. Colorful rings appear due to the varying optical
retardation from the center to the periphery. Four extinction
branches originating from the center of each droplet areoriented along the polarizer axes. This suggests that thedirector is radially symmetric and the long molecular axes are
orthogonal to the liquid crystal–PDMS interface. Figure 3(b)shows a schematic of the director orientation in a microdroplet
with a radial hedgehog defect at the center. We focusedon an isolated microdroplet and excited it with laser light(532 nm) just inside the polymer–liquid crystal interface as
the resonant modes are formed inside. The irradiation point isindicated by the arrow in Fig. 3(c). The dye molecules emit

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PHYSICAL REVIEW E 96, 022702 (2017)

FIG. 1. Schematic of the scanning near-field optical microscope
setup.

fluorescence and a thin light ring around the circumferenceof the microdroplet is observed, indicating the excitationof WGM resonance. The restricted fluorescence within the
microdroplet suggests that the dye molecules are not diffused
across the interface. The light emitted from the microdroplet
is collected by the same objective and sent to the spectrometer.
Figure 4(a) shows the characteristic resonance spectra obtained
from a microdroplet at two temperatures. It is shown that thepeak intensity of the modes decreases and the full-width at
half-maximum increases with the temperature. The separation
between any two consecutive modes is also increased at higher
temperatures. Liquid crystals, being birefringent, possess
different refractive indexes (ordinary and extraordinary) for
different polarizations of light. The optic axis is radial, as
the droplets exhibit a radial director structure. Hence the TE
mode senses the ordinary (no ) and the TM mode senses theextraordinary (ne ) refractive index. For 5CB, ne > no and they

FIG. 2. Polarizing optical micrograph showing suspended microdroplets of 5CB liquid crystal in PDMS at room temperature.

FIG. 3. (a) Polarizing optical micrograph of a focused 5CB
microdroplet. (b) Schematic showing the molecular orientation inside
the microdroplet. (c) A DCM dye–doped micro-droplet illuminatedwith a focused laser beam (λ = 532 nm). The red arrow indicates theirradiation point.

show opposite temperature dependences, i.e., ne decreases,
whereas no increases, as the temperature is increased towards
the isotropic-nematic phase transition. On the other hand, therefractive index of PDMS remains constant (ns = 1.43). For asmaller droplet size (diameter, <15μm) it can be shown that
only TM modes with radial mode number q = 1 are supported[15,21].
To characterize the resonance spectra quantitatively, wemeasured the Q factor as a function of the temperature fromthe resonance line width of the mode with the highest intensity.
It is given by Q = λ/λ, where λ is the resonance line widthcorresponding to the resonant wavelength λ. Figure 4(b) shows
the variation of the Q factor as a function of the temperature.
The Q factor at room temperature is large (1150) and decreases
rapidly as the temperature is increased towards the nematicisotropic phase transition. For a spherical microresonator, theQ factor can be expressed as [14,22] 1/Qtot = 1/Qmat +1/Qrad + 1/QLC + 1/Qss + 1/Qcoupling , where Qmat refers to
the intrinsic material loss due to absorption and scattering in
the dielectric medium of the resonator. It is given by Qmat =2π nr /λαmat , where αmat is the absorption decay constant andnr is the refractive index of the resonator. The radiation loss
Qrad is due to the curvature of the microcavity. Qss refers
to the surface scattering loss due to roughness, QLC refers
to the loss due to the thermal fluctuations of the director,
and Qcoupling refers to the losses involved in coupling into
the external devices. The contribution of the intrinsic material
loss Qmat is directly proportional to the refractive index of theresonator and hence strongly depends on the temperature. The
loss (QLC ) due to the thermal fluctuations of the director isalso expected to increase slightly with increasing temperature.
Since the size of the droplet is very small and it supports only
TM modes corresponding to the radial mode number q = 1,
we can consider nr  ne . The temperature dependence of ne
is given by ne = Ae − Be T − Ce (1 − T /Tc )β , where Ae , Be ,Ce , and β are constants and Tc is the isotropic-nematic phasetransition temperature. The exponent β is about 0.2 for 5CB[23]. Assuming that the major contribution to the temperature

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FIG. 5. Schematic showing the director field and resonancemodes in a microdroplet at (a) zero, (b) small, and (c) large electricfields. Blue and yellow ellipsoids represent modes; red lines, thedirector. ITO, indium-tin-oxide.

factor remains unchanged up to a certain value and decreases
rapidly with increasing voltage. The decrease in the Q factor
can be understood based on the effective refractive index seenby the light due to the elastic distortion of the director. It is
known that the LC director deforms inside the droplets beyond
a particular voltage [14,25]. The effective refractive index of
the nematic LC is given by [24]n2e n2o,(2)neff = nr =n2o sin2 θ + n2e cos2 θ

FIG. 4. (a) Representative WGM spectra recorded at two temperatures, namely, at 27.5◦ C (black line) and 30.5◦ C (red line). (b)Variation of Q factor with temperature. The solid line is the best fit
to Eq. (1). Droplet diameter: 9 μm.

variation of Q comes from Qmat , we can write


T β
.
Q ≈ A − BT − C 1 −Tc

where θ is the angle between the propagation direction andthe director (optic axis). This relation is usually appropriate
for the uniform director field but this can be adapted to thepresent situation, especially near the equator where the modes
are located (blue ellipsoids in Fig. 5). At zero field θ = 90◦
and neff = nr  ne . In this case the director (near the equator(1)

The fitting of the Q factor with the temperature is shown in
Fig. 4(b). The fit parameters obtained from the best fit are
A = 19 776, B = 67, C = 3129, and β = 0.2. Interestingly,
this exponent (β) is comparable to the exponent of thetemperature dependence of the extraordinary refractive index
of 5CB [23]. It further confirms that the major contribution to
the temperature dependence of Q comes from the change in theextraordinary refractive index. We also note that a very smallchange in the extraordinary refractive index can give rise to a
substantial change in the Q factor (e.g., ne /Q ∼ 3 × 10−5at room temperature).
We further studied the effect of an ac electric field on
the Q factor of the optical resonance. The electric fieldcreates elastic distortion of the director in the microdroplets
beyond a particular value [14,24–26]. The typical distortionsof the director under normal boundary conditions at zero,
low, and high electric fields are shown schematically in
Fig. 5. The change in the internal structure is expected to
impact the resonance properties. We kept the temperature
fixed in the nematic phase and collected the spectra at various
voltages, and the Q factor was estimated from the line widthof the mode with the highest intensity. Figure 6(a) shows
typical spectra obtained at two voltages. Figure 6(b) shows
the variation of the Q factor with the applied voltage. The Q

FIG. 6. (a) WGM resonance spectra at two voltages: 0.35 V(black line) and 6.19 V (red line). (b) Variation of the Q factor
with the applied voltage. The solid line is a guide for the eye. Droplet
diameter: 8 μm.

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FIG. 7. (a) Variation of free spectral range (FSR) with applied
voltage. The straight line is the best fit (linear) with a slope of
0.17 nm/V. (b) Variation of FSR with temperature.

of the droplets where the modes are located) tilts toward the
direction of the electric field and neff (nr ) decreases. Since
Qmat is linearly proportional to nr , it is expected to decrease
with increasing field.
Careful observation shows [Fig. 4(a) and Fig. 6(a)] thatthe free spectral range, i.e., the separation of wavelengths
between two consecutive modes (δλSM ; denoted by the FSR)of the resonance also changes with increasing temperature andvoltage. To quantify this we measured δλSM as a function of thetemperature and applied voltage and the results are presented
in Fig. 7. It is shown that the FSR increases linearly with theapplied voltage, and at the highest applied voltage (i.e., 11 V)
the increment is about 2 nm [Fig. 7(a)]. Figure 7(b) shows thevariation of the FSR with temperature. It also increases almost
linearly with the temperature and changes slope at ≈31◦ C.
The increase in FSR with increasing temperature and voltage
can be expected due to the change in the effective refractive
index of the microresonator assuming that the size remains
unchanged. The FSR of a spherical microresonator is given by[1]
FSR = δλSM =λ2,2π nr R(3)

where R is the radius of the microresonator and λ is theresonant wavelength. When the temperature or the voltage
is increased the effective refractive index (neff ) decreases as
discussed previously. Since neff (=nr ) is inversely proportional
to the FSR, it is expected to increase in both cases.

FIG. 8. (a) Variation of the diameters of microdroplets withrespect to (a) time and (b) temperature. Lines in (a) are the linear
best fits. The slope of the smaller droplet 33.5 (μm) is 9 nm/min.
Dotted lines in (b) show the deviation from the linear variation.

A quantitative estimation of the FSR can be made from
Eq. (3). For example, at T = 26.5◦ C and T = 33.5◦ C, theeffective refractive indexes are given by neff = 1.702 andneff = 1.672, respectively [27]. The radius of the microdroplet
is 4.5 μm and the resonant wavelengths at these two temperatures corresponding to the highest peaks are 603.3 nm and602.3 nm, respectively [see Fig. 4(a)]. The calculated FSRs atthese two temperatures are 7.5 nm and 7.65 nm, respectively.
Thus the increase in FSR due to the increase in temperature(from 26.5◦ C to 33.5◦ C) is about 0.15 nm. But experimentally
we find that the increase in FSR is about 2.3 nm [Fig. 7(b)].
Similarly, we estimate the FSR at zero and at the highest
applied voltage. At zero voltage θ = 90◦ , and from Eq. (1)we get neff = ne = 1.702 [27]. Assuming that the director on
the equator is completely reoriented along the field direction[i.e., θ = 0◦ ; see Fig. 5(c)] at the highest voltage we getneff = no = 1.53 [27]. The radius of the droplet is 4.0 μm
and the resonant wavelengths are 604.3 nm and 598 nm,
respectively [see Fig. 6(a)]. The estimated corresponding FSRs
at the two voltages are 8.54 nm and 9.3 nm, respectively.
Hence the maximum increase in the FSR due to the electricfield is about 0.75 nm. But from the experiment we find that
the FSR is increased by about 2 nm [Fig. 7(a)]. Therefore theexperimentally observed enhancement of the FSR cannot be
understood in either case from the decrease in the effective
refractive index alone. This discrepancy motivated us to checkthe size of the droplets with time, as the FSR is inversely

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proportional to the droplet radius (R). For this purpose wemeasured the diameters of a few microdroplets as a functionof the time and temperature.
Figure 8(a) shows the variation of diameters of two different
microdroplets with time. It is observed that the diameters ofthe microdroplets decrease with time and the rate is higher
for larger microdroplets. For example, the initial diameterof a microdroplet with D = 33.5 μm decreases to 32.5 μm
over 140 min. Hence the radius (R = D/2) decreases at therate of nearly 4.5 nm/min (from the slope). At the samerate a microdroplet of radius 4 μm [Fig. 6(a)] is expected
to decrease to 3.42 μm during the experiment. Taking the
reduced size into account (i.e., R = 3.42 μm) and average
refractive index n = 1.587 [i.e., (ne + 2no )/3], the estimated
FSR from Eq. (3) is about 10.5 nm, which is similar tothat shown in Fig. 7(a). Therefore the size reduction during
the course of the experiment has a significant effect on the
FSR. Figure 8(b) shows the variation of the diameters of
two different microdroplets with the temperature. Initiallythe diameter decreases linearly with the temperature andbeyond 31◦ C it decreases much more rapidly. The effect of
the rapid size reduction beyond this temperature is reflected
clearly in the temperature dependence of the FSR [Fig. 7(b)].
In particular, initially the FSR increases linearly with thetemperature and at 31◦ C it changes slope and increases further.

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In conclusion, we have studied the effect of temperatureand electric field on the quality factor and free spectral rangeof nematic liquid crystal microdroplets embedded in PDMS.
The quality factor decreases significantly with increasing
temperature due to the decrease in the extraordinary refractive
index. With increasing electric field the quality factor also
decreases beyond a particular voltage due to the decrease in
the effective refractive index. The free spectral range increases
with increasing temperature and voltage due to the reduction in
the effective refractive index. Thus both the quality factor andthe free spectral range can be tuned electrically and thermally.
We find that the size reduction with time has a significant effecton the free spectral range. The experimental change in thefree spectral range is in good agreement with the quantitative
estimation based on the time-dependent size reduction of the
microdroplets.
ACKNOWLEDGMENTS

We gratefully acknowledge the support from SERB (Grant
No. EMR/2015/001566) and DST (Grant No. DST/SJF/PSA02/2014-2015). J.A.S. acknowledges UGC for the BSR
fellowship. We acknowledge the support of the Center for
Nanotechnology, University of Hyderabad.

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