PHYS-159 Department of Nuclear and Atomic Physics September, 2014 Physics

# Coherent random lasing in a
disordered array of amplifying
microresonators

Anjani Kumar Tiwari

Prof. Sushil Mujumdar

DECLARATION

This thesis is a presentation of my original research work and has not been
submitted earlier as a whole or in part for a degree/diploma at this or any
other Institution/University. Wherever contributions of others are involved,
every effort is made to indicate this clearly, with due reference to the
literature, and acknowledgement of collaborative research and discussions.

The work was done under the guidance of Prof. Sushil Mujumdar at the Tata
Institute of Fundamental Research, Mumbai.

Anjani Kumar Tiwari

[Candidate’s name and signature]

In my capacity as supervisor of the candidate’s thesis, I certify that the
above statements are true to the best of my knowledge.

Prof. Sushil Mujumdar

[Supervisor’s name and signature]

Date: 21/04/2014

###### Acknowledgements.

First and foremost, I am deeply grateful to my adviser, Prof. Sushil Mujumdar
for his scientific guidance, valuable advice, support and encouragement in the
journey towards my Ph.D. Working at the Nano-Optics and Mesoscopic Optics
Laboratory (NOMOL) has been a great learning experience for me. I have been
extremely fortunate to have some very helpful labmates, and it was a great
pleasure working together in a group. I sincerely thank Balu, Bikas, Girish,
Randhir, Ravitej, Swapnesh, Shadak and Sreeman. I particularly acknowledge
Ravitej for his valuable help in the Monte Carlo simulations (Chapter 2) and
FDTD calculations (Chapter 8). During this work, I have benefited from the
discussion on various topics with Dr. Rajesh Nair and Dr. Shivakiran. I also
thank Prof. Amol Dighe and Prof. R. Palit for having a keen interest in my
research progress throughout my Ph.D. I express my gratitude to the DNAP
office, workshop, library, photography section and university cell for their
help. I thank my friends and colleagues Amaresh, Bhupender, Deepak, Pankaj,
and Prashant for making my stay at TIFR a delightful and memorable experience.
Last but not least, I would like to thank my parents for their constant
support over the years.

###### Contents

  1. 1 Introduction
    1. 1.1 A short history of the random laser
    2. 1.2 Conventional lasers and resonant feedback
    3. 1.3 Laser-like emission from disordered media
    4. 1.4 The coherent random laser
    5. 1.5 Attempts of frequency control in random lasers
    6. 1.6 Light localization
    7. 1.7 Thesis
  2. 2 Frequency fluctuations in coherent random laser, inter-scatterer vs . . .
    1. 2.1 Photon transport Monte Carlo simulations
    2. 2.2 Sparse system: Resonant amplifying scatterers
    3. 2.3 Dense scattering system
    4. 2.4 A practical proposal
  3. 3 Practical implementation of an aerosol random laser
    1. 3.1 Experimental setup and characterization of the amplifying aerosol
    2. 3.2 Coherent random lasing from the polydisperse droplet array
      1. 3.2.1 Longitudinal and transverse emission spectra
      2. 3.2.2 Spectral image, threshold and frequency fluctuations
    3. 3.3 Emission properties of a dye-scatterer random laser in a microchannel . . . .
  4. 4 A monodisperse aerosol random laser
    1. 4.1 Generation of monodisperse amplifying aerosol
    2. 4.2 Whispering gallery modes for characterization
    3. 4.3  Collective lasing from the monodisperse microdroplet array
      1. 4.3.1 Discretization of frequency distribution
    4. 4.4 Characteristics of the collective modes
      1. 4.4.1 Threshold behavior
      2. 4.4.2 Angular distribution of the collective emission
      3. 4.4.3 Polarization anisotropy
  5. 5 Lasing from a quasi-one-dimensional, amplifying period-on-average . . .
    1. 5.1  The transfer matrix method
      1. 5.1.1  Scattering and phase matrices
      2. 5.1.2  A single dielectric layer
      3. 5.1.3  Multiple dielectric layers
    2. 5.2 Origin of collective lasing modes
      1. 5.2.1 Passive array
      2. 5.2.2 Active array
    3. 5.3 Frequency distribution of the modes
    4. 5.4 Dependence of lasing threshold on system size and disorder
    5. 5.5 Verification of the quasi-1D approximation
  6. 6 Single-mode, stable coherent random lasing
    1. 6.1 aPARS system under weak and strong disorder
    2. 6.2 The concept of spectral mode-matching
    3. 6.3 Experimental observations: mode-matched and mode-mismatched coherent .
  7. 7 Anderson localization-induced random lasing
    1. 7.1 Ensemble-averaged transmittance and localization length
    2. 7.2 Field distribution inside the PARS system
    3. 7.3 Experimental observation, intensity profile of the lasing modes
    4. 7.4  Mode extent and IPR distribution
      1. 7.4.1 IPR distribution of the gap states
      2. 7.4.2 Experimentally measured IPR distribution of the random lasing modes
  8. 8 Collective modes in direct-coupled, bent chains of active spherical . . . . .
    1. 8.1 Experimental setup
    2. 8.2 Collective modes in bent arrays
    3. 8.3 2D FDTD simulations
  9. 9 Epilogue

### Introduction and motivation

A conventional laser consists of a gain medium and a carefully designed
resonant cavity [1]. The most common configuration of a laser cavity is a
Fabry-Perot (FP) resonator, made of two parallel mirrors. The consequent
lasing intensity and frequency depend on the inversion and the physical
parameters of the cavity. If the gain overcomes all the cavity losses, the
system crosses a threshold and lases. Any inhomogeneities causing scattering
within the cavity introduce additional losses in the lasing process and hence
are considered detrimental for lasing action. In 1968, Letokhov [2] proposed
the amplification of light in a random scattering environment instead of a
well-defined set of mirrors. This proposal was motivated by the extension of
the lifetime of a light wave in a multiply-scattering medium, similar to that
created by a resonator cavity but without the wavelength-selectivity. Earlier
experiments on this proposal were performed on laser powders, and the
existence of stimulated emission was confirmed. A breakthrough result was
reported by Lawandy et al. [3], who observed a significant collapse of
emission linewidth of a dye-solvent medium with the addition of titanium
dioxide nanoparticles. This system has been termed as a ‘random laser’ since
it exhibits laser-like characteristics of intensity enhancement and linewidth
collapse after a pump threshold.

Random lasers are exciting optical sources where amplification is coupled with
multiple scattering [4]. Multiple scattering increases the interaction time of
light with an inverted medium, and this results in enhanced stimulated
emission. The structural disorder needed for scattering is realized by
creating a random variation of refractive index in the gain medium. Since
there are no prescribed restrictions on the nature of scatterers and the gain
medium, a variety of different materials have been reported to show random
lasing [5, 6, 7, 8, 9, 10]. For example, Neodymium-doped powders, dye-doped
microparticles, colloidal suspensions of titanium dioxide, zinc oxide,
polystyrene in various gain media, etc. are used for this purpose.

Based on the feedback mechanism, random lasers are classified into two
categories: (1) Diffusive random laser and (2) Coherent random laser. The
former describes a system that emits a stable spectrum with a fixed number of
modes (usually one or two) and a bandwidth that is narrowed by about an order
of magnitude (typically reaching $4-5$ nm) compared to the homogeneous gain
medium. In a diffusive random laser, the propagation of light is modeled as
the diffusion of light with gain, and the phase information and the
interference effects are neglected. This feedback mechanism involves the
transport of intensity and is known as nonresonant or incoherent feedback.
This kind of feedback is relevant in the disorder regime characterized by
$L>\ell>\lambda$, where $L$ is the system size, $\ell$ is the mean free path
of light of wavelength $\lambda$. In diffusive random lasers, the spectral
peak position is predicted by the gain maximum. On the other hand, coherent
random lasing differs significantly in emission characteristics. Under certain
conditions of gain and pumping, the spectra from the samples exhibit several
ultranarrow peaks (bandwidth $\sim 0.1$ nm) riding on an incoherent pedestal.
These ultranarrow modes are temporally coherent, and strongly fluctuate in
both intensity and frequency. Early studies of this phenomenon attributed
these modes to Anderson-localized resonant modes of the disordered medium.
Such a feedback, where the light wave follows a closed loop path over a
randomly formed resonator, is termed resonant feedback. Clearly, this feedback
is based on the electric field transport and requires the phase information of
the scattered light wave. The phase shift requirement of $2\pi$ along the loop
determines the lasing frequency [6, 11]. In a disordered system, the
interference effects become prominent when $\ell\sim\lambda$. In weaker
samples, though, the ultra-narrow modes can be understood without the
requirement of light interference under conditions of high gain and small
sample size [8].

Theoretically, random lasing has been studied using various techniques. In the
intensity transport regime, the diffusion equation was extensively used with
concomitant gain [12, 13, 14]. Since diffusion can be numerically simulated as
random walks of particles, this technique was utilized by assuming a light
pulse to comprise a packet of ‘photons’ to understand the spectro-temporal
features of random lasing [15, 8]. Resonant (field) feedback has been treated
by solving Maxwell equations in a disordered environment using Finite-
Difference Time-Domain (FDTD) calculations in one and two dimensional systems
[16, 17]. The Transfer Matrix Method (TMM) has also been vastly used to study
one-dimensional random lasers, with a particular focus on quasi-periodic
systems with amplification [18, 19].

The important characteristics due to which random lasers particularly stand
apart are their strong fluctuations in various physical parameters such as
output intensity, lasing frequency, spatial mode extent etc. [20, 21, 22]. Out
of these, frequency fluctuations are the major impediment in practical
applications of random lasers. This thesis investigates one roadmap towards
minimization of frequency fluctuations. There exist a few mechanisms by which
some control on random lasing frequency has been achieved. Of these, two
prominently successful techniques involve (a) shaping the optical pump profile
to select a particular mode [16, 23] and (b) usage of monodisperse amplifying
Mie scatterers [9]. The former technique requires apriori knowledge of the
spatial profile of a particular mode at the desired wavelength. Thus this
technique, successfully demonstrated in coherent random lasers, achieves
wavelength selectivity more than wavelength control. The latter technique, on
the other hand, actually controls the location of modes in frequency.
Hitherto, however, this technique has been demonstrated only in diffusive
random lasers. In this thesis, we address the frequency behavior of coherent
random lasing in a medium with monodisperse, resonant scatterers.

This synopsis is organized in the following manner. Section A describes Monte
Carlo simulations wherein we demonstrate efficient and frequency-controlled
coherent random lasing in an ensemble of monodisperse, resonant scatterers
with intra-scatterer gain, and propose a practical implementation of such a
system. Next, experimental results are presented for polydisperse (Section B)
and monodisperse (Section C) amplifying scatterers arranged in a disordered
fashion in a linear array and the consequent frequency behavior. Motivated by
the current literature on chains of resonators, the practical applicability of
this system as a coupled-resonator optical waveguide (CROW) is examined
(Section D). After that, we model our system using transfer matrices (Section
E) and trace the origin of the modes to the gap states of an amplifying
periodic-on-average random superlattice (aPARS). Next, we exploit spectral
properties of such systems to achieve single-mode stabilized coherent random
lasing (Section F). Finally, the statistics of the spatial extent of modes is
further studied (Section G) where direct evidence of localization is obtained.

### Monte Carlo simulations of monodisperse random medium

We employ Monte Carlo simulations to study the spectral features of a coherent
random laser comprising monodisperse spherical scatterers. The algorithm
essentially tracks the propagation of photons in a multiply scattering medium,
coupled with inversion dynamics. The simulation has two parts: excitation and
emission. During the excitation process, pump photons undergo three-
dimensional random walks and are absorbed by ground state molecules which
cause a local population inversion. In the next stage, spontaneously emitted
fluorescent photons also perform random walks and interact with the excited
medium and experience gain proportional to their dwell-time in the inverted
region. Our system consists of randomly distributed monodisperse spherical
scatterers, whose size ranges over $\sim 1-5~{}\mu$m. With these sizes, the
particles are able to sustain resonances with appropriate quality factors. In
this system, it is instructive to note the comparative behavior of inter-
scatterer and intra-scatterer dwell-time ($\tau_{d}$) distribution, since the
gain is proportional to the $\tau_{d}$, thereby influencing the spectral
features. Figure 1[A] shows two representative spectra from a conventional
random laser, made of randomly shaped scatterers. The spectra show peaks at
random frequencies which differ from pulse-to-pulse. Figure 1[B] shows four
spectra for a particle size of $1.09~{}\mu$m (refractive index contrast,
$n=1.8$) as a function of excitation pump energy for resonant scatterers with
intra-scatterer gain. The spectra confirm persistent lasing with stable (well-
defined frequency range), efficient (larger intensity in the coherent mode)
lasing modes, the lasing frequency being determined by the Mie resonances of
the scatterers.

Figure 1: [A] Two numerically calculated spectra showing coherent modes from a
conventional random laser [B] Emission spectra from a random laser with
resonant scatterers with intra-scatterer gain. Inset: Mode exaggerated for
clarity. [C] Emission spectra from amplifying Mie scatterers. Red curves:
Monodisperse scatterer system. Black curve: With $12~{}\%$ polydispersity.

We, then, propose an experimentally realistic system capable of generating
coherent random lasing, comprising monodisperse Mie scatterers with intra-
scatterer gain. We conceive a high gain system that is made of a laser dye
(gain cross-section $\sigma_{g}~{}\sim~{}10^{-16}$ cm2). We assume that
spherical microdroplets are created out of this dye, which have a refractive
index contrast of $1.33$ (corresponding to the solvent for the dye) with the
surrounding medium (air). Thus each microdroplet is an amplifying Mie
scatterer. We study the spectral features of an ensemble of such microdroplets
in the air or an aerosol. A polydisperse system is first simulated by assuming
$12~{}\%$ spread in scatterer diameter (which was assumed to be $3.6~{}\mu$m).
The black line in Figure 1[C] depicts the spectrum from this system at a pump
energy of $30~{}\mu$J. Lasing peaks appear at random positions. Next, we
simulate spectra from the monodisperse system pumped at $7~{}\mu$J as shown in
Figure 1[C]. Multimode emission is observed at specific wavelength intervals.
The spectra change from shot to shot indicating that fluctuations continue to
exist, but the lasing peaks are restricted to the resonance band of the
individual scatterer, which shows that the fluctuations are reduced. The
measurements of the lasing peak distribution quantify the control achieved on
the lasing wavelength.

### Practical implementation of an aerosol random laser

Motivated by the calculation, we created microdroplets using a device called
the Vibrating Orifice Aerosol Generator (VOAG). The technique is based on the
fracture of mechanical perturbation of an unstable liquid jet [24]. The
schematic of the experimental setup is shown in Figure 2[A]. In our case, the
liquid is a solution of Rhodamine 6G dissolved in methanol. Using pressurized
N2 gas, this solution is forced through a micro-capillary having an in-built
piezoelectric gate, the regulation of which creates polydispersity or
monodispersity in the sizes and separations of the microdroplets. The
microdroplet stream is irradiated by the second harmonic ($\lambda=532$ nm) of
an Nd: YAG laser to optically pump the thus-formed scatterers. Importantly,
this system assumes the form of a linear array, and the longitudinal emission
and the transverse emission have differing character. In the setup, the mirror
M2 was placed at an angle of about $45^{\circ}$ from the vertical axis to
redirect the longitudinal emission towards the spectrometer. The lens L1 was
placed on a translation stage to collect and focus both the transverse and
longitudinal emissions onto the entrance slit of the spectrometer, as depicted
in the inset of Figure 2[A]. A CCD camera (CCD2) was employed to image the
microdroplet array.

Figure 2: Coherent random lasing from polydisperse array of amplifying
scatterers. [A] Experimental setup for the observation of coherent emission.
Legend: V: Vibrating Orifice Aerosol Generator; M1, M2: Mirrors; L1, L2:
Lenses; BD: Beam Dump, S: Spectrometer [B] Representative emission spectra in
longitudinal (top panels) and transverse (bottom panels) direction. [C]
Continuous broadband distribution of lasing wavelength. [D] Observed lasing
threshold behavior: Solid circles are the output coherent intensity, empty
circles are the probability of random lasing.

Figure 2[B] shows two representative longitudinal spectra in the top panels
and two transverse emission spectra in the bottom panels. Ultra narrowband
coherent random modes are observed in the longitudinal direction. Every peak
in the longitudinal direction has a corresponding weak mode in the transverse
direction, as marked by the arrows. These peaks, when analyzed in the spectral
images (not shown here), confirm the extent of the coherent modes over the
entire array. The rest of the low-intensity peaks in the bottom panels can be
traced to individual resonances. We irradiated a continuous microcolumn of the
same laser dye with a comparable length and diameter and did not observe
generation of coherent modes. This implies that discretization of refractive
index plays a vital role in generating optical feedback. The lasing peaks
fluctuate in both frequency and intensity. $500$ spectra were taken to analyze
the frequency distribution of these lasing modes. As shown in Figure 2[C], the
coherent peaks occur at random wavelengths over a range of $\sim 10$ nm. This
band is approximately determined by the width of the gain profile and the
excitation energy. The variation of coherent intensity in the longitudinal
direction as a function of pump energy is shown in Figure 2[D]. Dash line is a
linear fit to the coherent intensity, which is used to find the threshold
energy. A threshold behavior is observed at $E_{p}=0.5~{}\mu$J, endorsing the
lasing phenomenon.

### A monodisperse aerosol random laser

After that, we study the behavior of the above system in a resonant scattering
environment. The monodisperse scatterers are generated by applying a periodic
perturbation on the piezocrystal inside the capillary, which fractures the
liquid jet into equal-sized microdroplets. The monodispersity is confirmed
from the Whispering Gallery Modes (WGM’s) of the individual microdroplets
[25]. One of the embodiments of the monodisperse droplet array is shown in
Figure 3[A]. In this case, the average size of the droplet is estimated to be
$17.1~{}\mu$m. Across the array, we estimated a size variation of $\sim 100$
nm, which confirmed a strong monodispersity ($\sim 0.6~{}\%$). The separations
were allowed to remain random as seen in the image.

Figure 3: Coherent random lasing from monodisperse scatterers. [A] One
manifestation of the microdroplet random array and three representative
longitudinal emission spectra at a pump energy $E_{p}=0.7~{}\mu$J. [B]
Histogram of the lasing wavelengths shows discrete bunches. [C] Output
intensity as a function of excitation pump energy, the collective mode shows a
lasing threshold at $E_{p}=0.12~{}\mu$J.

Three representative longitudinal spectra captured from the monodisperse array
are shown in Figure 3[A]. The peaks are strongly coherent as the incoherent
pedestal is absent. The distribution of random lasing peaks is shown in Figure
3[B], which shows that these peaks appear only in discrete, specified
intervals of wavelengths. Such bunching was observed whenever the scatterers
were monodisperse. The width of each bunch is $\sim 2$ nm compared to $10$ nm
of the polydisperse array, or of a conventional random laser. Thus, the
monodisperse aerosol shows a reduction in frequency fluctuations. Notably, the
bunch separation matches with the Fabry-Perot (FP) free spectral range of an
individual scatterer and not the whispering gallery modes. Figure 3[C] shows
the threshold behavior in these monodisperse arrays, the threshold is almost
an order of magnitude smaller than the polydisperse configuration.

### Collective amplified modes in a bent array of spherical microresonators

At this stage, we refer to the current literature on linear chains of
microsphere resonators. In that regard, a linear array of microspherical
resonators has been a system of interest in another field of integrated
photonic circuits. Such arrays have been studied as candidates for
waveguiding, as the so-called coupled-resonator optical waveguides (CROW’s)
[26]. The motivation here is to transmit an optical signal via coupling of
successive resonators. Accordingly, linear chains of microspheres has been
studied, wherein the coupling is mostly via evanescent modes. Radiative
coupling occurs through nanojet-induced modes [27]. Such modes incur bending
losses and hence are not considered candidates for bent CROW’s [28]. In this
project, we considered our system as a potential candidate for active CROW’s.
The participating resonances are the Fabry-Perot modes and not the surface
(whispering gallery) modes, and hence the coupling is direct far-field. We
examine the ability of the gain to overcome bending, and other radiative,
losses in bent arrays of the amplifying microdroplets. In the experiment, we
employ a tapered nozzle which gently blows air on the the droplet stream to
bend it in a regulated fashion. The pressure is maintained so as to just bend
the stream, and not distort it completely. We achieve continuous bending up to
$\sim~{}25^{\circ}$. Figure 4[A] shows one such configuration where the array
is bent at an angle of $\sim~{}10^{\circ}$.

Figure 4: [A] An experimental realization of the bent array at an angle of
$\sim 10^{\circ}$. [B] Experimentally observed spatial profile of the extended
mode using an ICCD attached to a spectrometer, each dot is a droplet and the
array acts as a carrier of light. [C] Two mode profiles calculated using FDTD
in a linear and bent arrangement.

Figure 4[B] shows experimentally captured coherent modes for the straight and
the bent array. It is clear from the image that all the resonators coherently
participate and the bending does not prohibit the existence of the carrier
mode. The dependence of the carrier mode on the bending angle was quantified
by counting the number of modes at a given excitation energy, which roughly
was halved when the bending angle was about $25^{\circ}$. Nonetheless, the
gain in the system overcomes the far-field losses as well as bending losses
and excites the mode across the entire array, which can even be upto $60$
microspheres long. Since it was difficult to experimentally measure the
transmitted intensity and hence the bending losses, we carried out two-
dimensional FDTD calculations to get better insights on the transmission.
Figure 4[C] shows a propagating mode excited at FP wavelength in the passive
system. The bending loss is evident, and increases with angle. We have
observed the reduction of these bending losses by adding gain in the
simulations. Further, we find that there is an advantage created by the
spherical surface as against conventional plane FP resonators. As can be seen
in the image, the mode is not necessarily centered on the axis of the array
after the bend, but propagates via off-axis lensing from the microspheres.
Such CROW’s could be of interest in optofluidic devices.

However, the experimentally observed intensity variation along the array
differs from the expected behavior. This difference can be traced to the
origin of the collective longitudinal modes.

### Origin of the longitudinal modes

We then carried out theoretical activity to trace the origin of these random
lasing modes. Since only the longitudinal emission possessed these unusual
characteristics, and besides, since the frequency signatures in the emission
were of a Fabry-Perot character, we only modeled the transport in one
dimension. To that end, the droplet array was assumed to be a one-dimensional
amplifying dielectric multilayered structure. The propagation of the
electromagnetic field is then accurately described by the transfer matrix (TM)
method. As described in the Introduction, the TM method has been extensively
used to understand localizing and amplifying properties of one-dimensional
random media [18]. In this method, successive multiplications of these
transfer matrices provide transmission properties of the entire multilayer.
Frequency characteristics of both passive and active systems can be analyzed
using this method.

Figure 5: Transfer matrix calculations. [A] Spectra showing passive
transmittance from an ordered and two random configurations. The magenta curve
shows configurationally-averaged transmittance over $1000$ realization for the
strongly random system. [B] Spectral profile for the same configuration under
amplification, the inset shows the output intensity as a function of gain
level $n^{\prime\prime}$ to identify the threshold. [C] Histogram of lasing
wavelength for (a) a monodisperse system with weak disorder in spacing (b) a
monodisperse system with strong disorder in spacing and (c) for polydisperse
system with strong disorder in spacing.

Figure 5[A] shows the transmission spectra from passive multilayers with a
varying degree of randomness for a system of $20$ unit cells. Each unit cell
has a dielectric layer of thickness $A=20~{}\mu$m with an inter-layer spacing
$B$ of $6~{}\mu$m. For a perfectly ordered configuration (red curve), two stop
bands appear in the illustrated region of interest. Next, we introduce
disorder in the interlayer spacing (quantified by $\delta B$, such that the
separations are uniformly distributed over [$B-\delta B/2,B+\delta B/2$])
while keeping the thickness monodisperse. For weak randomness, the earlier
band-edge modes migrate into the stopband region. At larger degree of
positional disorder, several new modes are generated in the gap with randomly
distributed quality ($Q$) factors. The frequency sensitivity of the passive
system is revealed by the magenta curve which shows the configurationally
averaged transmittance over $1000$ realizations. The separation between the
two high transmission peaks is equal to the free spectral range (FSR) of an
individual layer. Next, we introduced optical gain to the same system, and the
spectra are shown in Figure 5[B]. In the perfectly ordered configuration, the
band edges have the highest $Q$ and hence the smallest threshold for lasing.
In the weakly periodic configuration, the highest $Q$ modes are the perturbed
band-edge modes. Under stronger disorder, multiple lasing modes are generated
in the stopband region. We identified the frequency distribution of these
lasing modes over $500$ realizations (Figure 5[C]). A weakly disordered system
shows a double-peaked bunch centered at the stopband of the underlying ordered
configuration; these two bunches originate from the perturbed band-edges. When
the sample is made slightly more disordered, the memory of the band-edges is
washed out. Finally, when both the sizes and separations are strongly
randomized, the frequency-sensitivity is lost, and the lasing peaks are
distributed over a wide, continuous range. Comparison with experimental data
shows that the separation between histogram bunches is equal to that seen in
the experiments under identical layer thicknesses. Thus, the TM calculations
reveal the origin of the experimentally observed lasing modes as arising from
the gap states of these systems, known as amplifying periodic-on-average
random systems.

### Single-mode, stable coherent random lasing

With the identification of the origin of the lasing modes, we explore the
possibility to obtain single mode and stable coherent random lasing. In this
scenario, stability implies a minimum fluctuation in lasing frequency, while
single mode means that only one mode in the entire frequency band crosses the
threshold and lases. In a gain environment, the lasing frequency depends on
two main parameters, namely (i) Spectral position of high $Q$ modes and (ii)
location of the gain maximum. The latter is a molecular property and cannot be
altered, but for minor changes caused by the concentration of the lasing dye.
Our study on aPARS shows that the highest Q modes appear at the minima of the
underlying FP profile of the periodic system, and can hence be controlled by
using appropriate system parameters.

Figure 6: [A] Blue curves show the gain profile. Top left: Black curve is the
spectrum from a periodic system with later thickness $17.7~{}\mu$m. Bottom
Left: Red curve is the Fabry-Perot profile of a single dielectric layer; the
histogram shows that the first lasing modes are equally populated in both
bunches. Top right: Black curve shows the spectrum from a periodic system with
layer thickness of $18.0~{}\mu$m. Bottom right: Red curve: Fabry-Perot profile
of a single layer; the histogram shows that the lasing peaks are highly
populated at the gain maximum. [B] Experimentally stabilized single mode
coherent random lasing. [C] Histogram of the lasing wavelength, the
fluctuations have been reduced to $\sim 1.2$ nm.

Figure 6 illustrates the idea to achieve stable coherent random lasing via
mode matching. Mode matching refers to matching one of the minima of the FP
profile with the maximum of the gain curve. (In conventional lasers with
resonators, the cavity length is chosen so that one resonant mode matches with
the gain maximum. In our case, the FP-profile of the single cell determines
the stopband positions.) In Figure 6[A], the blue lines show the frequency
dependence of the gain profile. The black transmission spectrum (top left
panel) arises from 30 monodisperse layers arranged in a perfectly periodic
fashion. The size of each layer is $17.7~{}\mu$m with an inter-layer spacing
of $8.5~{}\mu$m. The red line (bottom left) indicates the FP profile of a
single layer. $500$ spectra are analyzed to study the frequency behavior of
first lasing mode with $\delta A=100$ nm and $\delta B=1000$ nm. The
distribution of lasing peaks (bottom left) shows two equally populated
bunches, implying that the lasing emission randomly toggles from one frequency
interval to another. Then we changed the slab width to $18~{}\mu$m to match
the minimum of FP profile with the maximum of the gain curve (right panels).
The resulting histogram shows the lasing distribution where $76~{}\%$ of the
modes populate in the center bunch. Thus, a single mode is emitted per pulse
within a range of $\sim 2$ nm with a probability of $76~{}\%$. It should be
noted that, since the gain exists only in the droplet and not outside, any
randomness in the separations does not affect this behavior, apart from mildly
widening the central bunch.

In the experiment, we changed the droplet diameter by varying the applied
pressure and the perturbation frequency of the piezocrystal at the gate of the
microcapillary. Figure 6[B] shows a set of experimentally observed spectra
which depict single mode lasing, with a very small variation in the lasing
wavelength within a narrow band of $\sim 1.2$ nm. We found that over a small
ensemble of about $20$ spectra, $\sim~{}90\%$ yield stable modes within the
same band. In larger ensembles of over $100$ spectra (Figure 6[C]), a
stability of over $75~{}\%$ was obtained. This technique also requires the
gain coefficient to be appropriately maintained, so as to avoid multimode
emission over the lower quality resonators as well. Thus using a mode-matched
aPARS, we have experimentally achieved stable, single mode random lasing over
$1.2$ nm, compared to a $10$ nm band as seen in coherent random lasers.

### Spatial extent and mode profile of random lasing modes in aPARS

Transfer matrix calculations reveal that, in a periodic-on-average random
superlattice (PARS) system, the localization length $\xi$ is sensitive to the
frequency. In the vicinity of the band gap, $\xi<L$, and the system is capable
of localizing the fields. The variation in the $\xi$ can be expected to appear
in the spatial extent of the modes too. Figure 7[A] shows the calculated
transmission spectrum from a PARS consisting of $40$ unit cells. The right
panel exaggerates the relevant spectral region. The magenta curve shows the
band-structure of the underlying periodic system. The transmission spectrum of
the random system has several peaks, out of which the three highest quality
modes appear at $\lambda=558.2,558.9$ and $557.9$ nm. Figure 7[B] shows the
calculated normalized intensity distributions for these high $Q$ modes. The
colour of the depicted modes corresponds with the marker arrows in the
exaggerated spectrum.

Figure 7:  [A] Calculated spectrum from a passive PARS system. Right Panel:
Exaggeration of the region of interest, showing three highest quality lasing
peaks. Purple plot shows the underlying band-structure of the periodic system.
[B] Spatial mode profiles (intensity) of these three lasing peaks, identified
by the dotted arrows in same color. [C] Top: Image of the microdroplet array,
and the axial refractive index profile. Bottom: Spectral images and intensity
profiles of three lasing modes from the experimental aPARS system at a single
disorder configuration.

We see that the tightly localized mode (blue profile) originates from the
state deep inside the gap, while the weakly localized mode (black) is close to
the passband region. A mode within the passband region is almost extended over
the entire sample. Importantly, within one configuration, several lasing modes
with strongly varying spatial extents can occur. Figure 7[C](b)-(d) shows the
experimentally observed mode profiles from a weakly random system, all
observed from the same configuration under uniform pumping. Similar
characteristics of varying spatial extent are seen, and some clear tightly-
localized modes (b) are also observed. Thus, we conclude that the quasi-one-
dimensional aPARS is capable of localizing light, and the observed lasing
modes are obtained over amplification of localized states in a quasi-one-
dimensional disordered medium. We, then, quantified the spatial extent of the
modes by constructing the distribution of the Inverse Participation Ratios
($P_{2}^{\prime}$s) of the localized modes, and we found that the weakly
disordered system exhibited a broader distribution compared to the strongly
disordered one. This behavior is consistent with the behavior of calculated
localization lengths in one-dimensional random systems.

### Summary

In summary, this thesis provides an experimental demonstration of a scheme to
control the lasing frequency of a coherent random laser. This has been
achieved by creating a linear disordered array of monodisperse
microresonators. The co-operative longitudinal emission in the system obeys
the effective quasi-one-dimensional multilayer formation along the axis. Given
the control on sizes and separations of these resonators, we have successfully
created an amplifying periodic-on-average random system. Such systems have
been extensively studied in theory mostly in the passive domain, and
experimental demonstrations are few, that too only in passive systems [29, 30,
31]. Further, the experimental demonstrations in a multilayer do not allow for
configurational averaging due to the non-availability of a large number of
samples. Ours would be the first experimental demonstration of an amplifying
PARS system. The inherent dynamic nature of our system allows for
configurational averaging, and studying averaged quantities that are of prime
interest in the physics of disordered systems. Thus, the experimental
achievement of such a system, that is also easy to implement (not requiring
sophisticated fabrication techniques), is a prime message of this thesis.
Furthermore, a significant goal in the field of random lasers, namely, lasing
over photon localized modes, has been conclusively achieved in these
experiments. Novel physical results related to the distribution of the spatial
extent of localized modes in amplifying disordered systems have been provided.
We hope these results trigger interest in other groups worldwide to further
study random lasing in such nearly periodic systems.

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#### Publications contributing to this thesis

  * •

Anjani Kumar Tiwari, Ravitej Uppu and Sushil Mujumdar, “Frequency behavior of
coherent random lasing in diffusive resonant media,” Photonics and
Nanostructures - Fundamentals and Applications, 10, Issue 4, pp. 416-422
(2012).

  * •

Anjani Kumar Tiwari, Ravitej Uppu and Sushil Mujumdar, “Aerosol-based coherent
random laser,” Optics Letters, 37, 1053 (2012).

  * •

Anjani Kumar Tiwari, Balu Chandra, Ravitej Uppu and Sushil Mujumdar,
“Collective lasing from a linear array of dielectric microspheres with gain,”
Optics Express, 20, 6598 (2012).

  * •

Anjani Kumar Tiwari and Sushil Mujumdar, “Random lasing over gap states from a
quasi-one-dimensional amplifying periodic-on-average random superlattice,”
Physical Review Letters, 111, Issue 23, 233903 (2013).

  * •

Anjani Kumar Tiwari, K. Shadak Alee, Ravitej Uppu and Sushil Mujumdar,
“Single-mode, quasi-stable coherent random lasing in an amplifying periodic-
on-average random system,” Applied Physics Letters, 104, 131112 (2014).

  * •

Anjani Kumar Tiwari, et al., “Direct observation of Anderson localization-
induced random lasing,” Manuscript to be prepared.

  * •

Anjani Kumar Tiwari, Ravitej Uppu, and Sushil Mujumdar, “Experimental
demonstration of small-angle bending in an active direct-coupled chain of
spherical microcavities,” Applied Physics Letters, 103, 171108 (2013).

#### Other publications

  * •

Ravitej Uppu, Anjani Kumar Tiwari and Sushil Mujumdar, “Identification of
statistical regimes and crossovers in coherent random laser emission,” Optics
Letters, 7, 662 (2012).

  * •

Rajesh V. Nair, Anjani K. Tiwari, Sushil Mujumdar and B. N. Jagatap, Photonic-
band-edge-induced lasing in self-assembled dye-activated photonic crystals,”
Physical Review A, 85, 023844 (2012).

  * •

Rajesh V. Nair, Anjani K. Tiwari, Sushil Mujumdar and B. N. Jagatap,
“Inhibition and enhancement of spontaneous emission using photonic band gap
structures,” Advanced Material Letters, 4, Issue 6, pp. 497-501 (2013).

##  Chapter 1 Introduction

###  1.1 A short history of the random laser

Wave propagation in a random scattering environment is a subject with a long
history [ishimaru78, hulst81, ping90]. Many natural media like the atmosphere,
the ocean surface, clouds, fog, snow, etc. have inherent spatial
inhomogeneities which vary randomly in space and time. Due to this inherent
disorder, wave propagation becomes chaotic. The concept of disorder in optics
was historically related to these media and related phenomena. However, in
recent years, a field by the name mesoscopic optics has taken form, which
relates more to the optical counterpart of disorder-oriented phenomena in
condensed matter physics. Perhaps the most exotic of these phenomena are
Anderson Localization. In 1958, Philip Anderson introduced the concept of
electron localization in disordered systems and explained the disorder-induced
metal-insulator transition [anderson58, lagendijk09]. He revealed that
localization arises due to the quantum interference effect of electrons and
the impurities assist coherent multiple scattering and subsequent interference
of electron wave. As the localization phenomenon arises only due to the wave
nature of electrons, it holds true for any kind of wave propagating in the
random scattering environment. Accordingly, in the recent past, there have
been extensive studies of disorder-induced localization of classical waves
such as electromagnetic radiation, sound waves, elastic waves, etc. [ping90,
john83, graham90]. Of these, the larger body of literature exists in the
optical domain.

Over conventional electron localization, the study of light localization has
additional advantages. For example, in the case of electrons, a sub-Kelvin
temperature is required to produce interference effects, while optical
experiments can be carried out at room temperature. Further, as photons are
non-interacting unlike electrons, which interact through Coulombic forces,
pure localization effects can be studied using optical waves. Moreover, the
intensity and wavelength of photons can be controlled to very high finesse
which is not possible for electrons in a metal. Another huge advantage with
optical waves is the possibility to directly measure various properties like
statistical distribution of intensity, temporal response, etc. Hence, the
experimental study of localization with optical waves is much easier to
perform than with electrons and has thus attracted considerable interest in
the optics research community. In 1984, S. John first introduced localization
of optical waves [sajeev84]. At the same time, coherent backscattering (which
is the signature of localization) was observed by Albada and Lagendijk
[albada85] and Maret and Wolf [wolf85]. In 1987, S. John proposed strong
localization of photons in the disordered dielectric superlattices [sajeev87,
sajeev91].

The entry of structured disorder in lasers is a rather intriguing story. At
the heart of a conventional laser is an amplifying medium and a resonator. An
optical resonator plays the most crucial role in lasing action and is
responsible for coherence. The significance of resonators triggered a huge
research activity in the 1960s [siegman99a, siegman99b]. It was believed for a
long time that any inhomogeneity in the amplifying medium is detrimental to
laser action as they induce scattering losses. However, in 1966, a group led
by Letokhov demonstrated a new type of amplifying phenomenon where feedback
was provided by a ‘bad’ resonator [Ambartsumyan66]. Following that, in 1968,
Letokhov combined the idea of multiple scattering and gain and theoretically
predicted the amplification of light from a random scattering medium
[letokhov68]. In such systems, the light was fed back into the amplifying
region by multiple scattering via ‘nonresonant feedback,’ a term which will be
clarified later. This idea received a massive boost in 1994 when Lawandy et
al. reported laser-like emission and linewidth narrowing from an alcoholic
solution of Rhodamine 6G by adding TiO2 microparticles [lawandy94]. They
observed that upon increasing the pump energy, the emission intensity diverged
and the bandwidth collapsed to $\sim 10$ nm. It was during this time that the
system acquired the name random laser. This work stimulated the area of random
lasers and similar observations were made by several groups in various dye-
scatterer and other systems like laser crystal powders, etc. [gouedard93,
noginov95, siddique96]. A variety of new phenomena in these systems were
discovered, making the field of random lasers an integral component of
mesoscopic optics.

###  1.2 Conventional lasers and resonant feedback

Conventional lasers consist of a pumping source, a gain medium, and a well-
defined resonator cavity [siegman86]. The pumping source is required to
establish the population inversion condition in the gain medium. Amplification
occurs via stimulated emission process when a photon interacts with the
inverted gain medium. A photon, while propagating in the inverted gain medium,
gathers an intensity exponentially proportional to its propagation length
$I\propto\text{exp}(L/l_{g}$). Here, $l_{g}$ is the gain length and is defined
as the path length over which the initial intensity is amplified by a factor
of $e$.

Figure 1.1: [A]: Schematic representation of a laser. Legend: G: gain medium.
Flashlamp: pumping source to excite the gain medium. M1 and M2: two mirrors to
form a Fabry-Perot cavity. [B]: Allowed modes of the cavity. [C]: The
threshold behavior of a typical laser, inset shows the intense peak in the
spectrum.

Understandably, the optical resonator plays the most important role in the
generation of lasing action. It increases the output intensity by feeding the
light back into the gain medium and thereby increasing the propagation length
$L$. As depicted in Figure 1.1[A], two parallel mirrors M1 and M2 form a
cavity. The operation frequency of the laser depends on the physical
parameters of the cavity such as its dimensions, reflectivity of the mirrors,
etc. The constructive interference condition is given by

$2L_{\text{cav}}=n\lambda$ (1.1)

where, $L_{\text{cav}}$ is the length of the cavity, $n$ is an integer
multiple and $\lambda$ is the wavelength of light. Clearly, the resonant
wavelength gets selectively chosen by the cavity as shown in Figure 1.1[B].
The cavity fed back light at these wavelengths into the gain medium. If the
total gain overcomes the resonator losses, the system crosses threshold and
begins to lase. At threshold, the output intensity shows a sudden growth, as
shown by the arrow in Figure 1.1[C] which marks the threshold. It is known
that any inhomogeneities within the cavity introduce additional scattering
losses and hence are considered detrimental for lasing action.

###  1.3 Laser-like emission from disordered media

Clearly, in a conventional laser, the length of the cavity determines the
lasing frequency and hence it is sensitive to the operational conditions like
mechanical vibration, thermal expansion and so on. The group led by Letokhov
proposed a new type of laser which was based on a unique feedback mechanism
devoid of resonances [Ambartsumyan66, letokhov68]. Figure 1.2[A] depicts the
idea where one of the FP mirrors is replaced by a diffused scattering surface.
This configuration forbids the possibility of forming a stable resonance. In
fact, the rough surface is responsible for many low-quality modes which
overlap in frequency and thus wash out the possibility of any specific
resonance. Interestingly, even in this situation, under strong pumping, the
emission intensity gets enhanced at the gain maximum. Figure 1.2[B] depicts
another schematic where the inner wall of an inhomogeneous circular cavity
provides similar feedback.

Figure 1.2: Design of different cavities with non resonant feedback [A]: A
Fabry-Perot cavity formed by a mirror and a scattering surface. [B] A circular
cavity with scattering inner wall and a small outlet. [C] Randomly arranged
scatterers in an amplifying medium, the gain region is indicated by the yellow
color.

Letokhov proposed the amplification of light in a random scattering
environment with gain (see Figure 1.2[C]). He considered an amplifying random
scattering environment where the spontaneously emitted light experiences both
multiple scattering and amplification. The relevant length scales to quantify
the scattering phenomenon are the scattering mean free path and the transport
mean free path. As these length scales are frequently used in this thesis, we
first define these quantities. The scattering mean free path $l_{s}$
determines the average distance between two scattering events and is defined
as

$l_{s}=\frac{1}{N\sigma_{s}}$ (1.2)

where $N$ is the scatterer density and $\sigma_{s}$ is the scattering cross-
section of the individual scatterer. Since $l_{s}$ does not contain the
information of scattering anisotropy a more relevant parameter in the multiple
scattering environment is transport mean free path ($l_{t}$). It is defined as
the average distance traveled by the photons before their direction of
propagation gets completely randomized and is given by

$l_{t}=\frac{l_{s}}{1-g}$ (1.3)

where $g$ is the angular anisotropy parameter in scattering and given by $g$ =
$\langle cos\theta\rangle$, where $\theta$ is the scattering angle. In a
random scattering medium, the actual path length traveled by the photon is
much larger than the chord distance between the starting and the ending
points. The total distance between the starting and the ending points of
photons in the gain region determine amplification. This frames the concept of
amplification length ($l_{\text{amp}}$). If the gain length of the photon is
$l_{g}$, then the amplification length is given by

$l_{\text{amp}}=\sqrt{\frac{l_{t}l_{g}}{3}}$ (1.4)

In the diffusive scattering regime ($\lambda<l_{t}<L_{\text{sys}}$, where
$L_{\text{sys}}$ is the length of the system), the propagation of light can be
described by the random walks of photons, and the dynamics is governed by the
diffusion equation

$\frac{\partial^{2}I(r,t)}{\partial
t}=D\nabla^{2}I(r,t)+{\frac{v}{l_{\text{amp}}}I(r,t)}+S(r,t)$ (1.5)

where $D=vl_{t}/3$, is the Boltzmann diffusion coefficient and $v$ is the
transport velocity of light in the medium. The second term on the right side
describes the amplification and $S(r,t)$ represents the source term. Letokhov
found that the solution of the above equation diverges at some value of gain.
He showed that, in a diffusive amplifying medium, if the volume gain is larger
than the surface losses, the system crosses the threshold of random lasing
[letokhov68]. In a slab geometry, the critical thickness $L_{cr}$ above which
the system starts to lase is given by [wiersma96]

$L_{cr}=\pi l_{\text{amp}}=\pi\sqrt{\frac{l_{t}l_{g}}{3}}$ (1.6)

This innovative idea of mirrorless resonator thus opened a new possibility of
lasing with nonresonant feedback. Early experiments on this proposal were
performed on laser powders [varsanyi71, fork74, fork79] and the existence of
scattering-enhanced stimulated emission was confirmed. The first breakthrough
result of significant gain narrowing at threshold was reported by Lawandy et
al. [lawandy94]. They observed a substantial spectral and temporal narrowing
in a dye solution with the addition of titanium dioxide nanoparticles. This
system eventually has been termed as a ‘random laser’, since, after a certain
pumping threshold, it exhibits laser-like properties of intensity enhancement
and bandwidth narrowing assisted by randomness. The feedback generated by the
randomness is termed ‘nonresonant’ feedback.

###  1.4 The coherent random laser

While the above-mentioned system was termed a laser, it still lacked a high
degree of spatial or temporal coherence. The existence of temporal coherence
was first reported by the group led by H. Cao. They used ZnO semiconductor
powder to realize the essential conditions of high gain and strong scattering
[cao99, cao02]. They observed ultranarrow lasing peaks (bandwidth $\sim 0.5$
nm) on top of an incoherent background. The origin of coherent random lasing
was attributed to the recurrent scattering of light (see Figure 1.3[A]),
motivated by the fact that the scattering mean free path in the system was
less than the emission wavelength. The recurrent scattering arises due to the
coherent feedback of light between the scatterers which are the key factor for
photon localization in the Anderson model. Such a feedback, where the light
wave follows a closed loop path over a randomly formed resonator, is termed
resonant feedback. Clearly, this feedback is based on the electric field
transport and requires the phase information of the scattered light wave. The
phase shift requirement of 2$\pi$ along the loop determines the lasing
frequency, essentially, involving resonant feedback in the random medium.

Figure 1.3: [A] A schematic diagram of the formation of a closed loop path
through the recurrent scattering of light in ZnO powder. [B]: Trajectories of
photons under diffusive propagation modeled as random walks.

However, subsequent experiments with dye-scatterer systems showed that such
sharp lasing peaks can also be observed in very weakly scattering systems that
fall in the diffusive transport regime [cao00]. The explanation for this
behavior was provided by Mujumdar and Wiersma et al. [sushil04]. They showed
that diffusion can enhance the lifetime of spontaneous emission photons to an
excessive degree. They modeled diffusive propagation as random walks of
fluorescent photons that gather gain during propagation. The diffusion process
in the finite-sized system, realizes a few long and rare paths(Figure 1.3[B]),
which realize an excessive gain on the virtue of their path length. Since
these are statistically rare, the peaks created by the paths do not get
averaged out during the complete relaxation of the inverted medium. Thus, they
create an intense peak in the spectrum. This model did not involve
interferences and explained the coherent peaks in a diffusive scenario, thus
involving nonresonant feedback. The frequency of the coherent modes is decided
by the spontaneous emission events that excite the amplified extended modes.
Nonetheless, further discussions based on resonant feedback persisted in the
field, wherein the origin of the modes was attributed to the Fabry-Perot
resonances that form between two nanoparticles resulting in equispaced modes
[wu06]. Theoretically, resonant (field) feedback has been implemented by
solving Maxwell equations through Finite-Difference Time-Domain (FDTD)
techniques in one and two dimensional random systems [sebbah01, soukoulis00].
The Transfer Matrix Method (TMM) has also been vastly used to study one-
dimensional random lasers, with a particular focus on quasi-periodic systems
with amplification [soukoulis99, soukoulis02].

Over the years, a variety of materials were identified to show random coherent
lasing [polson04, tulek10, turistsyn10, strangi06, noginov05]. In most cases,
the scatterers are separately added to the amplifying medium. For example,
when nanoparticles like polystyrene or TiO2 are added in a laser dye solution,
the nanoparticles act as scatterers, and the dye solution serves as the
amplifying medium. On the other hand, in some samples, both scattering and
amplification can be provided by the same entity. For example, Nd-doped
powders, dye-doped microparticles, etc. serve this purpose. Regardless of the
constituent materials, the emission spectra from such systems exhibit strong,
temporally coherent peaks that fluctuate in frequency, intensity, spatial mode
extent, etc. [zaitsev07, zaitsev06, hackenbroich05, hcao01, beenakker96,
wiersma07, sushil07, lagendijk07, uppu10]. Out of these, frequency
fluctuations are the major impediment in practical applications of random
lasers. Hence, there is considerable interest in the community in developing
techniques to overcome them.

###  1.5 Attempts of frequency control in random lasers

One of the important properties of a laser device is good stability in the
intensity and frequency of the lasing modes. The resonator design exactly
determines the lasing frequency. Accordingly, a device can be fabricated at
any desired wavelength, assuming the gain profile covers it. In contradiction,
in the random laser, the inherent spatial disorder rules out this possibility.
In general, the emission spectrum of a coherent random laser has distinct
multimode lasing peaks which fluctuate over a range of $10-15$ nm determined
by the gain profile. For most of the real-life applications, it is desirable
for the random laser to have a better control on the number of lasing modes
and their wavelength. In that regard, various techniques such as, usage of
monodisperse Mie scatterers, shifting the effective gain curve of the
amplifying material, shaping the optical pump profile and deliberately
creating a point defect, etc. have been demonstrated to control the lasing
frequency [gottardo08, lagendijk11, sebbah12, sebbah13, shiva12, leonetti13,
fujiwara13]. Here, we briefly review these techniques.

In one of the earliest attempts of frequency control, Gottardo et al. used
self-assembled amplifying Mie scatterers and exploited their resonant features
[gottardo08]. In these experiments, Polystyrene microspheres of diameters
larger than the wavelength of light such as $0.9~{}\mu$m, $1.0~{}\mu$m and
$1.2~{}\mu$m were used. The inherent polydispersity in the diameters was $\sim
2~{}\%$. The amplification was incorporated by infiltrating DCM dye (dissolved
in ethanol) into the sample. The packing fraction of the sample was $\sim
55~{}\%$, and the typical thickness of a sample was nearly $100~{}\mu$m.
Multiple scattering events were ensured as the thickness of the sample was
much larger than the scattering mean path ($2-3~{}\mu$m). Under optical
pumping, samples with particle diameters of $0.9~{}\mu$m, $1.0~{}\mu$m and
$1.2~{}\mu$m exhibited lasing at $602.9$ nm, $637.5$ nm and $616.5$ nm
respectively which were all away from the gain maximum. These lasing peaks
appeared close to the transmission minimum of the samples. Thus, the lasing
wavelength was successfully tuned by changing the diameter of the spherical
scatterers showing wavelength control of diffusive random lasers.

Lagendijk et al. presented a way of tuning the diffusive lasing wavelength by
absorption engineering [lagendijk11]. In their experiment, they used a
solution of Rhodamine 640P dye in methanol as the gain medium and Titania
particles as elastic scatterers. A non-fluorescent dye (Quinaldine Blue,
quantum efficiency $0.1~{}\%$) was added into the system to shift the lasing
peak towards the blue wavelength. By changing the concentration of the non-
fluorescent dye, they showed a shift of $14$ nm in the peak emission
wavelength from $606$ nm to $592$ nm.

Sebbah et al. used gain engineering to achieve frequency control. First, they
numerically demonstrated that any desired mode of a coherent random laser can
be extracted from the emission spectrum [sebbah12]. This involved choosing a
correct spatial profile of the pump. To this end, they used an iterative
optimization algorithm. The basic idea behind this technique was to feed the
gain into a preselected mode. They successfully showed that the emission of a
coherent random laser can be brought into the single mode regime by this
selective excitation process. Furthermore, the threshold of the selected modes
was lower when the system was excited with the optimized pump, as compared to
a uniform pump profile. This method was shown to work also for weak scattering
media, where the modes were spatially extended and therefore strongly
overlapping. Later on, they carried out experiments to demonstrate this
technique [sebbah13]. The active control on the pump was achieved by a spatial
light modulator (SLM). Single mode operation was shown in an optofluidic
random laser of 3 mm long structured PDMS [shiva12]. Clearly, this method is
applicable only for static systems and requires a priori knowledge of the
spatial profile of a particular mode. Furthermore, the basic principle behind
this technique is wavelength selectivity rather than wavelength control.

In another technique, Fujiwara et al. proposed that the random lasing
properties can be controlled by intentionally creating a defect site in the
random sample of monodisperse scatterers [fujiwara13]. Their random sample
contained submicrometer sized ZnO nanoparticles which act as scatterer and
gain medium. To incorporate the point defects, a small amount of green
fluorescent polystyrene particles ($900$ nm) were added into the solution. The
fluorescence of polystyrene allows locating the point defects. Upon optical
excitation, the system showed unique lasing properties like suppression of the
number of lasing modes, lowering of the threshold, and control of lasing
wavelength. The wavelength control in the random laser was attributed to the
average resonant properties of the ZnO particle film, whereas, the location of
the defect determined the spatial position of the lasing sites.

###  1.6 Light localization

Anderson et al. pointed out that the dimensionality of the disordered system
is a crucial aspect in the physics of localization [abrahams79]. Based on the
scaling theory, they suggested that, if the dimension of the system is less
than or equal to two, all the states can be localized, provided that the
system is sufficiently large. On the other hand, the 3D system requires
certain critical disorder strength. The condition on the critical disorder is
known as Ioffe-Regel criterion ($kl_{s}\leq 1$, where $k$ is the wave vector
of photon and $l_{s}$ is the scattering mean free path) [iofee60].

In the recent years, there have been many attempts to observe Anderson
localization of light. Here, we briefly discuss some of the important results.
To observe strong localization in 3D, Wiersma et al. prepared a strongly
scattering system from micrometer-sized GaAs semiconductor particles
(refractive index = $3.48$) [wiersma97]. By grinding semiconductor powders,
samples with different particle sizes ($10~{}\mu$m, $1~{}\mu$m and $300$ nm)
were prepared, which offered a range of scattering strengths. In the
experiment, the transmission coefficients as a function of sample thickness
were measured in diffusive as well as in the localized regime. It was observed
that, in the weaker sample, the transmission coefficient decreased linearly
implying diffusion. On the other hand, in the stronger samples, it followed an
exponential decay, a hallmark of Anderson localization. While this was the
first experimental report on Anderson localization in the near visible regime,
it also generated a discussion regarding the contribution of absorption in the
sample, which also realizes an exponential decay of transmittance [maret99].
To overcome the issues related to absorption, the group led by Genack used a
different approach to identify localization. They reported that the variance
of the normalized total transmission (var(sa)) can serve as a tag for
localization. They claimed that, regardless of absorption, if
g${}^{{}^{\prime}}\leq 1$, the system achieves localization, where,
g${}^{{}^{\prime}}$ was a new localization parameter and given by
g${}^{{}^{\prime}}\equiv 2/3$ var(sa) [chabanov00]. These results were
observed in the microwave domain. In the microwave frequencies, it is easier
to achieve localization because the large scatterer sizes allows the
fabrication of scatterers with a good control on randomness. Besides, with
larger $\lambda$, the Ioffe-Regel criterion is more accessible. Indeed, the
same group had already demonstrated photon localization in the microwave
regime [garcia91, genack91], wherein they had studied transmission and
diffusion coefficients and intensity correlations in a random mixture of
aluminum and Teflon spheres. Recently, the group of Maret reported visible
light transport in three-dimensional bulk samples comprising of TiO2
nanopowders [sperling13]. They measured the time-dependent broadening of an
incident laser pulse in the transverse direction of the sample. The evolution
of transverse width with time provided a direct measure of the localization
which was independent of absorption effects. However, these results need to
clarify issues relating to inelastic scattering of light realized by nonlinear
interactions [schffold13]. This inelastically scattered light arrives late and
can create an impression of long-time delay of scattered light.

Lower-dimensional systems have already been investigated for light
localization. In the microwave regime, localization effects in a two-
dimensional system have been reported [rachida91] where the scattering was
provided by the collection of dielectric cylinders randomly placed between two
parallel aluminum plates on a square lattice. Sharp peaks in the transmission
spectrum were observed. These sharp peaks had high electric field regions and
were attributed to the localized states. More recently, strongly confined
Anderson-localized cavity modes were reported in disordered photonic crystal
waveguides. Essentially, the system consisted of one-dimensional waveguides
fabricated in two-dimensional lattices. By engineering the disorder in the
photonic lattice in the vicinity of the waveguide, high quality-Anderson-
localized modes (Q$\sim$105) have been demonstrated [vollmer07, lodahl10].

In the current times, a different kind of localization, called transverse
localization, is being vigorously investigated. This concept, introduced by
Lagendijk et al., describes a situation where the wave propagates in one
direction but is confined in the transverse direction by disorder
[lagendijk89]. This is relatively easier to observe and elucidates a lot of
physics of Anderson Localization. Lahini et al. demonstrated transverse
localization in a one-dimensional lattice of coupled optical waveguides
patterned on an AlGaAs substrate ($5$ mm long disordered photonic lattice)
[lahini08]. In their experiment, a light beam was injected at the input end of
a disordered photonic lattice. The beam then evolved as it propagated along
the structure and coherently tunneled between the neighboring waveguides. They
showed that, in an ordered structure, the light beam follows normal ballistic
propagation properties. However, as the disorder strength increases, light
tends to localize around the input position. At even higher disorder, they
observed an exponentially decaying transverse profile. Segev et al.
demonstrated Anderson localization of light in two-dimensional disordered
nonlinear photonic lattices [segev07]. The disordered photonic lattice was
constructed from a $10$ mm-long SBN:60 crystal using optical induction
technique. By varying the disorder strength of the sample, they showed the
transition from the ballistic regime to diffusive transport regime. They
observed that, as the disorder strength was increased, the inverse
participation ratio (a quantity that indicates the mode spatial extent, also
defined later in this thesis) and its relative fluctuations increased.
Further, when Anderson localization occurred, the transverse field acquired an
exponentially decaying tail from the center of the incident beam.

In the scenario of amplification, the occurrence of light localization had led
to the concept of mirrorless lasing via random resonators [pradhan94]. The
earliest claim to the same was made by Cao et al. when they observed
ultranarrow coherent lasing modes from ZnO nanopowders. The origin of the
ultranarrow modes was attributed to the recurrent scattering of light and
Anderson localization, though the claim of Anderson localization was not
explicitly supported [cao99]. In another experiment, V. Milner and A. Z.
Genack demonstrated localization induced low-threshold random lasing
[genack05]. The sample consisted of a stack of partially reflecting cover
slides of random thickness ($80~{}\mu$m - $120~{}\mu$m). The cover slides were
arranged in a one-dimensional fashion, and the amplifying layers (Rhodamine 6G
solution) were placed between them. When the stack was pumped with a pulse
laser, the emitted light gets localized due to multiple scattering between the
interfaces. They observed low threshold ultranarrow lasing when the system
size was much larger than the localization length.

###  1.7 Thesis

This thesis describes the experimental study of coherent random lasing from a
unique system consisting of an amplifying aerosol. The aerosol is achieved in
a quasi-one-dimensional fashion with random spacing between the individual
liquid microdroplets. We implemented a roadmap towards minimization of
frequency fluctuations in this coherent random laser. We show that by using
monodisperse scatterers, the fluctuations can be reduced into a single band of
$\sim 1.2$ nm. Transfer matrix analysis and finite difference time domain
techniques are used to explain the experimental data. By studying the spatial
profile of the random lasing modes, we attribute the observed random lasing to
the Anderson localization states of the periodic-on-average random system
(PARS).

This thesis is organized as follows.

Chapter 2: In this chapter, Monte Carlo simulations based on the propagation
of photons were performed, which showed that efficient and intense lasing
modes can be obtained with a system of intra-scatterer gain. This is studied
in the diffusive scattering domain. We observed frequency-controlled coherent
random lasing in such a system. Based on these simulations, we propose a
practical implementation of an aerosol, comprising liquid microdroplets in
air. Monodispersity in the microdroplets can be expected to induce frequency
control.

Chapter 3: In this chapter, we describe the generation process of the aerosol
and its characterization. The aerosol is in the form of arbitrarily shaped
dye-doped microdroplets. We show that, upon optical excitation, the
longitudinal emission spectrum consists of coherent random lasing (collective)
modes. Using direct spatiospectral imaging of the modes, we confirm that they
are extended over several microdroplets which coherently participate in the
emission.

Chapter 4: In this chapter, we implement monodispersity in the above system
and carry out the experimental study of random lasing modes from a linear
array of monodisperse microspheres. We confirm that the wavelength of the
collective modes is sensitive to the size of microdroplets. Our analysis of
lasing bands indicates that the Fabry-Perot resonances of individual
microdroplets participate in the collective emission. Furthermore, we discuss
various properties of the collective lasing modes like threshold behavior, the
effect of system size, angular distribution and polarization anisotropy in
this system.

Chapter 5: This chapter contains a theoretical discussion of plane wave
propagation in a one-dimensional structure comprising a dielectric multilayer.
We model the microdroplet array as an amplifying periodic-on-average random
superlattice (aPARS) system. We perform transfer matrix calculations with gain
to analyze the origin of the lasing modes and their properties. Based on our
calculations, we conclude that the observed random lasing modes originate from
the high-quality gap states of the underlying periodic system. The ensemble-
averaged analysis reveals that the gap states are restricted in the specific
frequency range governed by the size of the individual layer.

Chapter 6: Next, we exploit spectral properties of PARS systems to achieve
single-mode, stable coherent random lasing. To that end, we use the concept of
spectral mode-matching which refers to matching the gain maximum with the
spectral realm of gap states. In the experiment, we used the mode matching
technique and reduce the frequency fluctuations of the random lasing modes
into a single band of $\sim 1.2$ nm.

Chapter 7: In this chapter, we study the spatial properties of the modes using
transfer matrix calculations. We then present direct evidence of Anderson
localization of light in the microdroplet array. To quantify the spatial
extent of the modes, we calculate localization length and the Inverse
Participation Ratio (IPR) and discuss the statistics of the IPR. This chapter
conclusively proves the occurrence of Anderson localization in the amplifying
PARS system.

Chapter 8: Motivated by the current literature on chains of resonators, we
examine the practical applicability of this system as a coupled-resonator
optical waveguide (CROW). To that end, we experimentally demonstrate
collective amplified modes along a bent chain consisting of microdroplet
resonators. Furthermore, to get a better insight of the system, we present
finite-difference time domain calculations in a bent array of infinite
cylinders. We observe that the light propagation is realized by the filaments
of the nanojet-induced modes (NIMs) which were formed at the edge of the
cylinders due to the lensing effect induced by the curved surfaces. This
chapter emphasizes the efficacy of the system as a CROW.

##  Chapter 2 Frequency fluctuations in coherent random laser, inter-scatterer
vs intra-scatterer gain

In this chapter, we address the issue of frequency fluctuations in coherent
random lasers and show that the fluctuations can be reduced by using resonant
scatterers. Our calculation, based on Monte Carlo simulations, shows that an
intra-scatterer gain system provides more efficient and cleaner lasing
spectra. The chapter is organized as follows: In section 2.1, we discuss the
details of the photon transport Monte Carlo technique and parameters involved
in studying the dye-scatterer random lasing system using this technique. In
section 2.2, we consider a sparse system of monodisperse scatterers and
compare the emission properties of an inter-scatterer gain system (the gain
being present between the scatterers) versus an intra-scatterer gain system
(gain inside the scatterers). In section 2.3, we consider a dense scattering
system and show that stable and nearly thresholdless lasing can be achieved
for intra-scatterer gain system. Finally, in section 2.4, we propose a
practical system made of an amplifying aerosol to control the frequency
fluctuations and discuss its emission properties.

###  2.1 Photon transport Monte Carlo simulations

To study the spectral behavior of coherent random laser, we perform the same
Monte Carlo simulation implemented in references [mujumdar00, sushil04,
mujumdar10, ruppuaip, ruppu11, anjpnfa]. The simulation space is a box of
dimensions $10$ mm $\times~{}10$ mm $\times~{}10~{}$mm which contains a
uniform distribution of scatterers in an amplifying environment, akin to the
dye scatterer random system. We assume that the box contains N dye molecules,
decided by the concentration of the amplifying medium. The system was
discretized into $10~{}\mu$m $\times~{}10~{}\mu$m$\times~{}10~{}\mu$m boxes
and the number of excited ($N_{1}$) and ground ($N_{0}$) state molecules in
each box was registered to accurately calculate the local gain. We model light
propagation as random walk of photons inside the computational box.
Interference effects are not accounted for in the simulation. This assumption
is motivated by the fact that the scattering mean free path of photons is much
larger than their wavelength. Thus, the photons are in the diffusive transport
regime where we can neglect the possibility of recurrent scattering between
the scatterers. In the simulation, we incorporate the exact scattering
parameters and calculate total dwell-time of light due to multiple scattering.
The simulation consists of two major parts, the first part handles the
excitation process and the second part deals with the subsequent emission.

In the first process, a pump beam is incident on the front surface of the
computational box. The pump beam is assumed to consist of an ensemble of
photons or energy packets traveling along the longitudinal direction. The
total number of such packets determines the pump energy. We set the wavelength
of the incident photon bunch at $532$ nm, corresponding to a frequency-doubled
Nd: YAG laser. Upon entering the front surface of the sample, the photon bunch
suffers multiple scattering from various scattering sites and undergoes a
three-dimensional random walk. The random walk was implemented by translating
the position of the photon packet after the $i^{th}$ scattering event as

$x_{i+1}=x_{i}+p_{i}$ (2.1)

where,

$|p_{i}|=-\ell_{s}\ln\Sigma,$ (2.2)

$\Sigma$ is a uniformly distributed random number between ($0$, $1$) and
$\ell_{s}$ is the scattering mean free path. Thus, $|p_{i}|$ belongs to an
exponential distribution of path lengths between successive scattering events,
with the mean $\ell_{s}$. The angular coordinates ($\theta_{0}$, $\phi_{0}$)
of all photon packets before the first event were taken to be ($0$, $0$) to
simulate a collimated beam. For subsequent scattering events, the angular
coordinates ($\theta_{i}$, $\phi_{i}$) were picked from a uniform distribution
over $4\pi$ steradian. During the random walk, the photons interact with the
ground state molecules and experience absorption as determined by its
absorption cross-section $\sigma_{abs}$. The ground state ($N_{0}$) and
excited state ($N_{1}$) populations in each unit cell of the discretized
volume are accordingly updated and recorded. This phenomenon terminates when
either the photon bunch is completely absorbed or scattered out from the
sample. At the end of excitation stage, the simulation records the final
distribution of ground and excited state molecules. In the second stage, the
fluorescence is initiated. Spontaneous emission events generate energy packets
with unit intensity, which corresponds to one photon. The spatial distribution
of the spontaneous events is determined by the spatial excitation profile. The
frequency of the spontaneously emitted photons is picked from a cumulative
distribution of wavelengths constructed from the emission cross-section
profile $\sigma_{em}$. In our simulation, we maintained the spectral
resolution at $0.1$ nm. Similar to the pump light, the spontaneously emitted
photons also perform three dimensional random walks through the active random
medium. During the random walk, each spontaneously emitted photon experiences
amplification or absorption depending on the local population inversion. The
random walks in the amplifying medium are terminated when either the photons
exit the sample or are completely reabsorbed by the ground state molecules.
After the complete de-excitation process, total number of emitted photons is
recorded as a function of wavelength to construct the emission spectra.

###  2.2 Sparse system: Resonant amplifying scatterers

Figure 2.1: Schematic representation of the inter-scatterer [A] and intra-
scatterer [B] gain systems. Orange color represents the gain region, the
volume fraction of the scatterers is $1~{}\%$.

Figure 2.1 shows the schematic representation of the inter-scatterer and
intra-scatterer gain systems with monodisperse resonant scatterers. An inter-
scatterer gain system (Figure 2.1[A]) has the amplification between the
scatterers while in the intra-scatterer gain system (Figure 2.1[B]), the gain
is present inside each scatterer. We first consider a system with a volume
fraction of $1~{}\%$, which implies that the scatterers are sparsely
distributed. Various parameters of the resonant scatterer system are shown in
Figure 2.2. These numbers were calculated for a system comprising TiO2
(refractive index $n=2.4$) spheres of diameter $1.09~{}\mu$m with a refractive
index contrast with respect to the ambient medium (here, methanol) of $n=1.8$.
The black and the green curves in the top panel are the absorption and
emission spectra of the Rhodamine dye. The blue curve in the top panel depicts
the calculated Mie scattering efficiency ($Q_{scat}$) within the spectral band
of interest [matzler02]. The amplitude of these resonance profiles determines
the scattering mean free path, while the width determines the quality factor.
We first extract these resonances by fitting a Lorentzian profile
$L$($\lambda$) which is defined by

$L(\lambda)=\frac{1}{\pi}[\frac{\Delta\lambda/2}{{(\lambda-{\lambda}_{0})}^{2}+({\Delta\lambda/2})^{2}}]$
(2.3)

where $\Delta$ $\lambda$ is the full width at half maximum (FWHM) of the
Lorentzian profile and $\lambda_{0}$ specifies the location of the peak. The
red curve shows these extracted high quality modes where the dwell-time of the
photons is relatively large. The $Q_{scat}$ profile also has various low
quality modes, but the quality factor of these modes is few orders of
magnitude smaller and hence do not contribute strongly to the overall
transport.

Figure 2.2: [A] Black and green curves: absorption and fluorescence spectra of
the gain medium. Blue curve: Mie scattering efficiency for a particle of
$1.09~{}\mu$m with a refractive index contrast of $1.8$ with ambient medium
(here, alcohol, $n=1.329$). Red curve: Extracted high quality modes from the
scattering profile. [B] Blue curve: Variation of scattering mean free path as
a function of wavelength. Black curve: Number of scattering event before the
photons exit from the three dimensional computational box. [C] Extracted mode
centered at $559.8$ nm.

The blue curve in Figure 2.2[B] is the variation of transport mean free path
($l_{t}$) with wavelength. By definition, $l_{t}$ is inversely proportional to
the $Q_{scat}$ and is given by

$l_{t}=\frac{2d}{3(1-g)\phi Q_{scat}}$ (2.4)

where, $d$ is the particle diameter, $\phi$ is volume fraction of the
scatterers and $g=\langle\text{cos}\theta\rangle$, where $\theta$ is the
scattering polar angle [This equation is consistent with the definition of
$l_{t}$ given in section 1.3]. Thus, in a monodisperse scattering environment,
the transport mean free path $l_{t}$ is smaller at the positions of the Mie
resonances indicating stronger diffusion. Furthermore, at each resonance,
stronger scattering enhances the number of scattering events encountered by
the fluorescent photons. The black curve shown in the bottom panel is the
total number of scattering events encountered by the fluorescent photons
before they exit from the computational box as registered by the simulation.
At each scattering event, the fluorescent photon spends some resident time
inside the scatterer, which is determined from the quality factor of the
corresponding resonance. In a multiple scattering environment, $\tau_{intra}$
is calculated by the cumulative resident times within the multiple scatterers,
while $\tau_{inter}$ is obtained from the total path-lengths of the photons.
Figure 2.2[C] shows the parameters of the extracted resonance, which will be
discussed later regarding calculations of the intra-scatterer dwell-times.

Figure 2.3 shows the consequence of resonance on the emission spectrum. Figure
2.3[A] depicts the emission spectrum for inter-scatter gain system at the said
pump energies. The blue curve is the variation of $Q_{scat}$ with wavelength.
At low pump energy ($E_{p}$), the system yields the typical fluorescence
spectrum of Rhodamine dye. As we increase the pump excitation energy, lasing
peaks develop in the spectrum. The peak positions are determined by the Mie
resonances and the spectrum shows maximum enhancement at the highest quality
mode. In this system, we observed that, in addition to the coherent modes in
the vicinity of $\lambda_{res}$, multiple low intensity modes also appeared
across the spectral range. The overall emission spectrum shows gain narrowing
because of multiple amplified modes. Figure 2.3[B] shows the representative
spectra for intra-scatterer gain system. Here, the high intensity coherent
modes appear only at the Mie resonance. Furthermore, in this system, the
spectra are much cleaner and more stable111By ‘stable’, we mean that the
coherent lasing peaks have less frequency fluctuations under optical pumping.
compared to the inter-scatterer gain system as seen in the inset.

Figure 2.3: [A] Inter scatterer gain system: Blue curve is the Mie scattering
efficiency curve. Four typical emission spectra at different pump energies are
shown. The lasing peaks appear near one resonance of an individual resonator.
Further, multiple modes are generated at the resonance. [B] Intra-scatterer
gain system: Here, the spectra are cleaner and more stable. The inset is the
exaggerated region for clarity. Black curve: inter-scatterer gain system. Red
curve: intra-scatterer gain system.

To understand the emission in the inter-scatterer and intra-scatterer gain
systems, we estimated $\tau_{inter}$ and $\tau_{intra}$, which are the dwell-
times between and within scatterers, respectively. The $\tau_{intra}$ is
extracted from the Mie scattering profile by estimating the quality factor of
each resonance, defined as $Q=\frac{{\lambda}_{0}}{\Delta\lambda}$, where,
$\Delta\lambda$ is the FWHM of the resonance and $\lambda_{0}$ specifies the
center wavelength. The lifetime of the photon inside a scatterer is
proportional to the quality factor and is given as
$\tau=\frac{Q{\lambda}_{0}}{2\pi c}$. At the central resonant wavelength, the
dwell-times in the scatterers are exponentially distributed with a time
constant $\tau$. The total intra-scatterer dwell-time in the sample was
calculated by the cumulative sum of all these exponentially distributed
variables. At another wavelength in the resonant band, the dwell-time in the
scatterers will be exponentially distributed with a time constant of
$\tau^{{}^{\prime}}$. Intuitively, this decay will be faster than that at the
central wavelength, implying thereby that the $\tau^{{}^{\prime}}$ is smaller,
the best scaling of which is given by
$\tau^{{}^{\prime}}=(h_{2}/h_{1})\times\tau$, where $h_{1}$ is the height of
the extracted Lorentzian at the central wavelength (Figure 2.2[C]) and $h_{2}$
is the height of the Lorentzian at the other wavelength.

By incorporating these relations, we estimate the dwell-time distribution of
photons before they exit the computational box. Two wavelengths are chosen to
estimate the dwell-time distributions, the resonant $\lambda_{res}=559.8$ nm,
and the nonresonant $\lambda_{nres}=563.3$ nm. Figure 2.4[A] shows the total
dwell-time distribution at these wavelengths. From the total dwell-time, we
separate the distribution of $\tau_{inter}$ and $\tau_{intra}$. As shown in
Figure 2.4[B], $\tau_{inter}$ has a slight increase at the resonant
wavelength. Note that the difference is only modest, because the large number
of scattering events encountered by the photons are compensated by the small
$l_{t}$. This leads to a smaller disparity in the dwell-time distribution at
$\lambda_{res}$ and $\lambda_{nres}$.

Figure 2.4: [A] Distribution of total dwell-time of photons at resonant
$\lambda_{res}=559.8$ nm (red) and at nonresonant $\lambda_{nres}=563.3$ nm
(blue) wavelengths. [B] Distribution of inter-scatterer dwell-time at the same
wavelengths. [C] Distribution of intra-scatterer dwell-time at three
wavelengths $\lambda_{res}=559.8$ nm, $\lambda^{{}^{\prime}}=560.3$ nm and
$\lambda_{nres}=563.3$ nm.

Figure 2.4[C] discusses $\tau_{intra}$ at three wavelengths, namely, resonant
$\lambda_{res}=559.8$ nm, nonresonant $\lambda_{nres}=563.3$ nm, and an
intermediate wavelength at the half max $\lambda^{{}^{\prime}}=560.3$ nm of
the resonance. Here, obviously the disparity between $\tau_{inter}$ and
$\tau_{intra}$ is large, clearly because the nonresonant light is not
sustained in the scatterer. Importantly, in this sparse system, the total time
distribution is dominated by the $\tau_{inter}$ (note the difference in the X
axis scale between 2.4[B] and 2.4[C]). The cleaner and more stable spectra in
the intra-scatterer gain system can now be attributed to a larger disparity in
the $\tau_{intra}$ at $\lambda_{res}$ and $\lambda_{nres}$. The modes at
$\lambda_{nres}$ experience only a marginal amplification. Hence, only those
modes which lie at the resonant frequencies are preferentially amplified. On
the other hand, in an inter-scatterer gain system, as the disparity between
$\lambda_{res}$ and $\lambda_{nres}$ is small, occasionally the extended modes
at $\lambda_{nres}$ can also experience gain and contribute in lasing.

###  2.3 Dense scattering system

In the previous section, we discussed a sparse system with a volume fraction
($\phi$) of 1$\%$. Small $\phi$ results in larger $\tau_{inter}$ and hence,
the emission favors the inter-scatterer gain configuration. In this section,
we consider a system with denser packing, which results in reducing the
scattering mean free path. Thus, naturally the $\tau_{inter}$ and
$\tau_{intra}$ distributions are altered. Here, we assume a volume fraction of
$10~{}\%$ which corresponds to a mean free path of about $5-7~{}\mu$m.
Clearly, the diffusion approximation still holds in the system as the mean
free path is several times larger than the wavelength.

Figure 2.5: [A] Left panel: Distribution of inter-scatterer dwell-time of
photons at $\lambda_{res}=559.8$ nm (red) and at $\lambda_{nres}=563.3$ nm
(black). Right panel: Distribution of intra-scatterer dwell-time at
$\lambda_{res}=559.8$ nm, $\lambda_{\text{FWHM}}=560.3$ nm and
$\lambda_{nres}=563.3$ nm. [B] Variation of lasing peak power with the
excitation energy for inter-scatterer (black) and intra-scatterer (red) gain
system with two representative spectra in the inset at the energies marked by
the arrows.

Fig. 2.5[A] shows the distribution of both $\tau_{inter}$ (left panel) and
$\tau_{intra}$ (right panel) at resonant and at nonresonant wavelengths for
the dense system. Here, $\tau_{intra}$ shoots up (few picoseconds) due to the
increased number of scattering events, and the inter-scatterer and intra-
scatterer dwell-times are now comparable. Fig. 2.5[B] further depicts the
behavior of the lasing peak intensity as a function of excitation intensity.
The black curve corresponds to the inter-scatterer gain system. A clear
threshold is observed at $E_{p}~{}\sim~{}0.3~{}\mu$J. The bottom inset shows a
representative spectrum at $E_{p}=0.4~{}\mu$J. The spectra consist of coherent
random lasing peaks riding on a broadband pedestal. With higher excitation
energy, multiple coherent modes start to grow. On the other hand, in the
intra-scatterer gain system, the lasing intensity enhances rapidly even with a
very small excitation energy as shown by the red curve. This is due to the
excessive residence times of the photons inside the amplifying scatterers. At
$\lambda_{res}$, the absorbed energy is rapidly channeled into the resonant
modes and thus the incoherent pedestal is suppressed. Consequently, very
clean, ultra-narrowband coherent random lasing is observed in this system. A
representative spectrum at $E_{p}=0.2~{}\mu$J is shown in the top inset. Upon
further excitation, gain saturation occurs and the maximum intensity of the
coherent peaks shows a saturation behavior. Thus, based on these analyses, we
conclude that very low or almost thresholdless random lasing can be achieved
from a dense ensemble of amplifying resonant scatterers.

###  2.4 A practical proposal

A random ensemble of resonant scatterers has already been used in the context
of random lasers [gottardo08]. In those experiments, photonic glasses
consisting of monodisperse spherical scatterers (diameter $\sim~{}1~{}\mu$m,
polydispersity $<2~{}\%$) were used. The gain was incorporated by adsorbing
the dye molecules on the surface of the spheres. Thus the gain was present on
the surface of spheres. Under optical pumping, diffusive random lasing was
observed that was sensitive to the resonances of the system. In this system,
coherent random lasing was not reported. This could be because the gain was
not present inside the spheres. On the other hand, active scatterers like ZnO
powders can serve the requirement of high gain. However, the polydispersity of
$5-8~{}\%$ inherently present 222This is the expected polydispersity created
in the synthesis process for a particle size of about $1~{}\mu$m. in the
microparticles washes the overall resonant effect of the system. Thus, despite
their high gain, ZnO powders are not suitable for frequency sensitive coherent
random lasing.

Figure 2.6: [A] Scattering efficiency curve ($Q_{sca}$) for a methanol
microdroplet. [B] Red curve: Three calculated spectra from a mondisperse
aerosol for a volume fraction of $10~{}\%$. Black curve: Calculated spectrum
of the aerosol with a polydispersity of $12~{}\%$. [C] Gray and red histograms
are the distribution of the lasing wavelengths from the aerosol system.

To observe frequency control in a coherent random laser, we propose a
potentially realistic system comprised of Mie scatterers. The fact that the
dye molecules have high quantum efficiency in alcohol ($\sigma_{g}\sim
10^{-16}$ cm2) makes them the right material to fulfil the requirement of high
gain. A possible mechanism to create the amplifying scatterers is by creating
microdroplets from the said dye solution. Such microdroplets will have a
refractive index contrast of $1.329$ (of the solvent, with respect to air) and
will offer a high gain for the light resident inside it. A random ensemble of
such microdroplets, technically called an aerosol, will act as an amplifying
disordered medium. We simulated optical transport in such an aerosol system,
and the results are illustrated Figure 2.6. The red curve in Figure 2.6[A]
shows the Mie scattering profile ($Q_{sca}$) of a spherical microdroplet with
$d=3.6~{}\mu$m and $n=1.329$. Compared to the earlier system, the resonances
are broader due to the lower refractive index of these scatterers. Three
representative simulated spectra from a monodisperse system pumped at
$7~{}\mu$J are shown by the red curves in Figure 2.6[B]. The effect of the
broadband nature of the resonance is evident. Multimode emission is observed
at each resonance location. Furthermore, these modes fluctuate within the
broad resonance band. A summary of frequency fluctuations is obtained by
creating a histogram of the lasing frequencies over $75$ spectra. The red
columns are the frequency distribution of lasing modes from the monodisperse
aerosol system. It is clear that the distribution is discretized and the
lasing peaks appear at each resonance (red histogram).

In our calculation, we also studied the effect of polydispersity. The black
line in Figure 2.6[B] is a representative spectrum from the polydisperse
system by assuming a $12~{}\%$ spread in the scatterer diameter (at
$E_{p}=30~{}\mu$J). Note that large excitation energy is required to trigger
the coherent modes, which is also a consequence of the polydispersity. The
gray columns in Figure 2.6[B] are the probability distribution of lasing
frequencies from the polydisperse system. Here, the lasing peaks appear at
random positions, and the distribution is broadband and continuous. This
histogram exemplifies the normal frequency fluctuations seen in coherent
random lasers. In contrast, a monodisperse amplifying scatterer system emits
modes only in discrete frequency intervals. Thus, it can serve to control
frequency fluctuations in a coherent random laser.

Summary: We have simulated coherent random laser emission from monodisperse
amplifying scatterers and presented a comparative behavior of two systems,
namely, inter-scatterer and intra-scatterer gain systems. We find that the
intra-scatterer gain system provides intense, much cleaner and stable random
lasing in a very narrow band of wavelengths, determined by the underlying Mie
resonances. In a system of dense scatterers, the lasing phenomenon starts at
very low excitation energies and the system shows almost thresholdless lasing.
Based on these conclusions, we discuss the possibility of an experimentally
realistic system comprising an amplifying aerosol.

##  Chapter 3 Practical implementation of an aerosol random laser

In this chapter, we follow up our previous simulation work with experimental
demonstration of coherent random lasing from an amplifying aerosol. The
aerosol consists of a linear array of polydisperse, arbitrarily shaped and
randomly placed dye-doped microdroplets. Under optical excitation,
ultranarrow, coherent random lasing is observed in the longitudinal direction
of the array. Spectral imaging and threshold behavior confirms the coherent
participation of each microdroplet. This chapter is organized in the following
way: In section 3.1, we discuss the experimental setup and characterize the
aerosol. We, then, discuss the frequency fluctuations and threshold behavior
of coherent random lasing in section 3.2. Finally, in section 3.3, we compare
the emission properties of the aerosol system with that of Rhodamine dye
filled in a microcolumn.

###  3.1 Experimental setup and characterization of the amplifying aerosol

To generate an aerosol of dye-doped microdroplets, we use a glass
microcapillary attached to a Teflon tube. The other end of the tube is
connected to a liquid chamber, filled with a $1$ mM Rhodamine dye solution
(dissolved in methanol). Using a non-interacting gas ($N_{2}$) supply, the
liquid chamber is pressurized to about $15$ atmospheres which forces the
liquid into the capillary at a constant rate. Under this pressure, the exiting
liquid jet, which is mechanically unstable, breaks into an array of fine
microdroplets.

Figure 3.1: Schematic representation of experimental setup. Legend: MC:
microcapillary, F: filter, S: spectrometer, M1, M2: mirrors, BD: beam dump,
CCD1: CCD for spectral measurement, CCD2: CCD for imaging the scatterer, L1,
L2: lens. Inset shows the schematic to obtain simultaneous spectra from the
transverse (black ray) and longitudinal (red ray) direction.

The schematic of the setup to observe coherent random lasing from the aerosol
is shown in Figure 3.1. The droplet array is excited by the second harmonic of
a Nd: YAG laser ($\lambda=532$ nm, pulsewidth $\sim 25$ ps) using the steering
mirror M1. The beam is focused to a focal spot of $\sim 1$ mm to pump a
sufficient number of droplets ($\sim 20$). Lens L2 (focal length $f=50$ mm) is
used to image the array on an imaging CCD (CCD2). The filter F is used to
attenuate the scattered green light from the droplets. Mirror M2 is placed at
an angle of about $42^{\circ}$ with respect to the axis of the array. This
allows us to redirect the longitudinal emission towards the entrance slit of
an imaging spectrometer (S). Lens L1 ($f=150$ mm) is mounted on an XYZ
translation stage to simultaneously collect the emission in the longitudinal
(along the array axis) and transverse direction (perpendicular to the array
axis) as shown in the inset.

Figure 3.2: [A] Random manifestations of the microdroplet array, the arrays
are uncorrelated to each other. [B] Probability distribution of the
longitudinal size (left panel) and the spacing (right panel) over $500$
realizations of the array; the most probable size and spacing are $25~{}\mu$m
and $40~{}\mu$m respectively.

Figure 3.2[A], shows a few realizations of the highly polydisperse
microdroplet array. It is clear that, every configuration is uncorrelated to
each other. The size and spacing between the droplets were estimated using the
imaging CCD which has a rectangular $1024\times 768$ pixel sensor, the size of
each pixel being $4.65~{}\mu$m $\times~{}4.65~{}\mu$m 111As mentioned in the
camera specifications.. In the experiment, the droplet array was magnified by
a factor of about $10$. By counting the total number of pixels covered by the
droplet, we estimated its size. Note that, the number of pixels occupied by
the image depends on its magnification. For the said magnification,
$1~{}\text{pixel}=4.65/10~{}\mu$m $=465$ nm. Furthermore, as the surface of
the microdroplets in the image does not have a sharp boundary, there is an
uncertainty of about $1~{}\mu$m in the measurement 222Although this
characterization technique gives only approximate estimates, we note that this
chapter does not require exact parameters for analysis. A more rigorous
technique will be described in the next chapter..

Figure 3.2[B], shows the distribution of the longitudinal size of droplets and
the inter-droplet spacing (surface to surface) over $500$ configurations. The
droplet size varied from $10~{}\mu$m to $50~{}\mu$m and the spacing varied
from $5~{}\mu$m to $180~{}\mu$m. The size and spacing depended on various
physical parameters like viscosity of the liquid, flow rate of the solution
and pressure applied on the chamber etc.

###  3.2 Coherent random lasing from the polydisperse droplet array

####  3.2.1 Longitudinal and transverse emission spectra

Two representative spectra of the longitudinal emission at $E_{p}=4.2~{}\mu$J
are shown in the top panel of Figure 3.3. The spectra are multimode and no
periodicity is observed in the coherent peaks. At higher pump energies, the
coherent peaks exhibited a self-averaging behavior. The bottom panel
illustrates the simultaneously measured transverse emission spectra, which
show a multitude of random, low-intense coherent peaks on a broad fluorescence
background. The intensity in the longitudinal direction is approximately $15$
times stronger as compared to the transverse direction. As indicated by the
arrows in Figure 3.3, whenever there is a coherent peak in the longitudinal
direction, there is a corresponding low intensity peak in the transverse
direction. The linewidth of the intense peaks in the longitudinal direction
was measured to be resolution limited to $\sim 0.25$ nm, which corresponds to
a minimum coherence length of $\sim 1.6$ mm. Due to the changing configuration
of the array, the coherent random lasing peaks show shot-to-shot fluctuations
in both frequency and intensity.

Figure 3.3: Simultaneously captured longitudinal and transverse emission
spectra. Whenever there is a peak in the longitudinal direction, a
corresponding low intensity peak is also observed in the transverse direction.
Arrows identify the correlated transverse and longitudinal peaks.

####  3.2.2 Spectral image, threshold and frequency fluctuations

We extract further details of the random lasing modes from spatiospectral
imaging as shown in Figure 3.4. The top panel [A] illustrates the longitudinal
spectrum. Middle panel [B] shows the corresponding longitudinal spectral image
from which the spectrum in [A] is extracted. Two intense spots are observed in
[B] which corresponds to the two coherent peaks in the emission. The bottom
panel [C] is the simultaneously captured transverse spectral image. The
transverse image also consists of many bright spots, which represent the
internal resonances of the microdroplets. Since the droplets in the array are
polydisperse, these intra-particle resonances do not occur at the same
wavelengths. The low-intensity peaks seen in transverse direction (Figure 3.3)
are the result of these intra-particle resonances. We found that the
longitudinal emission does not show any correlation with the intra-particle
resonances, which confirm that the coherent peaks do not originate from the
single microdroplets.

Figure 3.4: Two shot-to-shot spectra from the droplet array, [A] Longitudinal
emission spectra [B] Corresponding spectral image [C] Simultaneously captured
transverse spectral image. Every bright spot in the transverse direction has a
vertical bright streak, which confirms the coherent participation of droplets.
Side panel: magnified image of the area enclosed by the dotted rectangle.

Importantly, every intense spot in the longitudinal direction (seen in [B])has
a corresponding low intensity vertical streak in the transverse emission (seen
in [C]). These are labeled by yellow dots underneath. For clarity, the right
panel shows the magnified image of the region marked by the dotted rectangle.
It is clear that, each vertical streak is extended over the entire pumped
region and there is a perfect positive correlation between the transverse
extended mode and the high intensity longitudinal peak. This observation
confirms that the lasing peak observed in the longitudinal direction
originates from the cooperative behavior of all droplets present in the pumped
array.

Figure 3.5: [A] Frequency behavior of coherent random lasing peaks. The peaks
appear over a range of $10$ nm. [B] Threshold behavior of lasing emission. The
black circles show the longitudinal emission intensity as a function of pump
energy. The blue circles show the probability of lasing. The emission
intensity shows a lasing threshold at $E_{p}=0.6~{}\mu$J.

To study the frequency fluctuations of the random lasing modes, we collected
$500$ spectra. The frequency distribution of the coherent peaks is shown in
Figure 3.5[A]. Like a conventional random laser, the distribution is
continuous and centered at the gain maximum of Rhodamine dye. The width of the
distribution is about $10$ nm indicating that the lasing frequency fluctuates
over $10$ nm. Figure 3.5[B] shows the threshold behavior exhibited by the
aerosol system under excitation. The longitudinal emission intensity (black
circles) has a clear lasing threshold at $E_{p}=0.6~{}\mu$J, after which the
intensity grows linearly on the average, with strong fluctuations. It is known
that, in a coherent random laser, the system does not emit random lasing modes
at each excitation pulse and a probability of random lasing can be measured
[fallert09]. This probability quantifies the ratio of spectra exhibiting
coherent modes to the total number of spectra. In our system, the lasing
probability also shows this threshold behavior at the same energy (blue
circle). We find that at threshold, about $4~{}\%$ of the spectra show lasing.
At a higher pump power ($4~{}\mu$J) the lasing probability rises to about
$25~{}\%$, and beyond that it shows a saturation behavior.

###  3.3 Emission properties of a dye-scatterer random laser in a microchannel
geometry

In this section, we compare the emission properties of the aerosol with the
same dye solution, with suspended scatterers, in a microchannel geometry. We
show that the discretization of the liquid jet in the form of microdroplets
plays a vital role in the coherent random lasing. First, we briefly discuss
the fabrication procedure of a microchannel geometry. To that end, we used a
commercial micropipette puller (Sutter Instruments, P$-2000$). It consists of
a $20$ W CO2 laser with a beam diameter of $3.5$ mm. The CO2 laser beam is
internally redirected by a retro-reflective mirror to provide relatively
uniform intensity on a mounted capillary. A quartz glass capillary (inner
diameter: $700~{}\mu$m and outer diameter: $1000~{}\mu$m) is loaded in the
puller and the laser light scans a specified longitudinal area. During the
scan, the CO2 laser heats and melts the exposed area of the capillary. At the
same time, an internal tension device pulls the capillary along its major axis
which results in the filament formation. The filamentation speed of the
capillary can be controlled by adjusting the velocity of initial tension.
Next, the laser is deactivated and after a preset delay, a hard pull is
applied. Due to the hard pull, the capillary breaks into two separate pieces.
The parameters affecting the size and shape of the pulled capillary include
heating temperature, scanned area, pulling strength, etc.

Figure 3.6: Image of the pulled capillary. The internal diameter of the
capillary was measured to be $24~{}\mu$m.

The image of a fabricated microcapillary is shown in Figure 3.6. The uniform
region of the capillary ($\sim 1$ mm in length) has an internal diameter of
about $24~{}\mu$m and the aperture formed at the tip was measured to be about
$10~{}\mu$m (not shown in the figure). The dye solution was filled by
capillary action. The image of the filled capillary is shown in Figure 3.7[A].
The uniform region of the microcapillary was pumped by the same laser with a
focal spot of about 1 mm. The longitudinal emission spectra at various pump
energies are shown as the black curves in Figure 3.7[B]. The neat dye solution
exhibited normal fluorescence. Next, we randomized the dye solution by adding
ZnO nanoparticles (refractive index $2.0$, size $\sim 10$ nm, concentration
$7.5\times 10^{14}$/cc). The spectra at the same excitation energy are shown
by the red curves in Figure 3.7[C]. Spectral narrowing due to the randomness
is immediately evident from the figure.

Figure 3.7: Longitudinal emission spectra from the dye solution filled in the
microchannel geometry [A] Image of the microchannel [B] Emission spectra of
the pure dye at various pump excitation energy. [C] Emission spectra of the
dye solution with ZnO nanopowder. [D] Spectral features of emission: square
markers- from pure Rh6G/methanol solution, circle markers- with ZnO
nanopowder. Inset shows the corresponding spectrum at $E_{p}$ = $0.53~{}\mu$J.

The longitudinal spectral response of the dye in the microchannel geometry is
quantified in Figure 3.7[D]. The black and the red markers show the respective
response of pure and disordered dye solution as a function of pump excitation
intensity. The pure dye solution has normal fluorescence properties like
steady emission enhancement (see the black squares) and gradual bandwidth
narrowing. At higher power ($1.2~{}\mu$J), the bandwidth was observed to be
$\sim 20$ nm. On the other hand, the disordered dye solution exhibits the
signature of diffusive random lasing. Here, the output intensity shows a rapid
growth and the bandwidth collapses above the threshold energy of
$E_{p}=0.3~{}\mu$J. Importantly, in the microchannel geometry, coherent random
lasing was not observed.This confirms that merely a microcolumnar random
lasing geometry does not realize coherent modes. In the aerosol system, the
breakage of the continuous solution into tiny droplets is responsible for the
coherent lasing modes. Adequate optical feedback assisted by the high gain of
the dye realizes coherent random lasing in the microdroplet array.

Summary:
In summary, we have demonstrated coherent random lasing from an amplifying
aerosol. The emission spectra from the aerosol array consist of ultra-
narrowband random lasing modes. The transverse spectral image confirms that
the random lasing emission arises due to coherent participation of the
droplets. These modes are not the resonances of individual droplet as the
optical feedback is extended over the entire illuminated array. The lasing
emission is strongly directional along the array and exhibits a clear
threshold behavior. Furthermore, the lasing peak fluctuates in both frequency
and intensity due to fluctuating configuration of the aerosol array.

##  Chapter 4 A monodisperse aerosol random laser

In the previous chapter, we studied coherent random lasing from a random array
of polydisperse microdroplets. The lasing action was the consequence of
cooperative participation of all microdroplets present in the pumped region.
We discussed the properties of collective emission like ultranarrow bandwidth,
frequency fluctuations and threshold behavior. In this chapter, we
experimentally study the optical emission behavior of the array when the
amplifying microdroplets are monodisperse, well-separated and randomly
arranged. We find that the collective lasing phenomenon happens at specific
wavelength intervals. The analysis of lasing frequency indicates the
participation of Fabry-Perot resonances of the individual microdroplets. We
further report on the angular distribution and polarization anisotropy of the
collective emission. This chapter is organized as follows: Section 4.1 briefly
describes the generation process of monodisperse microdroplets. In section
4.2, we discuss the well-known Whispering Gallery Resonances of an individual
spherical scatterer and their utility in sizing the droplets. Then in section
4.3, we discuss the frequency quantization in the collective emission. Finally
in section 4.4, we present the general properties of the cooperative emission
like angular distribution, polarization anisotropy and threshold behavior.

###  4.1 Generation of monodisperse amplifying aerosol

The basic process of microdroplet formation using a narrow capillary was
described in the previous chapter. When a liquid is forced out of a narrow
aperture under pressure, the exiting liquid jet, which is mechanically
unstable, breaks spontaneously into an aerosol of fine microdroplets. Without
an external influence, the jet disintegrates into a stream of polydisperse,
irregularly shaped microdroplets. To break the liquid jet into equal sized
droplets, we perturb the liquid stream with an appropriate periodic
perturbation. The drop formation process can be controlled by the frequency
and amplitude of the perturbation. In our microcapillaries, the perturbation
on the stream is done by using an embedded piezo crystal. An electric square
wave signal of $3$ V peak-to-peak at a few hundred kHz applied to the piezo
generates monodisperse microdroplets. In this configuration, the
microcapillary functions as a vibrating orifice aerosol generator (VOAG), a
device which has been traditionally used to study Mie lasers [lin90,
berguland73, qian86, chang96]. The schematic of the experimental setup is
shown in Figure 4.1. The highlighted dotted rectangle marks the VOAG assembly.
PG is the pressure gauge to monitor the pressure applied on the liquid chamber
(C) and MC is the microcapillary having an embedded piezo crystal. The right
panel shows the representative embodiments of the array. The size of the
microdroplets depends on various physical parameters such as viscosity of the
liquid, diameter of the orifice, pressure applied on the liquid, etc. The
droplet diameter ($2a$) can be estimated using the following relation
[berguland73]

$2a=(\frac{6Q}{\pi f})^{1/3}$ (4.1)

where $Q$ is the volumetric flow rate of liquid through the orifice and $f$ is
the driving frequency. Various monodisperse configurations of the array of
droplets with diameter $16~{}\mu$m to $25~{}\mu$m can be achieved by scanning
the frequency ($500-1000$ kHz) and pressure ($10-20$ atmospheres). Figure
4.1[B] shows four representative embodiments of the array which we identify as
weakly periodic (i), aperiodic (ii and iii) and polydisperse (iv) array with
random spacings. Figure 4.1[C] shows the size variation of monodisperse
microdroplets as a function of perturbation frequency at four different
pressures. The size and monodispersity of the droplets are estimated using the
information of Whispering Gallery Modes (WGMs) which is discussed in the next
section. Importantly, at a given frequency and pressure, the array maintains
its monodispersity for a sufficiently long time ($30-40$ minutes), which
allows us to perform our experiments on various configurations with the same
degree of disorder. Figure 4.1[D] shows the distribution of interdroplet
spacing (surface to surface) over $500$ configurations. It is clear from the
histogram that the spacing between the droplets has a continuous distribution
with a spread of over $20~{}\mu$m 111The actual process of characterization is
explained in section4.2. It appears that smaller spacing is more probable than
larger, which could be a consequence of the fluid dynamic generation process.
But this variation is not the focus of our investigation. The size and the
spacing distributions can be changed by varying the perturbation frequency.

Figure 4.1: [A] Experimental setup to observe random lasing from the droplet
array; dotted rectangle highlights the piezo driven micro capillary. [B] Four
manifestations of the array. [C] Variation of droplet sizes as a function of
piezo frequency at various pressures. [D] Distribution of spacing between
nearest-neighbor droplets.

###  4.2 Whispering gallery modes for characterization

In an ideal condition, the VOAG generates highly monodisperse microdroplets
with a very narrow size distribution. By using simple imaging techniques, this
distribution cannot be resolved. In the current study, unlike in chapter 3, we
need a much more accurate method of measuring sizes and spacing of the
microdroplets. To this end, we use the transverse emission spectrum which
consists of Whispering Gallery Modes (WGMs) of the individual microdroplets.
The basic principle of the WGM formation is shown in Figure 4.2[A]. Let us
consider a ray of light (pointed by the arrow) propagating inside a sphere of
radius a with a refractive index of m relative to the ambient medium. If the
ray hits the surface at an angle of incidence $i$ $>$ $i_{c}$ =
$sin^{-1}$($1/m$), the phenomenon of total internal reflection takes place. A
resonance is hit if the totally internally reflected ray undergoes multiple
internal reflections and closes in on itself. In this eventuality, light gets
trapped inside the sphere. For large spheres (diameter $2a\gg\lambda$), the
trapped ray propagates close to the surface and travels a distance of $\sim
2\pi a$ in each round trip. Internal electric field within the sphere is
expected to be large whenever the ray satisfies the resonance condition, $\pi
a=n\lambda$, where $n$ is an integer multiple. Thus, the spectral position of
these resonances depends on the sphere size and its refractive index contrast
with the ambient medium.

Figure 4.2: [A] Schematic representation of the formation of the WGMs [B]
Spectral image of the droplet array, the showing TE and the TM modes [C] Blue
curve: Experimentally observed spectrum from the array; Red curve: Mie
scattering efficiency curve from a single scatterer of $16.5~{}\mu$m with a
refractive index contrast of $1.34$. [D] Field distribution of the TE103,3
mode at the equatorial plane.

Figure 4.2[B] shows the spectral image of a representative monodisperse array,
captured by the spectroscopic ICCD. The image consists of several vertical
bright stripes of alternating high and low intensity. These are the Transverse
Electric (TE) and Transverse Magnetic (TM) WGM modes, depending on whether the
electric or the magnetic field is parallel to the spherical surface. Whenever
the droplets in the array achieve monodispersity, the WGMs of all the droplets
occur at the same wavelength. The broadband emission spectrum of the dye
ensures that several resonant modes are excited. The spacing between two
consecutive TE or TM modes is given by [campillo88]

$\Delta\lambda=\frac{\lambda^{2}}{2\pi a}\Delta x$ (4.2)

where,

$\Delta x=\frac{tan^{-1}\sqrt{(m^{2}-1)}}{\sqrt{(m^{2}-1)}}$ (4.3)

The blue curve in Figure 4.2[C] is the spectrum of the droplet array which is
obtained by the vertical addition of the image. Here, the spacing between two
TE or TM WG modes is $4.81$ nm at a central wavelength of $555.63$ nm. Using
equation 4.2, this corresponds to a droplet size of $16.66~{}\mu$m. The
equation 4.2 is valid for large sphere diameters, and it is of interest to
validate its usage by explicit Mie theoretical calculations.

For a spherical scatterer, the scattering efficiency $Q_{sca}$ is given by the
Mie theory [bohren83].

$Q_{sca}=\frac{2}{x^{2}}\displaystyle\sum\limits_{n=0}^{\infty}(2n+1)(|a_{n}|^{2}+|b_{n}|^{2})$
(4.4)

where, $x$ is the size parameter defined by $x=ka$, $k$ being the wave number
($2\pi$/$\lambda$) in the ambient medium. The infinite summation in the series
can be truncated after $n_{max}$ terms which is given by

$n_{max}=x+4x^{1/3}+2$ (4.5)

The parameters $a_{n}$ and $b_{n}$ are the Mie coefficients and are given by

$a_{n}=\frac{m^{2}j_{n}(mx)[xj_{n}(x)]^{\prime}-\mu_{1}j_{n}(x)[mxj_{n}(mx)]^{\prime}}{m^{2}j_{n}(mx)[xh_{n}^{(1)}(x)]^{\prime}-\mu_{1}h_{n}^{(1)}(x)[mxj_{n}(mx)]^{\prime}}$
(4.6)

$b_{n}=\frac{\mu_{1}j_{n}(mx)[xj_{n}(x)]^{\prime}-j_{n}(x)[mxj_{n}(mx)]^{\prime}}{\mu_{1}j_{n}(mx)[xh_{n}^{(1)}(x)]^{\prime}-h_{n}^{(1)}(x)[mxj_{n}(mx)]^{\prime}}$
(4.7)

where, $\mu_{1}$ is the relative magnetic permeability of the sphere. For
nonmagnetic spheres, $\mu_{1}=0$. The function $j_{n}$ and $h_{n}$ are the
spherical Bessel functions of order $n$ and primes represent the derivatives
with respect to the arguments. To calculate the Mie scattering coefficients
and $Q_{sca}$, we used the Matlab functions by Matzler [matzler02]. The red
curve in Figure 4.2[C] is the plot of $Q_{sca}$ versus wavelength. To get the
best match, we first calculate the scattering efficiency curve for the
experimentally observed size parameters and then shift the curve by tweaking
the particle size. The WGM peak positions show an excellent agreement with the
experimental observation for a particle size of $16.5~{}\mu$m. This value is
within $\sim 100$ nm from the size calculated using equation 4.2, which
suggests the validity of the equation.

As discussed in section 3.1, we used an imaging CCD (CCD2) to estimate the
spacing between the droplet. The precise measurement of the droplet size also
helps us in this regard. Once we know the exact diameter of the droplets, we
can back calculate the effective pixel size of the imaging CCD more accurately
and hence the spacing. Figure 4.2[D] shows the field distribution at a
representative resonance wavelength at $\lambda=555.63$ nm (red color
corresponds to highest field density). From the Mie scattering coefficients,
we identify this WGM as the third order TE103,3 mode. It is clear that the
eigenmode is confined at the surface. The field exists in the form of a ribbon
about the equatorial plane. Roughly, this volume can be estimated to be $\sim
175~{}\mu$m3, assuming the radius vector spans $20^{\circ}$ about the
equatorial plane 222This angle is roughly estimated from FDTD calculation..
Since this is a very small volume, the lasing in the WGM modes is initiated at
very low threshold energies.

###  4.3  Collective lasing from the monodisperse microdroplet array

The emission properties of the monodisperse aerosol in the longitudinal
direction are summarized in Figure 4.3, which presents similar analysis as in
Figure 4.3, but for this monodisperse configuration.

Figure 4.3: Spectra from the monodisperse array, [A] Longitudinal spectra [B]
Two sets ‘a’ and ‘b’ of longitudinal (middle panel) and corresponding
transverse (bottom panel) spectral images, arrows indicate the WGMs. See text
for details.

Figure 4.3[A] illustrates six shot-to-shot emission spectra at a pump energy
$E_{p}=0.55~{}\mu$J. As discussed in section 3.2 both transverse and
longitudinal emission can simultaneously be collected by the lens L1. The
spectra consist of multimode coherent peaks with a resolutionlimited bandwidth
of $\sim 0.2$ nm. The number of modes increases with increasing excitation
energy. Figure 4.3[B] illustrates simultaneously captured spectral images in
both the longitudinal and transverse directions for two different excitation
pulses, labeled as ‘a’ and ‘b’. Within each (‘a’ or ‘b’) data set, the top
panel shows the longitudinal spectrum, the middle panel shows the longitudinal
spectroscopic image, while the bottom panel depicts the transverse
spectroscopic image. The transverse spectral image consists of WGMs (labeled
by the red arrows) and the scattered fraction of collective modes. The WGMs of
all the droplets occur at the same wavelength and remain fixed as long as the
array maintains its monodispersity. Small inclination of the WGMs observed in