DETERMINING THE CAVITY QUALITY FACTOR AND
COUPLING REGIME BY INPUT AMPLITUDE MODULATION
By
GREGORIO MARTINEZ JIMENEZ
Bachelor of Applied Science in Photonics
Benemerita Universidad Autonoma de Puebla
Puebla, Puebla, Mexico
2007
Submitted to the Faculty of the
Graduate College of
Oklahoma State University
in partial fulfillment of
the requirements for
the Degree of
MASTER OF SCIENCE
December, 2011
COPYRIGHT c⃝
By
GREGORIO MARTINEZ JIMENEZ
December, 2011
DETERMINING THE CAVITY QUALITY FACTOR AND
COUPLING REGIME BY INPUT AMPLITUDE MODULATION
Thesis Approved:
Dr. Albert T. Rosenberger
Thesis Advisor
Dr. Bruce J. Ackerson
Dr. Donna K. Bandy
Sheryl Tucker
Dean of the Graduate College
iii
TABLE OF CONTENTS
Chapter Page
1 Introduction 1
2 Optical Microcavities 3
2.1 Whispering Gallery Modes (WGMs) . . . . . . . . . . . . . . . . . . 4
2.2 Microcavity Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2.1 Ring Cavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.2 Dip Depth (m) . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.3 Response Regimes . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.4 Phase Detuning (δ) . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.5 Cavity Quality Factor (Q) . . . . . . . . . . . . . . . . . . . . 13
3 Amplitude Modulation 17
3.1 Types of Modulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Intensity Modulation using an AOM . . . . . . . . . . . . . . . . . . 17
3.2.1 Mathematical Description . . . . . . . . . . . . . . . . . . . . 18
3.2.2 Bragg Diffraction as a Scattering Process . . . . . . . . . . . . 19
3.2.3 Diffraction of an Optical Beam from an Acoustic Beam . . . . 20
3.3 Transient Cavity Response . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Experiment and Analysis 24
4.1 Experimental System . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2 Experimental Procedure . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3 Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
iv
4.3.1 Estimating Q . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3.2 Experimental Quantitative Data . . . . . . . . . . . . . . . . . 31
5 Conclusions 33
References 34
A Code for Mathematica of: The determination of mode frequency
width and quality factor from scan range and rate 38
B Code for Mathematica of: Estimating Q from amplitude modulation 40
Abstract 44
v
CHAPTER 1
Introduction
Optical microresonators have been of great interest to scientists for many years, be-cause
they can serve as sensitive probes for temperature, pressure, and composition
of their environment. For example, optical microresonators have been used in laser
absorption spectroscopy to detect molecules in the ambient gas or liquid [1–4], and
also have been used to detect temperature changes with a very high sensitivity [5–7].
All these applications depend on just a few parameters of the microcavity: its
coupling regime, quality factor, phase detuning, etc. This motivates the objective of
this thesis, which is to present a different method with more accuracy to measure the
coupling regime and quality factor Q for an optical microresonator, using amplitude
modulation.
In chapter two we begin with a description of optical microcavities. We review a
mathematical model for ring cavities, because we use a microsphere (a few hundred
μ m in radius) as an optical microcavity. The microspheres are fabricated by melting
the end of an optical fiber.
In chapter three, since in this method we need an optical modulator to modulate
the intensity from the laser, we review some theory about modulators, especially
acousto-optic amplitude modulators. Also we explain a mathematical model of the
transient cavity response, to finally get an equation for the throughput electric field
when the input is modulated in amplitude.
In chapter four, we explain the experiment where we use a microsphere resonator
in which a whispering gallery mode (WGM) is excited by the modulated input in
1
order to calculate its quality factor Q, for a specific wavelength. In the same chapter,
we explain the procedure followed to do the experiment and we show the quantitative
data measured for some experiments.
In chapter five, we present the conclusions of this work and set some ideas for
future work on this topic.
We include two appendices with the codes for the two Mathematica programs that
were used in estimating the quality factor.
2
CHAPTER 2
Optical Microcavities
Optical microcavities confine light to small volumes by resonant recirculation. The
name microcavities comes from the fact that they are often only tens or hundreds
of micrometers in diameter; therefore the cavity dimensions are on the order of a
relatively small multiple of a light wavelength.
An example of a microcavity is a microsphere, where the cavity boundary works
as a reflecting surface trapping the light inside. We can use a tapered fiber to couple
light into a mode of this microsphere, and by measuring the incoming power from
the fiber Pf and the throughput power Pt we can know the power trapped inside this
microsphere Ps.
Figure 2.1: Microsphere as an example of microcavities, where the red line indicates
laser light, Pf is the incoming power in the fiber, Ps is the power inside the microsphere
and Pt is the throughput power.
Not all wavelengths are going to be an integer submultiple of the circumference of
an experimental microsphere with a fixed optical path. Therefore we use a scanning
wavelength laser to find the right wavelengths to excite whispering gallery modes
3
(WGMs), where each different mode induces a dip in the throughput power with a
Lorentzian shape (to be discussed in more detail in section 2.2.1), where the maxi-mum
trapped power corresponds to the bottom of this Lorentzian function. We call
this particular wavelength the resonance wavelength, and its related frequency the
resonance frequency.
2.1 Whispering Gallery Modes (WGMs)
This phenomenon took its name from the whispering gallery which is located in the
dome of St Paul’s Cathedral, London, and has the property that if two people stand
at opposite sides of the gallery, and one whispers along the wall of the dome, then the
other person can hear what is said. However, if these messages are spoken across the
gallery using the normal volume, the sound waves will diverge and diffract, so that
the received message will be difficult to understand. The reason for this strange effect
is that the sound bounces along the wall of the gallery with very little loss, and so can
be heard at a great distance. In this case we are interested in optical WGMs, where
light is made to bounce around the edge of a fused-silica microsphere. When light is
traveling around the edge of a sphere it will be totally reflected at each bounce, and
propagate with little loss and almost without absorption from the fused silica. The
light will make many circulations around the sphere, undergoing interference with
itself. This means that only whole numbers of wavelengths of light can ’fit’ around
the edge of the sphere. This causes discrete modes, known as WGMs.
2.2 Microcavity Model
The main properties of interest in microcavities are the frequency linewidth (δν),
which is the full width at half maximum dip depth for the mode on resonance, and
the decrease in throughput power for a mode on resonance, that is, the dip depth
(m), defined at zero detuning as:
4
m ≡ Pf − Pt(0)
Pf
. (2.1)
Another important parameter that we are interested in is the coupling regime, specify-ing
response of the microcavity which could be undercoupled, overcoupled or critically
coupled depending on the ratio of coupling loss to intrinsic loss in the cavity.
The last important parameter that we are interested in is the ratio of the average
energy stored in the cavity (U), to the time rate of change in energy per cycle; this
is named the quality factor (Q) for any microcavity:
Q ≡ ω0
U
−dU
dt
, (2.2)
a higher Q indicates a lower rate of energy dissipation relative to the resonance
angular frequency (ω0), so the photons leak out more slowly. Discussions of each of
these properties will be extended in the next sections.
2.2.1 Ring Cavity
For this particular work we are interested in ring cavities, because we are using a
microsphere as an optical cavity. Given the spherical geometry of the cavity and the
planar optical path of a WGM, the most convenient model to be used is a ring cavity
[8, 9]. This choice is justified by the fact that it is possible to choose experimental
parameters of the fiber-microsphere system to ensure that a single fiber mode is
interacting with one WGM of the sphere at a time [10,11]. This ring cavity is treated
as four mirrors, with all but one mirror having perfect reflectance (see Fig. 2.2); In
this analogy we explicitly treat reflection as non phase accumulating and transmission
as accumulating a
2 phase shift. Therefore a field transmitted into the cavity making
any number of round trips on resonance and then transmitted out, accumulates a
total phase shift of π (modulo 2π) with respect to the incident field.
To find the intracavity sphere field Es we consider transmission into the cavity
5
Figure 2.2: Ring cavity.
from the fiber field Ef , after one round trip, at the location just before the field re-encounters
the partially transmitting mirror, and then sum over cavity round trips.
The amplitude transmission coefficient into and out of the cavity is taken to be
imaginary it, the amplitude reflection coefficient of the coupling mirror is real r, the
phase detuning from cavity resonance is δ, and the round trip cavity field loss is αL/2;
here α is an effective intrinsic power loss (scattering + absorption) coefficient, and
L = 2πRs is the circumference of the sphere of radius Rs. Thus the intracavity field
is:
Es = itEf e
− L
2 ei + r(e
− L
2 ei )2itEf + r2(e
− L
2 ei )2(e
− L
2 ei )itEf +
r3(e
− L
2 ei )2(e
− L
2 ei )(e
− L
2 ei )itEf + . . .
= itEf e
− L
2 ei (1 + (re
− L
2 ei ) + (re
− L
2 ei )2 + . . . ). (2.3)
This sum can be written in closed form as:
Es = itEf e
− L
2 ei
Σ∞
n=0
(re
− L
2 ei )n =
itEf e
− L
2 ei
1 − re
− L
2 ei
. (2.4)
6
Now we need to find the total throughput electric field (Et), because this one
would come from the experimental data, therefore we need to sum the reflected fiber
field and the out-of-phase cavity output:
Et = rEf + itEs = rEf − t2e
− L
2 ei
1 − re
− L
2 ei
Ef =
(
r − e
− L
2 ei
1 − re
− L
2 ei
)
Ef , (2.5)
where we have used :
r2 + t2 = 1. (2.6)
The measured throughput power Pt is then:
Pt ≡ Pf
Et
Ef
2
= Pf
(
r − e
− L
2 ei
1 − re
− L
2 ei
)(
r − e
− L
2 e−i
1 − re
− L
2 e−i
)
. (2.7)
Here Pf is the incident power from the fiber. After multiplying and using the Euler
identity for cosine we get:
Pt = Pf
1 + e Lr2 − 2e
L
2 rcos(δ)
e L + r2 − 2e
L
2 rcos(δ)
. (2.8)
In the usual experimental limit, where αL << 1, δ << 1, and T = t2 = 1−r2 <<
1, the dependence of observed throughput power on phase (or frequency) detuning
takes the structure of the negative Lorentzian function shown in Fig. 2.3, like that of
a damped, driven harmonic oscillator [12].
To see this explicitly, we use the following approximations: e L ≈ 1 + αL + ( L)2
2 ,
cosδ ≈ 1 − 1
2δ2 and r =
√
1 − T ≈ 1 − T
2
− T2
8 to rewrite Eq. (2.8) to second order in
small quantities as:
Pt = Pf
( L
2
− T
2 )2 + δ2
( L
2 + T
2 )2 + δ2
= Pf
(
1 − αLT
( L
2 + T
2 )2 + δ2
)
. (2.9)
7
Figure 2.3: Lorentzian lineshape of the analogous ring cavity on resonance.
m
Figure 2.4: Lorentzian structure of an experimentally observed resonance.
8
The observed experimental resonance structure shown in Fig. 2.4 coincides very
nicely with the model of Fig. 2.3; now in place the system can begin to be character-ized
in terms of experimental observables. Once beyond the general structure of the
resonance, which is observed to be Lorentzian, there are two prominent observables:
the decrease in throughput power on resonance (dip depth m), and the full spectral
width at half maximum of the resonance ( Δν).
2.2.2 Dip Depth (m)
To find the dip depth (m) we need to subtract the power throughput at resonance
(Pt(δ = 0)) from the power coming from the fiber (Pf ), and for convenience we
normalize this difference to get a dip depth 0 < m < 1:
m =
Pf − Pt(0)
Pf
, (2.10)
or:
m = 1 − Pt(0)
Pf
. (2.11)
Using Eq. (2.9), Eq. (2.11) can be expressed as:
m =
4αLT
(αL + T)2 . (2.12)
2.2.3 Response Regimes
Now the dip depth can be expressed in terms of the dimensionless variable x as
m =
4x
(1 + x)2 , (2.13)
where x is defined as:
x ≡ T
αL
. (2.14)
9
This has the physical significance that the on-resonance power observed in the
through fiber is a function of the ratio of coupling to intrinsic losses only. Further
physical insight can be garnered by further evaluating the intracavity power enhance-ment
in the sphere as a function of this ratio. We define the dimensionless power
enhancement ξ as the ratio of power within the sphere on resonance to input power,
which is expressed as the square modulus of Es in Eq. (2.4) with δ = 0, divided by
|Ef |2:
ξ =
Te− L
(1 − e− L
2 r)2
. (2.15)
Using the same approximations that led to Eq. (2.9), but in first order, ξ can be
written as:
ξ =
m
αL
. (2.16)
We recognize that the maximum obtainable dip depth is unity and this occurs when
αL = T; since the intrinsic loss is fixed, then we find:
ξmax =
mmax
αL
=
1
αL
. (2.17)
Physically we conclude that the maximum obtainable intracavity power enhance-ment
of the microresonator for any given intrinsic loss will be when the parameter x
is unity or rather when the transmission coefficient and intrinsic cavity loss are equal.
With the knowledge afforded by the parameter x, we can now make a rough anal-ogy
to the classical harmonic oscillator. Just as this type of oscillator [12] has three
distinct regimes of system response based on the strength of damping in the system
(underdamped, critically damped and overdamped) the optical oscillator will likewise
have three distinct damping regimes. However, we must make the distinction that in
our optical system the relevant damping is the coupling loss. With this the response
10
regimes are undercoupled: T < αL; critically coupled: T = αL; and overcoupled:
T > αL. Furthermore, these regimes clearly correspond to x < 1, x = 1 and x > 1
respectively. Notice that the dip depth m is quadratic in the parameter x. This then
leads to two solutions for x given any particular dip depth which is less than 100%.
These two solutions correspond to the under and overcoupled regimes. For further
analysis it becomes convenient to derive an analytic Lorentzian form for the power
spectrum of an observed WGM. Using the help of a new constant (γ) which will be
determined below and assuming that the intracavity power can be expressed in terms
of a familiar optical Lorentzian with resonance centered at zero detuning as [13],
Ps(δ) = Ps(0)
(
2 )2
(δ)2 + (
2 )2 , (2.18)
we can now compare forms with that of the analytic Lorentzian of Eq. (2.9) to identify
the damping parameter γ as:
γ = T + αL. (2.19)
Notice now that Eq. (2.16) for intracavity power enhancement on resonance can
be expressed in a more concise form as:
ξ =
4T
γ2 . (2.20)
Finally, the total expression for the sphere power can be written concisely as:
Ps(δ) = Pf
4T
γ2
(
(
2 )2
δ2 + (
2 )2
)
, (2.21)
and the detuning dependent power enhancement as:
ξ(δ) =
4T
γ2
(
(
2 )2
δ2 + (
2 )2
)
. (2.22)
With the microcavity power spectrum now expressed in a traditional Lorentzian
form the spectral width can be readily evaluated. By inspection the half width of
11
the resonance will be achieved when the denominator of Eq. (2.22) doubles, δ± =
±
2 . This will then produce a full width at half max power width (FWHM) of γ.
Notice that δ is dimensionless in the above expression and as such requires physical
interpretation.
2.2.4 Phase Detuning (δ)
Let us assume a plane wave propagating with angular frequency ω and wave vector k
along a circular path of length L = 2πRs where the cavity radius is again Rs. Further,
let us assume an adjacent natural resonance of the system with characteristics ω0 and
k0 is available in the cavity. The relative phase detuning of the plane wave from the
cavity resonance can then be expressed as the difference between the phases after one
round trip L , or rather,
δ = kL − k0L = (k − k0)L. (2.23)
The wave vector k can now be expressed as the ratio of angular frequency (ω) to
propagation velocity v. This allows the detuning to be expressed in terms of the
cavity modes effective index of refraction neff ≡ c
v and frequency ν as:
δ = (
ω
v
− ω0
v
)L = (
2πneffν
c
− 2πneffν0
c
)L =
2πneffL
c
(ν − ν0). (2.24)
Notice that the pre-factor term is the ratio of the cavity’s optical path length to
propagation velocity and therefore has the physical significance of the time associated
with a single round trip (τrt ≡ neffL/c). The round trip time is also the inverse of the
free spectral range of the cavity δν which is defined as the frequency spacing between
adjacent identical resonant modes, or rather a full rotation in the cosine term in Eq.
(2.8).
12
The FWHM in phase can then be written succinctly in measurable quantities as:
γ = δ+ − δ− = 2πτrt(ν+ − ν0 − ν− + ν0) ≡ 2πτrt(Δν) = 2π
Δν
δν
. (2.25)
However, from the previous analysis we also know the phase spectral width (or
phase linewidth) in terms of the cavity round trip loss. Equating these expressions
we have;
γ = T + αL ≡ 2πτrt(Δν) = 2π
Δν
δν
, (2.26)
solving for the measurable linewidth Δν we have;
Δν =
c
2πneff
(
T
L
+ α
)
. (2.27)
Let us now separate the contributions to the total linewidth into components
caused by coupling (c) and intrinsic loss (i) written in terms of partial linewidths as:
Δνi ≡ cα
2πneff
, (2.28)
Δνc ≡ cT
2πneffL
. (2.29)
The linewidth written in component form then becomes:
Δν = Δνi + Δνc. (2.30)
2.2.5 Cavity Quality Factor (Q)
The cavity quality factor Q can be thought of as the ratio of the energy of an un-damped
oscillation divided by the work done per cycle [14]. Formally, the Q can be
expressed as 2π times the ratio of time-averaged energy stored in the cavity to the
cavity’s energy loss per optical cycle. Mirroring the derivation outlined in [14], if U
13
is defined as the average energy in a cavity then the time rate of change of the energy
is the power dissipated per cycle and the quality factor can be expressed as:
Q = ω0
U
−@U
@t
. (2.31)
This provides us with a first order differential equation which is solved simply as:
∂U
∂t
= −ω0
Q
U,
U(t) = U(0)e
−!0
Q t. (2.32)
We can immediately recognize that the energy will therefore have a characteristic
cavity energy lifetime given by τc = Q
!0
. For further analysis we must now relate the
measured spectral width to the cavity Q. Let us define a cavity field with lifetime c
2
and single angular frequency ω0 as:
E(t) = e
− c
2 te−i!0t. (2.33)
Now taking the Fourier transform of the cavity field to shift to a frequency domain
we find:
E(ω) =
√1
2π
i
(ω − ω0) + i c
2
. (2.34)
By taking the square modulus we find an expression that is proportional to the power
spectrum:
E(ω)∗E(ω) =
1
2π
1
(ω − ω0)2 + ( c
2 )2 =
1
2π
1
(ω − ω0)2 + ( !0
2Q)2 . (2.35)
We can now see that the field initially at a single frequency ω0 has been broadened
by an amount Δω = ω0/Q. Solving for Q we have the bridge to interpret the spectral
width derived from the ring cavity model as:
14
Q =
ω0
Δω
=
ν0
Δν
. (2.36)
This parameter then gives one a dimensionless parameter that can be used to com-pare
resonant responses of cavities with similar or vastly different configurations, for
example a mass on a spring as compared to a capacitor or even a photonic crystal
cavity. In our case using the results of Eq. (2.30) we have;
Q =
ν0
Δνc + Δνi
(2.37)
Observing the structure of Eq. 2.37 allows one to rewrite the total Q into the
contributions due to the coupling and intrinsic losses separately as:
Q =
QiQc
Qi + Qc
, (2.38)
or more traditionally we may also write the inverse of Q as a sum of parallel impedances,
1
Q
=
1
Qc
+
1
Qi
. (2.39)
The contributions may be expressed in the same manner as Eq. (2.37) by sequentially
setting each loss term to zero;
Qi ≡ ν0
Δνi
,
Qc ≡ ν0
Δνc
. (2.40)
From the results of Eq. (2.32) we can express the Q as:
Q = 2πν0τc. (2.41)
With this we can now see the physical meaning of the lifetime by equating Eq. (2.37)
and Eq. (2.41):
τc =
1
2π(Δνc + Δνi)
=
1
2K
. (2.42)
15
As one would expect the cavity lifetime is inversely proportional to the total cavity
loss and this allows us to define a field decay rate K, or rather a power decay rate of
2K.
If dephasing is present the Q derived from spectral width in Eq. (2.36) will not
be the same as that explicitly derived from cavity lifetime as the lifetime is only
dependent on loss, not dephasing. The proper definition of Q is that derived in Eq.
(2.41) while Eq. (2.36) is an approximation, valid when dephasing is not present,
so this must be verified before use. The aforementioned dephasing is the result of
fluctuations in phase while the mode propagates within the cavity. For example, the
round-trip phase δ or time τrt may not have the same value for every round trip of
a photon, clearly leading to a broadening of the resonance without adding more loss.
This could come about due to thermal fluctuations, multiple scattering, or tunneling
into and immediately back out of the coupling fiber.
16
CHAPTER 3
Amplitude Modulation
3.1 Types of Modulator
Depending on which property of light is controlled, modulators are called intensity
modulators, phase modulators, polarization modulators, spatial light modulators, etc.
A wide range of optical modulators are used in very different application areas, such
as in optical fiber communications, displays, and in optical metrology.
An acousto-optic modulator (AOM), also called a Bragg cell, uses the acousto-optic
effect to diffract and shift the frequency of light using sound waves (usually
at radio-frequency). A piezoelectric transducer is attached to a crystal such as lead
molybdate. An oscillating electric signal drives the transducer to vibrate, which
creates sound waves in the crystal. These can be thought of as moving periodic planes
of expansion and compression that change the index of refraction. Incoming light
scatters off the resulting periodic index modulation and interference occurs similar
to that in Bragg diffraction. The interaction can be thought as a four-wave mixing
between phonons and photons.
3.2 Intensity Modulation using an AOM
The refractive index of an optical medium is altered by the presence of sound. Sound
therefore modifies the effect of the medium on light; the amount of light diffracted
by the sound wave depends on the intensity of the sound. Hence, modulation of
the intensity of the sound can be used to modulate the intensity of the light in the
17
diffracted beam. Typically, the intensity that is diffracted into the m = 0 order can
be varied between 15 to 99% of the input light intensity. Likewise, the intensity of
the m = 1 order can be varied between 0% and 80%.
Because the modulation frequency is much less than the optical frequency (Ω <<
ω), the frequencies of the incident and deflected waves are approximately equal (with
an error typically smaller than 1 part in 105). The wavelengths of the two waves are
therefore also approximately equal.
Figure 3.1: Acousto-optic modulator used as an intensity modulator.
3.2.1 Mathematical Description
Since optical frequencies are much greater than acoustic frequencies, the variations of
the refractive index in a medium perturbed by sound are usually very slow in compar-ison
with an optical period. There are therefore two significantly different time scales
for light and sound. As a consequence, it is possible to use an adiabatic approach in
which the optical propagation problem is solved separately at every instant of time
18
during the relatively slow course of the acoustic cycle, always treating the material as
if it were a static (frozen) inhomogeneous medium. In this quasi-stationary approx-imation,
acousto-optics becomes the optics of an inhomogeneous medium (usually
periodic) that is controlled by sound. The simplest form of interaction of light and
sound is the partial reflection of an optical plane wave from the stratified parallel
planes representing the refractive-index variations created by an acoustic plane wave.
A set of parallel reflectors separated by the wavelength of sound Λ will reflect light if
the angle of incidence (θ) satisfies the Bragg condition for constructive interference,
sin(θ) =
λ
2Λ
, (3.1)
where λ is the wavelength of light in the medium. This form of light-sound interaction
is known as Bragg diffraction, Bragg reflection, or Bragg scattering [15].
3.2.2 Bragg Diffraction as a Scattering Process
Light propagation through a homogeneous medium with a slowly varying inhomoge-neous
refractive-index perturbation Δn is described by the wave equation:
∇2E − 1
c2
∂2E
∂t2
≈ −δ, (3.2)
where
δ = −μ0
∂2ΔP
∂t2 = −2μ0ϵ0n
∂2
dt2(ΔnE), (3.3)
is a radiation source proportional to the second derivative of the product ΔnE. For
Bragg diffraction the perturbation Δn is created by the sound wave, so that the
scattering source is dependent on both the acoustic field and the optical field E, which
includes both the incident and scattered fields. One approximate method of solving
this scattering problem, called the first Born approximation, uses the assumption
19
that the scattering source δ is created by the incident field (rather than by the actual
field). Once we know the scattering source, we can solve the wave equation for the
scattered field. Assuming that the incident light is a plane wave:
E = Re
{
Aei(kr−!t)}
, (3.4)
and the perturbation caused by the acoustic wave is a plane wave
Δn = −Δn0cos(qr − Ωt), (3.5)
we substitute into Eq. (3.3), and reorder the terms of the product ΔnE to obtain:
δ = −(
Δn0
n
)(k2
rRe[Aei(krr−!rt)] + k2
sRe[Aei(ksr−!st)]), (3.6)
where ωr = ω+Ω, kr = k+q, kr = n!r
c ; and ωs = ω−Ω, ks = k−q, ks = n!s
c . We thus
have two sources of light frequencies ω ±Ω and wave vectors k ±q, that may emit an
upshifted or down shifted Bragg-reflected plane wave. Upshifted reflection occurs if
the geometry is such that the magnitude of the vector k +q equals nωr/c ≈ nω/c, as
can be easily seen from a vector diagram. Downshifted reflection occurs if the vector
k − q has magnitude nωs/c ≈ nω/c.
The intensity of the reflected light in a Bragg cell is proportional to the intensity
of sound, if the sound intensity is sufficiently weak. Using an electrically controlled
acoustic transducer, the intensity of the reflected light can be varied proportionally.
The device can be used as a linear analog modulator of light.
3.2.3 Diffraction of an Optical Beam from an Acoustic Beam
Suppose now that the acoustic wave itself is a beam of width Ds. If the sound
frequency is sufficiently high so that the wavelength is much smaller than the width
of the medium, sound propagates as an unguided (free-space) wave and has properties
analogous to those of optical beams, with angular divergence of:
20
δθs =
Λ
Ds
. (3.7)
This is equivalent to many plane waves with directions lying within the divergence
angle. The reflection of an optical beam from this acoustic beam can be determined
by finding matching pairs of optical and acoustic plane waves satisfying the Bragg
condition. The sum of the reflected waves constitutes the reflected optical beam.
There are many vectors k (all of the same length 2π/λ and many vectors q (all of the
same length 2π/Λ); only the pairs of vectors that form an isosceles triangle contribute.
If the acoustic-beam divergence is greater than the optical-beam divergence (δθs >
δθ) and if the central directions of the two beams satisfy the Bragg condition, every
incident optical plane wave finds an acoustic match and the reflected light beam has
the same angular divergence as the incident optical beam δθ. The distribution of
acoustic energy in the sound beam can thus be monitored as a function of direction,
by using a probe light beam of much narrower divergence and measuring the reflected
light as the angle of incidence is varied.
The time taken for the acoustic wave to travel across the diameter of the light
beam gives a limitation on the switching speed, and hence limits the modulation
bandwidth. The finite velocity of the acoustic wave means the light cannot be fully
switched on or off until the acoustic wave has traveled across the light beam. So to
increase the bandwidth the light must be focused to a small diameter (waist radius
w0 ≤ 0.5 mm) at the location of the acousto-optic interaction. This minimum focused
size of the beam is required for a high modulation speed.
3.3 Transient Cavity Response
Now we must determine if our treatment of cavity quality factor in terms of spectral
line width (Δν) is reasonable and further we must also verify that we indeed have
three separate coupling regimes. Fortunately, both can be found directly from the
21
cavity’s transient response to an abrupt (sub-cavity lifetime) incident power change.
In all of the preceding treatment we have analyzed a steady-state response; let us
now look into the transient response.
Clearly from the analysis performed for the ring cavity we would expect the intra-cavity
power to decay as a pure exponential and as such we could trivially treat its
transient response to a rapid decrease in pump power. While this is straightforward,
much more information can be garnered from the full response of the throughput
field to an arbitrary amplitude modulation event, meaning that Ef will now be time-dependent
[9]. Let us begin by assuming the cavity round trip losses are small and
we are sufficiently close to resonance such that the sum over cavity round trips in Eq.
(2.4), substituted into Eq. (2.5), can be replaced by an integral over all time in the
fractional time required to reach steady state t′′ ≡ nrtτrt:
Et = rEf − T
τrt
∫ ∞
0
Ef (t − t′′)e−K(1+i )t′′
dt′′, (3.8)
where nrt is the number of round trips, and the field decay rate K and cavity detuning
in units of half linewidths θ = 2 0−
Δ have been inserted explicitly using the relations
found from Eq. (2.24), Eq. (2.27) and Eq. (2.42). To aid in the analysis of the time
dependence of the integral, let us further change variables as t′ = t − t′′. With this
we find:
Et = rEf (t) − T
τrt
e−K(1+i )t
∫ t
−∞
Ef (t′)eK(1+i )t′
dt′, (3.9)
where the limits of integration have been redefined for consistency with the substitu-tion.
We can now take the time derivative of each side to find the differential equation
that describes the transient throughput response as:
22
dEt
dt
= r
dEf (t)
dt
+K(1 + iθ)
[
T
τrt
e−K(1+i )t
∫ t
−∞
Ef (t′)eK(1+i )t′
dt′
]
− T
τrt
e−K(1+i )tEf (t)eK(1+i )t. (3.10)
With simplification and substitution we arrive at the final form of the useful
differential equation for throughput transient response,
dEt
dt
= r
dEf (t)
dt
− K(1 + iθ)(Et(t) − rEf (t)) − T
τrt
Ef (t). (3.11)
When solved numerically with an approximate square-wave input fiber field, we find
not only the field decay rate but also three qualitatively distinct response regions
based on the coupling regime. It is important when solving the above differential
equation that a realistic fiber field switching and modulation (first term on the right-hand
side of Eq. (3.11)) is used and this can severely affect the observed response of
the system when the switching lifetime becomes comparable to the cavity lifetime.
We also solve Eq. (3.11) numerically with a sinusoidal function as input fiber field,
this because we will modulate the amplitude of the input field Ef with a sinusoidal
function using the AOM. The solution to the throughput field Et is a sinusoidal
function too. Its phase shift relative to Ef allows us to determine the coupling regime
and the value of Q, when sinusoidal input modulation drives a microsphere WGM on
resonance.
23
CHAPTER 4
Experiment and Analysis
4.1 Experimental System
The experimental setup, shown in Fig. 4.1, employs a free space light beam from
a tunable diode laser at a wavelength of about 1550 nm. Using a lens this beam is
focused to produce a narrow waist (w0 = 500 μm) at the acousto-optic modulator
(AOM). The AOM is fixed on a rotating table, used to find the best angle for intensity
modulation, then the deflected beam (see Fig. 3.1), is injected into an optical fiber
using a fiber coupler (FC). The other side of this optic fiber is connected to a tapered
fiber and the output optical signal is sent to a power detector (PD) at the end of
the system. Near to the thinnest part of this tapered fiber, we have placed the
microsphere on a translation stage; in this way we can get the throughput intensity
at the end for both cases, when the microsphere is making contact with the tapered
fiber and when it is not making contact.
The laser is frequency scanned across a WGM resonance using a function genera-tor
(FG1). A second function generator (FG2) drives the intensity modulation with a
frequency chosen based on the cavity lifetime τc of the mode. The throughput power
is detected and recorded. The detector signals are captured with the oscilloscopes
which are further connected to a laptop computer for additional data analysis. A
typical WGM has a factor Q of 1 × 108, or a cavity lifetime τc = 80 ns. Since the
AOM, when driven with a square wave, has a rise and fall time of about 25 ns for
w0 = 500 μm, cavity ringdown was not used to measure Q. Instead, a sinusoidal
modulation was used and the phase shift of the throughput, relative to the input,
24
was measured to determine Q.
Figure 4.1: Experimental system. Focusing lens (FL), acousto-optical modulator
(AOM), fiber coupler (FC), function generator driving the diode laser (FG1), function
generator driving the modulator (FG2), microsphere (sp).
4.2 Experimental Procedure
First we determine the dip depth and the frequency width for a chosen mode prior
to modulation. In the throughput power signal displayed on the oscilloscope we take
the difference between the off-resonance voltage and the on-resonance voltage. The
difference divided by the off-resonance voltage is the dip depth m. We find half of
this difference and there we measure the width of the dip which would be in seconds.
Then to get the frequency width we use the computer program shown in Appendix
A; we use this frquency width δν to calculate the factor Q using the next equation;
Q =
c
λδν
, (4.1)
where c is the speed of light and λ is the wavelength. We estimate the factor Q for the
cavity because we will use it to find the best modulation frequency, and also as a first
25
approximation to determine a more accurate cavity quality factor. Experimentally
we found that maximization of the observed throughput phase shift occurs for either
coupling regime when the period is approximately 25 cavity lifetimes (τmod = 25τc).
Using this relation we can calculate the best frequency of modulation (νmod), then set
up a function generator with this frequency and adjust the waveform to a sinusoidal
function and use this function generator to drive the modulator. The best νmod is
determined by the following equation:
νmod =
2πc
25Qλ
. (4.2)
Then the throughput signal from the detector with the microsphere in place is
recorded following this procedure: while scanning we place the mode exactly in the
center of the scan displayed on the oscilloscope. Then slowly the scan width is de-creased
by adjusting the function generator which is driving the laser until the mode
fills the entire oscilloscope window. Then the scanning is stopped and we turn on the
modulation. We make small manual corrections to the laser wavelength until we are
sure that the mode is on resonance (θ = 0). We can tell the modulated signal is on
resonance by observing the transmitted power on the oscilloscope being minimized
(the signal goes down which means the power is getting smaller) as we reach the
bottom of the dip. Now we take the sphere out and record this signal too because
this will be our incoming power. We plot both signals in Fig. 4.2. We can confirm the
dip depth (m) by measuring the distance between the means of these two sinusoidal
signals.
The data of the input and throughput power are then normalized (Appendix B);
to do this for each collection of data, we first subtract the smallest estimated value
of the signal from the experimental data. This offset will put the minimum values of
the power very close to 0 (the minimum power is not exactly 0 because we can just
estimate the minimum value, due to the uncertainty in the power detector). Then we
26
divide both offset signals by the new highest value of the input signal, in order to get
an ideal peak-to-peak power between 0 and 1. The input normalized data will have
a peak-to-peak amplitude of 1 (maybe with a small offset due to the estimation done
before), while the throughput power will have a peak-to-peak amplitude of about
1 − m, dependent on the dip depth m.
We normalize the data in order to remove the large dc background and improve
the precision when comparing with the model. The normalized data are plotted in
Fig. 4.3.
2000 4000 6000 8000 10 000 12 000 14 000
0.22
0.24
0.26
0.28
0.30
0.32
0.34
Figure 4.2: Data from oscilloscope. Blue, incoming power; red, throughput power on
resonance
2000 4000 6000 8000 10 000 12 000 14 000
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
Figure 4.3: Normalized data. Blue, incoming power; red, throughput power on reso-nance.
Finally we find the best fit curve for the incoming power, using a nonlinear model
27
fit (Appendix B) to a sinusoidal function like Eq. (4.3), with three fitting parameters,
the phase shift (φ), the peak-to-peak amplitude (A), and an offset (a) that may be
greater than 1.
Pf =
A
2
(Sin[(2π ∗ νmod ∗ t) + φ] + a); (4.3)
The signal of the power throughput Pt is fit to the same equation Eq. (4.3), using
different variables for the fitting parameters. The fits use the frequency of modulation
from the experiment νmod; both best fit sinusoidal functions are plotted in Fig. 4.4.
2000 4000 6000 8000 10 000 12 000 14 000
0.3
0.4
0.5
0.6
0.7
0.8
Figure 4.4: Best fit curves for: blue, incoming power; red, throughput power on
resonance.
Now we are ready to use the mathematical model.
4.3 Mathematical Model
The differential equation Eq. (3.11) for throughput field is solved numerically in
Mathematica using the same program as in [8] (the code is in Appendix B). This
program needs the following parameters as input: the first calculated Q (from the
linewidth measurement descibed at the begining of the previous section), the fre-quency
of modulation νmod, and the dip depth m for the mode on resonance. We use
the square root of the equation from the best fit curve of the incoming power Fig. 4.4
28
as the input field (Ef ). The model solution gives Et, and its square modulus is then
considered to be the throughput power Pt.
The modeled throughput power is plotted for both regimes by changing the value
nn in the code; if nn = evennumber then we have an undercoupled regime, as shown
in Fig. 4.5; otherwise, if nn = oddnumber we have an overcoupled regime, as shown in
Fig. 4.6. In this case the undercoupled and overcoupled cavity lifetimes are identical,
but the observed phase shifts are significantly different. Then we compare this with
the output best fit data Fig. 4.4 to determine the coupling regime first. In this case
looking to the model we know that we have an undercoupled regime.
0 2000 4000 6000 8000 10 000
0.2
0.4
0.6
0.8
tHnsL
PtHArbL
Figure 4.5: Mathematical model in undercoupled regime (blue input signal, red
throughput signal).
Once we have determined the coupling regime of the experimental data we com-pare
the best fit curve and the mathematical model, for the modulated incoming
power, Fig. 4.7, and for the throughput power, Fig. 4.8. We find that there is a little
dephasing between the model and the best fit line for the throughput power because
we are using the first estimation for Q.
29
0 2000 4000 6000 8000 10 000
0.2
0.4
0.6
0.8
tHnsL
PtHArbL
Figure 4.6: Mathematical model in overcoupled regime (blue input signal, red
throughput signal).
0 2000 4000 6000 8000 10 000
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Figure 4.7: Best fit line from experiment (red) and mathematical model for incoming
power (blue).
4.3.1 Estimating Q
We can calculate a more accurate cavity quality factor (Q) than the initial approxima-tion
by making small changes on the first estimated quality factor in the Mathematica
program. By trial and error and successive solutions of the differential equation, this
is repeated until both graphs match as closely as possible; an example is shown in
Fig. 4.9 where a better estimation of Q was done, and we can observe that the
shift between the curves is smaller than the shift in Fig 4.8 where we used the first
approximation of Q.
30
0 2000 4000 6000 8000 10 000
0.2
0.3
0.4
0.5
0.6
0.7
tHnsL
PtHArbL
Figure 4.8: Best fit line from experiment (red) and mathematical model for through-put
power (blue) with first Q approximation.
0 2000 4000 6000 8000 10 000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
tHnsL
PtHArbL
Figure 4.9: Best fit line from experiment (red) and mathematical model for through-put
power (blue) with best Q approximation.
4.3.2 Experimental Quantitative Data
We performed different experiments following the procedure above. We used the
measured linewidth in microseconds from the oscilloscope to calculate the frequency
width Δν, using the first part of the program in Appendix A, for different modes on
resonance. Different frequencies of modulation (νmod), were used depending on the
different first estimated quality cavity factor Q as we know from Eq. (4.2). We used
the steps from the last section to finally get a more accurate value for Q for each
mode. The results are shown in Table 4.1. We have an uncertainty of ±0.0001 × 108
31
in Q because this is the smallest value where we can see an actual change in the shift
between experiment and model (Fig. 4.9).
Table 4.1: Table of different data
Measured
linewidth
(μs)
Δν (MHz) νmod (kHz) First estimation
of Q
Final estimation of Q
360 3.62 920 0.53 × 108 (0.5582±0.0001)×108
352 3.54 930.01 0.54 × 108 (0.5232±0.0001)×108
270 2.71 890 0.70 × 108 (0.6857±0.0001)×108
32
CHAPTER 5
Conclusions
We have demonstrated that using sinusoidal input amplitude modulation we can find
the coupling regime and determine the cavity quality factor Q with better accuracy
for any mode in any cavity configuration, than for the regular methods where we used
the linewidth of the whispering gallery mode to calculate the cavity quality factor;
see Table 4.1.
In this method we used an AOM to modulate a tunable diode laser of wavelength
λ = 1550 nm. Then we measured the input and throughput signals of a microsphere.
After matching the mathematical model to these two signals, we were able to estimate
a better cavity quality factor Q factor and to recognize the coupling regime for a
certain WGM.
In future work we could use an electro-optical Mach-Zehnder amplitude modulator
because the greater frequency bandwidth of an electro-optical modulator can reduce
the experimental errors, so then a better fit curve can be calculated. Also using
an electro-optical modulator we can modulate with a square function, because of
the faster rise/fall time; this is another method to calculate coupling regimes and
quality factors. We could also confirm that this way to get a more accurate value for
the quality factor is viable for other different microcavity configurations, such as a
cylindrical cavity, etc.
33
34
References
[1] J. P. Rezac, “Properties and Applications of Whispering-Gallery Mode Reso-nances
in Fused Silica Microspheres,” PhD Diss., Oklahoma State University,
Stillwater, OK, 2002.
[2] G. Farca, “Cavity-Enhanced Evanescent-Wave Chemical Sensing Using Microres-onators,”
PhD Diss., Oklahoma State University, Stillwater, OK, 2006.
[3] A. T. Rosenberger, “Analysis of whispering-gallery microcavity-enhanced chem-ical
absorption sensors,” Opt. Express 15, 12959-12964 (2007).
[4] G. Farca, S. I Shopova and A. T. Rosenberger, “ Cavity-enhanced laser absorp-tion
spectroscopy using microresonator whispering-gallery modes,” Opt. Express
15, 17443-17448 (2007).
[5] V. S. Il’chenko and M. L. Gorodestkii, “Thermal nonlinear effects in optical
whispering gallery microresonators,” Laser Phys. 2, 1004-1009 (1992).
[6] T. Carmon, L. Yang. and K. J. Vahala, “Dynamical thermal behavior and
thermal self stability of microcavities,” Opt. Express 12, 4742-4750 (2004).
[7] H. Rokhsari; S. M. Spillane, and K. J. Vahala, “Loss characterization in micro-cavities
using the thermal bistability effect,” Appl. Phys. Lett. 85, 3029-3031
(2004).
[8] E. B. Dale, ”Coupling Effects in Dielectric Microcavities”, PhD Diss., Oklahoma
State University, Stillwater, OK, 2010.
[9] A. T. Rosenberger, (personal communication, 2009-2011).
35
[10] M. J. Humphrey, “Calculation of Coupling Between Tapered Fiber Modes and
Whispering-Gallery Modes of a Spherical Microlaser,” PhD Diss., Oklahoma
State University, Stillwater, OK, 2004.
[11] M. J. Humphrey, E. Dale, A. T. Rosenberger,and D. K. Bandy, “Calculation
of optimal fiber radius and whispering-gallery mode spectra for a fiber-coupled
microsphere,” Opt. Commun. 271, 124-131 (2007).
[12] S. T. Thornton and J. B. Marion, Classical dynamics of particles and systems,
5th ed. (Brooks/Cole, Belmont, CA, 2004), (Chapt. 13).
[13] E. Hecht and A. Zajac, Optics, 4th ed. (Addison-Wesley Pub. Co., Reading,
Mass., 2002), (Chapt. 11).
[14] J. D. Jackson, Classical electrodynamics, 3rd ed. (Wiley, New York, 1999),
(Chapt. 8).
[15] B. Saleh and M. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, John & Sons,
Incorporated, 2007). (Chapt. 19).
36
37
APPENDIX A
Code for Mathematica of: The determination of mode frequency width
and quality factor from scan range and rate
This is a code written to use in Mathematica software, where using a WGM linewidth
in microseconds measured from a throughput dip displayed on an oscilloscope (vari-able
width in the program), we can estimate a value for the cavity quality factor Q.
The other variables in the program are the voltage V and the frequency ν from the
function generator driven the diode laser. In this first part of the code we calculate the
frequency linewidth δν, we used a function derived from a fit to earlier measurements
to calculate the Scan Range.
Mode f r equency width
MHz =. ;
GHz = $1000$ MHz;
width = $352∗10ˆ−6 s$ ;
V = $50∗10ˆ−3$ ;
ν =20 1/s;
ScanRange [ V ] = ( $4 .9333305879661395 ‘ V + 2.0367879203843535 ‘ Vˆ2$ ) GHz ;
SR = ScanRange [V]
Rate = 2SR × ν
δν = width ×Rate
In this part of the code we estimate the quality factor Q using the frequency
linewidth δν given above.
38
MHz =1 × 106; c = 2.998 × 108; λ = 1550 × 10−9; ν = c/λ; Q = ν/δν
39
APPENDIX B
Code for Mathematica of: Estimating Q from amplitude modulation
This is a code written to use in Mathematica software, where using a modulated laser
we can estimate with more accuracy the cavity quality factor Q and coupling regime,
than the method showed in Appendix A. The program is used as described in Chapter
4.
Impor t ing data
datap = Import [
”C: / User s /Gr egor io /Desktop/Te s i sMa e s t r i a / datos / nsphere5 , 3 3 . i s f ” ,
”CSV” ] ;
datapr =
Import [ ”C: / User s /Gr egor io /Desktop/Te s i sMae s t r i a / datos /wsphere5 , 3 3 . \
i s f ” , ”CSV” ] ;
Data o f f r e sonanc e no normal iz ed
datapraw =
MovingMedian [
Table [ { datap [ $ [ i , $ 1 ] ] ∗ 1 0 ˆ 9 − 3700 , ( ( datap [ [ i , 2 ] ] ) ) } , { i , 3950 ,
Length [ datap ] − 2 5 0 0}] , 1 0 ] $ ;
40
P1 = Li s tPl o t [ datapraw , Frame −> True , Axes −> Fal se ,
PlotRange −> {Al l , Al l } , Pl o tSt y l e −> RGBColor [ $0 , 0 , 1$ ] ,
Joined −> True ]
Data on r e sonanc e no normal iz ed
dataprraw =
MovingMedian [
Table [ { datapr [ $ [ i , 1 ] ] ∗ 1 0 ˆ 9 − 3700 , ( ( datapr [ [ i , 2 ] ] ) ) } , { i , 3950 ,
Length [ datapr ] − 25 0 0}] , 1 0 ] $ ;
P2 = Li s tPl o t [ dataprraw , Frame −> True , Axes −> Fal se ,
PlotRange −> {Al l , Al l } , Pl o tSt y l e −> RGBColor [ $1 , 0 , 0$ ] ,
Joined −> True ]
NORMALIZED DATA
Data o f f r e sonanc e ( wi thout mi c rosphe r e ) normal i z ed
datap2 =
MovingMedian [
Table [ { datap [ $ [ i , 1 ] ] ∗ 1 0 ˆ 9 −
3700 , ( ( ( datap [ [ i , 2 ] ] ) − . 3 3 8 ) / . 0 0 6 ) } , { i , 3950 ,
Length [ datap ] − 2 5 0 0}] , 10$ ] ;
P3 = Li s tPl o t [ datap2 , Frame −> True , Axes −> Fal se ,
PlotRange −> {Al l , Al l } , Pl o tSt y l e −> RGBColor [ $0 , 0 , 1$ ] ,
Joined −> True ]
41
Data on r e sonanc e normal iz ed
datapr$2$ =
MovingMedian [
Table [ { datapr [ $ [ i , 1 ] ] ∗ 1 0 ˆ 9 −
3700 , ( ( datapr [ [ i , 2 ] ] ) − . 2 2 2 ) / . 0 0 6 } , { i , 3950 ,
Length [ datapr ] − 25 0 0}] , 10$ ] ;
P4 = Li s tPl o t [ datapr2 , Frame −> True , Axes −> Fal se ,
PlotRange −> {Al l , Al l } , Pl o tSt y l e −> RGBColor [ $1 , 0 , 0$ ] ,
Joined −> True ]
NONLINEAR FIT TO EXPERIMENTAL DATA
Exper imental modulation f o r AOM
νmod = 920 ∗ 103
Nonlinear fit to normalised data without microsphere
P5 = NonlinearModelFit[datap2, j/2 (Sin[( 2π ∗ νmod ×t ∗ 10−9) + a] + b), a, b, j,
t, MaxIterations -¿ 500000, ConfidenceLevel -¿ .99]
P5N = Normal [ P$5$ ]
Show[ Li s tPl o t [ datap2 ] , Plot [P5N, {t , $ 0 , 14000 $ } , Pl o tSt y l e −> Blue ]
, PlotRange −> {Al l , Al l } ]
P5 [ ” Bes tFi tParameter s ” ]
P5 [ ” ParameterEr ror s ” ]
Nonlinear fit to normalized data on resonance
P6 = NonlinearModelFit[datapr2,h/2 (Sin[(2π×νmod×t×10−9)+d]+g), d, g, h, t,
MaxIterations -¿ 500000, ConfidenceLevel -¿ .99]
42
P6N = Normal [P6 ]
Show[ Li s tPl o t [ datapr2 ] , Plot [P6N, {t , $ 0 , 14000 $ } , Pl o tSt y l e −> Red ] ,
FrameLabel −> {” t ( ns ) ” , ”\!\(\∗ Subscr iptBox [ \ (P\) , \( t \ ) ] \ ) ( Arb )”} ,
PlotRange −> {Al l , Al l } ]
P6 [ ” Bes tFi tParameter s ” ]
P6 [ ” ParameterEr ror s ” ]
MATHEMATICAL MODEL
Exper imental input s
c = 3 ∗ 108; λ = 1.55 ∗ 10−6; QL = .533 ∗ 108; θ = 0; νmod = 925 ∗ 103; m = .33;
tc = (λQL)/(2πc); .
K = 1/(2 t c ) ;
(∗nn=odd ove r coupl ed or c r i t i c a l l y coupled , nn=even undercoupled . ∗ )
nn = 0 ;
x = 1/m (2 − (−1)ˆRound [ nn ] 2 Sqr t [ 1 − m] − m) ;
Mathematical model
Ef [ t ] = \ [ Sqr t ] (0.33159043855024817(1.5594453412938587 ‘\+ Sin [2.415304039364746+ Ed = Table [ { t ∗10ˆ9 , Abs [ Ef [ t ] ] ˆ 2 } , {t , 0 , 10/\ nu {mod} , 1/(100 $\nu {mod}$ P7 = Li s tPl o t [Ed , PlotRange −> {Al l , Al l } ,
Pl o tSt y l e −> RGBColor [ 0 , 0 , 1 ] , Joined −> True , Frame −> True ,
Axes −> Fal s e ] ;
s o l u t i o n =
43
NDSolve [ {Et ’ [ t ] == Ef ’ [ t ] − K (Et [ t ] ) − K ( ( x − 1)/( x + 1 ) ) Ef [ t ] ,
Et [ $MachineEpsi lon ] == $MachineEpsi lon } , {Et [ t ] } , {t , 0 ,
10/\[Nu]mod} , MaxSteps −> 2 0 0 0 0 ] ;
EEr [ t ] = Et [ t ] / . s o l u t i o n [ [ 1 ] ] ;
Ed2 = Table[t ∗ 109, Abs[EEr[t]]2, t, 0, 10/νmod, 1/(100νmod)];
P8 = Li s tPl o t [ Ed2 + . 0 8 , PlotRange −> {Al l , Al l } ,
Pl o tSt y l e −> RGBColor [ 1 , 0 , 0 ] , Joined −> True , Frame −> True ,
Axes −> Fal s e ] ;
Show[P7 , P8 ,
FrameLabel −> {” t ( ns ) ” , ”\!\(\∗ Subscr iptBox [ \ (P\) , \( t \ ) ] \ ) ( Arb )”} ,
FrameLabel −> {” t ( ns ) ” , ”\!\(\∗ Subscr iptBox [ \ (P\) , \( t \ ) ] \ ) ( Arb)”
} ]
Best f i t l i n e and model f o r throughput s i g n a l
L1 = Li s tPl o t [{{1985 , 0} , {1985 , . 8 } } , Joined −> True ,
Pl o tSt y l e −> RGBColor [ 0 , 0 , 1 ] , Frame −> True , Axes −> Fal s e ] ;
L2 = Li s tPl o t [{{2013 , 0} , {2013 , . 8 } } , Joined −> True ,
Pl o tSt y l e −> RGBColor [ 1 , 0 , 0 ] , Frame −> True , Axes −> Fal s e ] ;
Determining Q by making the s h i f t sma l l e r
Show[P8 , Plot [ {P6N − . 1 5 } , {t , 0 , 10860}] ,
FrameLabel −> {” t ( ns )” ,” P { t }(Arb ) ” } ]
44
VITA
Gregorio Martinez Jimenez
Candidate for the Degree of
Master of Science
Thesis: DETERMINING THE CAVITY QUALITY FACTOR AND COUPLING
REGIME BY INPUT AMPLITUDE MODULATION
Major Field: Physics
Biographical:
Personal Data: Puebla, Puebla, Mexico on June 30, 1984.
Education:
Received the B.S. degree from BUAP, Puebla, Puebla, Mexico, 2007, in
Applied Physics
Completed the requirements for the degree of Master of Science with a
major in Physics Oklahoma State University in December, 2011.
Experience:
Employed by Oklahoma State University, Dept. of Physics, as graduate
teaching assistant from 2008 to present.
Name: Gregorio Martinez Jimenez Date of Degree: December, 2011
Institution: Oklahoma State University Location: Stillwater, Oklahoma
Title of Study: DETERMINING THE CAVITY QUALITY FACTOR AND COU-PLING
REGIME BY INPUT AMPLITUDE MODULATION
Pages in Study: 44 Candidate for the Degree of Master of Science
Major Field: Physics
We present a technique which allows us to determine the coupling regime and the
cavity Q factor, both important measurements used in microresonators, in a single
experiment, getting better accuracy than the normal technique where we use the spec-tral
width of the whispering gallery mode (WGM). Using an acousto-optic modulator
we modulate a laser with a sinusoidal input intensity, then we use this to excite the
WGMs of a microsphere and with a fast response detector, we detect the input and
output response. Using an oscilloscope we can collect the input and output signals
from the microresonator. Then using a nonlinear approximation we find the best fit of
these data to a sinusoidal function. We find a numerical solution to the mathematical
model for transient cavity response and using the experimental parameters, we gen-erate
the throughput power from the best fit experimental input power. By matching
this mathematical model curve to the best fit experimental throughput data, we can
determine if the coupling regime is undercoupled, overcoupled or critically coupled,
and we can estimate the cavity quality factor (Q). This technique has low uncertainty
compared with the technique used to measure the quality factor Q from a WGM’s
spectral linewidth (Δν).
ADVISOR’S APPROVAL: