Johns Hopkins University -...

Johns Hopkins University

Bachelor of Science Thesis in Physics and

Astronomy

Testing Infrared-blocking Filters usinga Fourier Transform Spectrometer

Zhuo Zhang

supervised byProf. Tobias Marriage

May 14, 2016

Abstract

Inflation is an inflationary stage of the universe moments after the Big Bang which ex-plains the homogeneity, isotropy and flatness of our universe. The theory postulatesdensity perturbations that have both scalar and tensor components, a prediction thatcan be tested by the existence of B-mode signals arising from the Cosmic MicrowaveBackground (CMB). CLASS, with its broadband frequency coverage, has the uniqueability to map the entire B-mode power spectrum of the CMB, including reionizationsignals at large angular scales. Detection of the faint CMB requires Infrared-blockingfilters to be placed at the receiver windows that eliminate IR from the enclosure whiletransmitting in-band signals. Upon improvements that reduced the random and sys-tematic errors, the author utilized the FTS to collect transmission rates of variousfilters which were then compared with simulations. Such measurements enable theCLASS team to select the filters most apt for installation.

Acknowledgements

Need a bit more time to properly thank everyone.

Contents

1 Introduction 21.1 Cosmology Background . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Hubble’s Parameter and the Expansion of the Universe . . . . 21.1.2 Friedmann Equation . . . . . . . . . . . . . . . . . . . . . . . 31.1.3 Multiple Component Universe . . . . . . . . . . . . . . . . . . 41.1.4 Epochs of the Universe . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Cosmic Microwave Background (CMB) . . . . . . . . . . . . . . . . . 8

1.3.1 Formation of CMB . . . . . . . . . . . . . . . . . . . . . . . . 81.3.2 Source of Polarization of the CMB . . . . . . . . . . . . . . . 81.3.3 Statistical Properties of the CMB . . . . . . . . . . . . . . . . 9

2 Cosmology Large Angular Scale Surveyor (CLASS) 122.1 Objective and Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Testing Infrared-Blocking Filters . . . . . . . . . . . . . . . . . . . . . 15

3 Theory 163.1 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.1.2 Frequency Resolution in Real-Life Systems . . . . . . . . . . . 17

3.2 Transfer Matrix Treatment of Optical Films . . . . . . . . . . . . . . 20

4 Diagnosis of the Fourier Transform Spectrometer (FTS) 224.1 FTS Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2 Output Side Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.2.1 Lock-in Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . 244.2.2 Testing with Thermal Source . . . . . . . . . . . . . . . . . . 28

4.3 Input Side Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.4 Voltage Controlled Oscillators . . . . . . . . . . . . . . . . . . . . . . 31

2

Contents

4.4.1 RF Switch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5 Experimentation 365.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.2.1 Sample Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.2.2 Verification of Si-powder Filter Parameters . . . . . . . . . . . 40

6 Conclusion 42

A Full Measurements 43

Bibliography 45

1

Chapter 1

Introduction

1.1 Cosmology Background

1.1.1 Hubble’s Parameter and the Expansion of the Uni-verse

The Big Bang theory the generally accepted model governing the expansion of theuniverse. The redshift in the majority of observed galaxies is strong evidence for theuniverse’s expansion. This phenomenon, undergoing time reversal, suggests that theuniverse grew from a singularity at the time of the Big Bang.

Suppose we observe a wavelength of λem from a galaxy that emits light of wave-length λob, then its redshift z is given by the formula

z ≡ λob − λemλem

. (1.1)

Astronomer Edwin Hubble, from observations in the early 20th century, devisedthe famous linear relation between z and the distance r of a galaxy known as Hubble’sLaw:

z =H0r

c. (1.2)

In the classical, non-relativistic limit z can be approximated by z ≈ v/c such thatHubble’s Law takes the form

v = H0r,

where H0 is Hubble’s constant. Set in the present time, it has a constant value of69.32 ± 0.80 km s−1 Mpc−1 [1]. This so called constant is in fact a parameter that

2

Chapter 1. Introduction

changes over time. For this reason, the parameter at times different from the presentmoment is more aptly titled Hubble’s parameter and is denoted by H.

Hubble’s parameter is a function of the scale factor of the universe a(t). Supposetwo galaxies at the present moment (t = t0) is separated by a distance r(t0). Overtime, their distance is multiplied by the scale factor a(t) such that the distance attime t becomes

r(t) = a(t)r(t0).

The cosmological principle states that the universe is isotropic and homogeneous onscale sizes greater than 100 Mpc. At these scales, every region in the universe hassimilar patterns of voids and superclusters that make one region indistinguishablefrom another. At scales that satisfy the cosmological principle, an observer from agalaxy will see the other galaxy receding at a speed of

v(t) =dr

dt= ar(t0) =

a

ar(t).

A comparison with Hubble’s Law shows that the proportionality constant betweenv and r is H = a

a.

The common phrase “telescopes are like time machines” holds a grain of truthin that the wavelength of light when emitted from a celestial body, λe at time teundergoes a shift of wavelength to λ0 when observed at the present moment (t0) dueto the expansion of the universe. The shift in wavelength is governed by the equation

λea(te)

=λ0

a(t0). (1.3)

Because the redshift and scale factor are related by the equation

1 + z =1

a(te), (1.4)

in a continually expanding universe as ours we can uniquely express the any pointin time te at which the universe was at scale a(te) using the language of redshift z.

1.1.2 Friedmann Equation

The cosmological principle is grounds for generalizing energy and matter propertiesof the universe at large scales. Based on this principle, Alexander AlexandrovichFriedmann, in 1922, derived from General Relativity and Einstein’s Field Equations

3


the Friedmann Equation that links together the scale and curvature of the universeto its energy density [2]. The Friedmann Equation takes the form

H(t)2 =8πG

3c2ε(t)− κc2

R20a(t)2

, (1.5)

where G is the gravitational constant, κ the curvature of the universe, R0 the radiusof curvature, and ε(t) the energy density. The three values that κ can take on —−1, 0 and 1 — respectively denote a negatively curved, flat and positively curveduniverse.

The critical density εc(t) describes the energy density in a spatially flat universe.By setting κ to 0, the critical density can be easily derived from the FriedmannEquation as

εc(t) ≡3c2

8πGH(t)2, (1.6)

used as a benchmark density compared to the actual energy density to gauge atthe curvature of the universe. A density larger than εc(t) is positively curved and auniverse with a smaller density is negatively curved.

The critical density at the present day can be found by setting H(t) to H0,rendering

εc,0 = 5063± 0.67 MeV m−3.

Of particular interest to cosmologists is the density parameter that denotes theratio between the actual density and the critical density:

Ω(t) ≡ ε(t)

εc(t). (1.7)

Using the density parameter, the Friedmann Equation is simplified to the form

κ

R20

=H2

0

c2(Ω0 − 1). (1.8)

1.1.3 Multiple Component Universe

We live in a multiple component universe that comprises matter, radiation and darkenergy.

The type of matter that cosmologists directly observe is baryonic matter, thematter that stars, planets and cosmic dust are made of. Another type of matternaked to telescopes is dark matter. The presence of dark matter was first evidenced

4


by Fritz Zwicky in the 1930s who observed that the Coma cluster waw held togetherby a attractive force too large for visible stars and gas to produce alone [3]. Studiesof rotational velocities of galaxy clusters allowed cosmologists to indirectly predictand measure the distribution of dark matter. Constituents of dark matter are spec-ulated to include an elementary particle known as the axion, primordial black holes,neutrinos, and Weakly Interacting Massive Particles (WIMPs) that only interact viagravity or the weak nuclear force [2].

In a universe where matter is neither destroyed or created, the energy densitydilutes by 1/a(t)3, in proportion to the increase in the volume of the universe. Thedominant interaction among matter is gravitation force that works to keep matterclumped together.

Radiation is another contributing factor to the contents of the universe. Likematter, radiation exerts a positive pressure that contributes to the negative acceler-ation of the scale of the universe. The energy density of radiation in relation to thescale factor is not only attributed to the number of photons per unit volume (n), butthe energy of a photon as a function of a(t). The wavelength of photons grows inproportional to a(t) as shown by Equation 1.3; hence the energy density of photonsεγ = nE = n hc/λ grows as a(t)−4.

The final component to the universe is dark energy. At the inception of itspostulation, Einstein introduced a new variable to the Friedmann Equation calledthe gravitational constant that granted the possibility of a steady-state universethat neither expands nor contracts [4]. However, Einstein conceded that not onlywas the cosmological constant (Λ) a deus ex machina that forced the universe tobe static, it moreover modeled a unstable universe such that minor changes in theenergy density would trigger a runaway expansion or collapse. The cosmologicalconstant was quickly abandoned, but it shed light on the nature of dark energy.We now believe that dark energy exerts a negative pressure that contributes to theaccelerated expansion of the universe [2]. The cause of dark energy is conjecturedto be vacuum energy, its energy density independent of the scale factor. These areattributes possessed likewise by the cosmological constant after which dark energyis modeled.

1.1.4 Epochs of the Universe

The energy density ε(t) can be decomposed into radiation, matter and dark energycomponents, denoted by εr, εm and εΛ. Per discussion in the previous section, thedensities of each component exhibits time dependencies of εr = εr,0/a

4, εm = εm,0/a3

and εΛ = εΛ,0.

5


Decomposing the energy density in this manner, we may rewrite the Friedmannequation in terms of respective density parameters (Eqn. 1.7) as

H2

H20

=Ωr,0

a4+

Ωm,0

a3+ ΩΛ,0 +

1− Ω0

a2(1.9)

The widely acknowledged Concordance Model provides observational data on theenergy density of each component that may be substituted into Equation 1.9. Theobserved density parameters in the present-day universe is listed in Table 1.1.

List of Ingredientsphotons: Ωγ,0 = 5.0× 10−5

neutrinos: Ων,0 = 3.4× 10−5

total radiation: Ωr,0 = 8.4× 10−5

baryonic matter: Ωbary,0 = 0.05nonbaryonic dark matter: Ωdm,0 = 0.23total matter: Ωm,0 = 0.28cosmological constant: ΩΛ,0 = 0.72total density: Ω0 ≈ 1.00

Table 1.1: List of Ingredients in the Concordance Model. Density parameters showthat the universe is nearly spatially flat and is at the present dominated by matterand dark energy. Radiation density taken from Ryden; matter and dark energydensities from Nine-year WMAP [1][2].

Substituting the parameters of the Concordance Model into Equation 1.9, onemay numerically compute the scale factor a(t) as a function of time. Shown inFigure 1.1, the universe in this model undergoes three distinct epochs: radiationdomination at the earliest times, matter domination at intermediary periods, anddark energy domination at the last stage. It is interesting to note that we live at thetransitional period between matter and dark energy domination.

1.2 Inflation

Physicist Alan Guth predicted that even before the radiation domination period theuniverse underwent an inflationary stage known as Inflation that was only momentsafter the Big Bang. This theory is significant for solving three quandaries unexplain-able by the Concordance Model.

6


Figure 1.1: Scale Factor as a Function of Time according to the Concordance Model.The scale factor grows in proportion to t1/2 at the radiation domination epoch, t2/3

at matter domination and exponentially at dark energy domination. The time ofradiation-matter and matter-Lambda euality are denoted by trm and tmΛ. Figurecourtesy Barbara Ryden.

Firstly, Inflation solves the ”flatness problem.” Data from the Concordance Modelpoint to a nearly flat universe but fails to offer a valid explanation for the occurrence.The flatness of space is a consequence of the inflationary expansion at this epoch.

Secondly, Inflation solves the ”horizon problem.” The fact that different portionsof the universe too far away to have been in causal contact with one another underthe Concordance Model have similar temperatures is explained by the minute size ofthe universe before Inflation that small enough to permit causal contact.

Thirdly, Inflation solves the ”monopole problem.” The magnetic monopoles thatwere created at the time of Inflation are believed to be diluted to levels unobservableby modern instruments [5].

The existence of Inflation and insights to its dynamics can be tested by thehypothesized primordial gravitational waves that arose in the epoch. The amplitudeof the waves is proportional to rate of expansion of the universe during Inflation andalso to the squared of the energy scale of Inflation. These signature traits render them

7


the ultimate probe to Inflation. Primordial gravitational waves with large enoughamplitudes are detectable by polarizations in the Cosmic Microwave Background(CMB) and have thereby motivated large-scale efforts in the astronomical communityto detect polarizations and temperature anisotropies in the CMB.

1.3 Cosmic Microwave Background (CMB)

1.3.1 Formation of CMB

The temperature of the early universe was high enough to ionize hydrogen atoms.This took place until a redshift of z = 1, 300. The early universe before prior toz = 1, 300 was occupied by an opaque soup of ionized hydrogen and electrons. Theopaqueness of the universe is due to the high density of ionized hydrogen of 1/a(t)3 ≈13003 or roughly 2 billion times that of the present universe such that photons couldonly travel a very short distance before scattered by an electron. The period at whichprotons and electrons combine to form neutral atoms is called the recombination.

The point at which the universe become transparent to photons occurred shortlyin the wake of recombination at a redshift of z = 1, 100. The first CMB photonsarrive from this period of decoupling from a spherical shell called the last scatteringsurface.

The formation of CMB takes places at another stage much later at roughly 200million years after the Big Bang in a period called reionization. This is when thefirst stars formed to reionize hydrogen. CMB photons at this epoch travel at a muchgreater mean free path and thereby contribute to polarizations at large angular scales.

1.3.2 Source of Polarization of the CMB

The polarization properties of the CMB, as will be shown in a later section, isimportant for characterizing the amplitude of primordial gravitational waves. Beforedelving into the statitiscal properties of the CMB, a discussion of the source ofpolarization is warranted.

Two mechanics must take place simultaneously in order to polarize the CMB:Thomson scattering and anisotropy.

Thomson scattering is the scattering of photons from different directions onto thesame electron to form a linearly polarized photon. Figure 1.2 illustrates the process.

Anisotropy is the second ingredient for polarization. Before recombination theabundance of Thomson Scattering had the effect of making radiation incident on elec-

8


Scattering.png

Figure 1.2: Thomson Scattering. Unpolarized light of different intensities scatter onan electron from different angles. As a consequence the scattered photon is linearlypolarized. Figure courtesy Wayne Hu.

trons isotropic, reducing the CMB polarization. Anisotropies come from two sources:the stochastic background of gravitational waves, and from density perturbations.

1.3.3 Statistical Properties of the CMB

Statistical properties of CMB temperature anisotropy yield a trove of valuable infor-mation. The correlation between the temperature fluctuation ∆T/T at two pointson the sky is described by the correlation function C(θ) that averages the productof temperature fluctuations at all points in the sky separated by the angle θ:

C(θ) =

⟨δT

T (n)

δT

T (n′)

⟩n·n′=cosθ

(1.10)

As the correlation function is defined over the celestial sphere, it is instructive toexpand it into a spherical harmonic series:

C(θ) =1

4π

∞∑l=0

(2l + 1)ClPl(cosθ), (1.11)

where Pl are the Legendre polynomials. The power of this expansion comes from thefact that Cl is a measure of temperature fluctuation on an angular scale roughly of180/l. This decomposition makes evident the contribution of different periods tothe CMB at different angular scales.

9


It is customary to plot the multipole l against a parameter ∆T which is definedas:

∆T ≡(l(l + 1)

2πCl

)1/2

〈T 〉. (1.12)

As shown in Figure 1.3, the shape of this plot provides a wealth of information aboutthe universe during recombination and reionization.

Figure 1.3: Multipole Expansion of Temperature Fluctuations in the CMB. Thereionization bump occurs at l ∼ 6 and the recombination peak at l ∼ 100. Themultipole l is related by the angular scale by l ∼ 180/θ. Figure courtesy BarbaraRyden.

Gravitational waves, as mentioned above, are one of the two ways of polarizingCMB that is of particular interest to cosmologists. They contribute to the tempera-ture anisotropy of the CMB at l ∼ 6 at reionization and l ∼ 100 at recombination.The amplitude of gravitational waves is characterized by the tensor to scalar ratior, the ratio between temperature anisotropies produced by gravitational waves andthose by density perturbations at l = 2.

Polarization of the CMB can be described by a degree of polarization P and an

10


angle (α) between the direction of polarization and a specified direction on the sky.Light can be shed on the statistical properties of polarization were one to expressthese two parameters via the Stokes parameter Q and U, defined by Q ≡ P cos(2α),U ≡ P sin(2α).

Figure 1.4: E and B mode polarization. E modes exhibit reflection symmetry whileB modes do not.

One may decompose the Q and U components of the polarizations at each pointinto what are known as E and B mode polarizations (Fig. 1.4). E mode polarizationsare symmetric across a line around its center, while B modes lack reflection symmetry.

Characterizing the polarization in this fashion proves useful for the detection ofprimordial gravitational waves. Polarizations in the CMB is generated from eitherdensity perturbations or gravitational waves. Those from gravitational waves lackreflection symmetry and hence produce B-modes [5][6]. Detection of B-modes istherefore crucial for probing the properties of gravitational waves.

11

Chapter 2

Cosmology Large Angular ScaleSurveyor (CLASS)

2.1 Objective and Strategy

The Cosmology Large Angular Scale Surveyor (CLASS) is a telescope array de-signed to measure B-mode polarization from the CMB. CLASS will operate at fourfrequency bands centered at 38, 93, 148 and 217 GHz and span multipoles that coversboth the reionization bump and the recombination peak. It is to be installed at theAtacama Desert in Chile at its completion.

CLASS is uniquely equipped with the technology to overcome barriers to de-tecting B-mode polarization generated from primordial gravitational waves, amongwhich include gravitational lensing of E-modes into B-modes and polarized Galacticemission from dust and synchrotron sources. Ground-based experiments as CLASSalso need to take into account atmospheric loading.

Gravitational lensing is the distortion of the path of light as it travels near amassive object such as a black hole. This complicates the detection of primordialgravitational waves as E-modes from the CMB can be lensed into B-modes whentraversing through massive objects as blackholes or galaxy clusters.

Figure 2.1 illustrates the complications of gravitational lensing to the detectionof primordial B-modes. The power of gravitational lensing remains roughly constantat l < 1000 before rolling off. However, the power from gravitational lensing exceedthat of primordial B-modes at large multipoles. The exact crossover value at whichthe lensing amplitude exceeds that of primordial B-modes is contingent on the tensor-to-scalar ratio r. Unique to CLASS is its sharp sensitivity at low multipoles able todistinguish primordial B-modes from lensed B-modes at l < 10 for a tensor-to-scalar

12

Chapter 2. Cosmology Large Angular Scale Surveyor (CLASS)

ratio as small as 0.001 [7].

Figure 2.1: B-mode Power Spectra Versus Multipole. Predicted power spectra forr = 0.001, 0.01, and 0.2 are shown. Horizontal lines show the upper limit of rat different multipoles from various experiments. The power spectra at large ls asmeasured by BICEP2, POLARBEAR and SPT experiments are shown here with1 − σ error bars. The dashed line is the power spectra of lensed B-modes. The redline designates CLASS sensitivity, lowest at large angular scales. Figure courtesyThomas Essinger-Hileman.

Polarized galactic foregrounds are a major source of contamination of primordialB-modes. At low frequencies, the dominant contaminant is from relativistic elec-trons accelerated by magnetic fields that produce synchrotron emission. At higherfrequencies the main contaminant is thermal emission from dust aligned with mag-netic fields. The two foregrounds are minimized at around 65 GHz. Shown in Figure2.2, CLASS receivers straddle frequencies at the foreground minimum. These bandsare also chosen to avoid strong atmospheric emission lines.

13


Figure 2.2: Multipole versus Frequency Coverage of Various Experiments. CLASSspans multipoles that cover the reionization bump at the lower end and the reion-ization peak. They in-band frequencies straddle the minimum of the combined dustand synchrotron minimum. Figure courtesy Thomas Essinger-Hileman.

Figure 2.3: Atmospheric Transmission Rates at CLASS Frequeny Bands. CLASSbands were chosen to avoid prominent oxygen and water emission lines. Figurecourtesy David Chuss.

The atmosphere is a dominant source of noise for ground-based experiment evenat premier millimeter-wave astronomy sites as the Atamaca Desert. CLASS choosesfrequency bands that avoid oxygen and water emission lines at the typical precipita-tion conditions in the region (Fig. 2.3) [8].

14


2.2 Testing Infrared-Blocking Filters

The author’s work involves testing the effectiveness of different infrared-blocking fil-ters placed before the cryogenic receiver windows of the CLASS telescopes. Cryogenicreceivers are required to detect the minute polarizations in the CMB, let along theB-mode polarizations that are at least a factor of r smaller than the total polariza-tion. The receiver on the focal plane, for example, is cooled down to a mere 70 mKsuch that minute fluctuations in temperature will offset the receiver’s functionality[7].

Infrared-blocking filters must be placed between the receiver window and the focalplane. We expect 40 W of infrared power incident of the receiver window. Whenreaching the focal plane, this power needs to be reduced to 100 -µW . The filtersmust thereby reduce the IR power by a factor of 400,000, while transmitting most ofthe in-band microwave signal [7].

A multitude of IR-blocking filters are being tested. A good candidate is thereflective metal mesh filter with grids lengths on the order of 100 µm. Anothercategory is the absorptive filter such as Nylon or polyethylene filters. The largeaperture size of the CLASS focal plane poses a challenge for absorptive heatersto cool down sufficiently to prevent emitting blackbody radiation in the IR range.Additionally, a type of scattering filter was manufactured at the CLASS laboratoryby distributing ultra-pure silicon powder in a thermoplastic substrate [9].

The efficacy of these filters are tested using a Fourier Transform Spectrometer(FTS) at the CLASS laboratory.

15

Chapter 3

Theory

3.1 Fourier Transform

3.1.1 Definitions

Raw interferograms from experiments are often plotted as a function of signal am-plitude over time or space. Fourier Transforms are performed to convert from theoriginal domain to the frequency domain to gauge at the signals frequency distribu-tion. Without loss of generality, we assume the the Fourier Transform is done on thespatial domain as this is the case in our experiments. It is formally defined as

F (ω) =1

2π

∫ ∞−∞

f(x)e−iωxdx. (3.1)

As F (ω) is a complex number, Fast Fourier Transform algorithms typically outputits modulus.

The inverse Fourier Transform converts from the frequency domain back to theoriginal spatial domain:

f(x) =

∫ ∞−∞

F (ω)eiωxdω. (3.2)

The Fourier Transform can be thought of as an unitary operator such that FF−1(F (ω)

)= F (ω)

and F−1F(f(x)

)= f(x). This property is evident if one explicitly works out the in-

tegrals for the unitary transformation and takes note of one particular representationof the Dirac-Delta function:

δ(k) =1

2π

∫ ∞−∞

e−ikxdx. (3.3)

16

Chapter 3. Theory

A concept closely related to the Fourier Transform is the Convolution. Theconvolution of two functions f and g is defined as

(f ? g

)(x) ≡

∫ ∞−∞

f(τ)g(x− τ)dτ. (3.4)

It is easy to show that interchanging the order of the two function produces the sameresult.

3.1.2 Frequency Resolution in Real-Life Systems

In real-life systems it is not possible to integrate over the entire spatial domain fromx = −∞ to x =∞. A consequence of the limited spatial range is the loss of frequencyresolution. This phenomenon can best be explained through convolution.

The product of two functions in one domain, when Fourier Transformed, becomesa convolution of the transformed functions:

F(f(x)g(x)

)=

1

2π

∫ ∞

−∞

(∫ ∞−∞

F (ω′)eiω′xdω′)(∫ ∞

−∞D(ω′′)eiω′′xdω′′

)e−iωxdx

=1

2π

∞∫−∞

∞∫−∞

F (ω′)D(ω′′)( ∞∫−∞

ei(ω′+ω′′−ω)x dx)dω′ dω′′

=1

2π

∞∫−∞

∞∫−∞

F (ω′)D(ω′′)(

2πδ(ω′ + ω′′ − ω))dω′ dω′′

=

∞∫−∞

F (ω′)D(ω − ω′)dω′

≡(F ? D

)(ω)

(3.5)

Likewise, the Fourier Transform of a Convolution in one domain is 2π times the

17

Chapter 3. Theory

product of the functions in the other domain:

F((f ? d

)(x))

=1

2π

∞∫−∞

( ∞∫−∞

f(τ)d(x− τ) dτ)

e−iωx dx

let u = x− τ such that

LHS =1

2π

∞∫−∞

f(τ)e−iωτ( ∞∫−∞

d(u)e−iωudu)dτ

=1

2π

(2πF (ω)

)(2πD(ω)

)≡ 2πF (ω)D(ω)

(3.6)

In a real-life system with limited spatial range, the raw interferogram f(x) ismultiplied by what is known as the apodization function d(x). An interferogramproduced by a moving mirror will be cut off at the begin and end position of themirror. The observed interferogram may be seen as a multiplication by the trueinterferogram f(x) multiplied by a square pulse that acts as the apodization functiond(x).

Multiplication of f(x) and d(x) in the spatial domain, from Eqn. 3.5, transformsinto a Convolution of F (ω) and D(ω). These transformations are coined the truefrequency spectrum and the instrument function. With respect to the above example,the Fourier TransformD(ω) of the square pulse is Sinc function which when convolvedwith the true spectrum distorts it.

To examine this effect, suppose we have two monochromatic light sources atfrequencies ω1 and ω2, which at an instant in time are respectively

cos(ω1x) and cos(ω2x).

Their Fourier Transforms are Dirac-Delta functions shifted by ω1 and ω2:

1

2

(δ(ω − ω1) + δ(ω1 − ω)

)and

1

2

(δ(ω − ω2) + δ(ω2 − ω)

).

The convolution of a Dirac-Delta function with the instrument function D(ω) isa shifted version of the instrument function itself. Illustrations in Figure 3.2 showthat when frequencies ω1 and ω2 lie too close to each other they become hard orimpossible to distinguish. It is for this reason that real-life systems have a limitedfrequency resolution.

18

Chapter 3. Theory

Figure 3.1: Frequency spectrum of monochromatic light of frequencies ω1 and ω2 inthe positive ω axis.

(a) Distinguishable fre-quencies

(b) Barely distinguishablefrequencies.

(c) Indistinguishable fre-quencies.

Figure 3.2: Frequency resolution of Sinc function.

Different instrument functions distort the frequency spectrum in different ways.The instrument function of a true spectrum is a Dirac-Delta function. Therefore,in real-life systems we look for instrument functions that as closely resemble theDirac-Delta function as possible. Often times, however, a compromise must often bemade between the sharpness of the peak of the instrument function and the size ofits sidelobes, meaning that the resolution is traded off with the smoothness of theplot. For instance, a triangular apodization function of the same width and peakheight as a square pulse transforms into Sinc2 with a peak height half of that of theSinc function transformed by the square pulse. The triangular apodization functiontransforms into an instrument function with a lower resolution that a square pulse,but has the advantage of smaller sidelobes the distort the frequency spectrum.

Candidates for apodization functions that strike balances among the peak height,width and size of sidelobes include the triangular function, Gaussian, and the Happ-

19

Chapter 3. Theory

Genzel functions1 [11].

3.2 Transfer Matrix Treatment of Optical Films

Many of the filters used in CLASS can be treated as single or multi-layered filmsunder normal incidence to the IR. Each film is thought of having an uniform mediumbetween two boundaries from which light enters and escapes. Using boundary condi-tions specified by Maxwell’s equations under normal incidence, one may describe theinput and output relations of the electromagnetic waves the transfer-matrix equationEI

HI

=

cosh(Ω + ik0h

)sinh

(Ω + ik0h

)/Υ

sinh(Ω + ik0h

)×Υ cosh

(Ω + ik0h

)EII

HII

. (3.7)

The parameters in Equation 3.7 are listed in Table 3.1.

List of Parameters in Equation 3.7EI , HI Incoming electric and magnetic fieldsEII , HII Outgoing electric and magnetic fieldsk0 Wave number of incident lightd Thickness of filmn0 Refractive index of surrounding mediumn Refractive index of filmγ attenuation factorh = n× dΩ = γ × dΥ0 =

√ε0µ0× n0

Υ =√

ε0µ0× n

Table 3.1: List of Parameters for Characteristic Matrix for Optical Films

The above 2 × 2 matrix is the characteristic matrix for optical films M. Asthis matrix comes into use for transmission calculations, it’s useful to abbreviate its

1A more detailed account of apodization techniques is described in T.Wei. ”The Design, Con-struction, and Testing of a Wide-Band Fourier Transform Interferometer.” 2012. Johns HopkinsUniversity.

20

Chapter 3. Theory

matrix elements as

M =

[m11 m12

m21 m22

](3.8)

The power of this matrix notation comes into play when multiple layers of filmsare stacked onto one another. If P is the number of layers, each with characteristicmatrix Mi, the subscript specifying its layer number, then the fields from the firstand last boundaries are related by the matrix equation:EI

HI

= MIMIIMI . . .MP

EP+1

HP+1

(3.9)

The ratio of the complex amplitude of the outgoing to incoming wave yields thecomplex transmission rate. If the filter contacts air at both sides, then at normalincidence the complex transmission rate as a function of frequency is reduce to:

t(k) =2 ∗Υ0

Υ0m11 + Υ20m12 +m21 + Υ0m22

. (3.10)

The physical transmission T, a real number between 0 and 1, is the modulus of thecomplex transmission [12].

21

Chapter 4

Diagnosis of the Fourier TransformSpectrometer (FTS)

4.1 FTS Overview

A Fourier Transform Spectrometer (FTS) is utilized in CLASS to test its opticalelements. In the author’s project, filters are tested under microwave signals at30 − 50 GHz and 70 − 90 GHz that reproduce the in-band frequencies of CLASS.The microwave signal goes through a Martin-Puplett interferometer that encodesthe signal’s amplitude onto the position of a moving mirror on the interferometer.The interferogram, upon a Fourier Transform, then encodes frequency informationof the signal. The frequency response of IR-blocking filters may be tested whenplaced inside the interferometer. The FTS needs to be produce and detect sufficientmicrowave signals at the CLASS in-band range to accurately test the efficacy ofIR-blocking filters. A setup of the interferometer is displayed in Figure 4.1.

The Martin-Puplett interferometer uses the wave optics mechanics to encodefrequency information onto the amplitude of the output wave. The incoming signalto the interferometer is split into two beams that reflect off of either a stationary ormoving dihedral mirror. When the beams are recombined and plane polarized, itsamplitude varies with the path difference between the stationary and moving mirrors.A non-polarized monochromatic beam generates an outgoing wave with amplitude:

< E0 >=A2

4(1 + cos∆), (4.1)

where ∆ is the phase difference of the diverging beams [10]. The Martin-Puplettinterferometer is the centerpiece of the FTS that splits the device into an input sideand an output side.

22

Chapter 4. Diagnosis of the Fourier Transform Spectrometer (FTS)

Figure 4.1: Martin-Puplett Interferometer Setup.

The purpose of the input side is to produce signals of desired amplitude andfrequency. Microwave signals are generated by voltage controlled oscillators (VCOs)generate frequencies up 10 GHz tunable by a control voltage. To generate a fre-quency band and not just a single tone, the control voltage is swept back and forthby a function generator. The VCOs alone cannot produce frequencies at the desiredin-band range, so the frequencies must be further tuned up by an Active MultiplierChain (AMC), a microwave source that inputs to the interferometer. Before con-necting the VCOs to the AMC, the signal first goes through a decoupler the splitsthe signal to two outputs – the secondary output leading to a microwave frequencycounter, the primary to a radio-frequency (RF) switch. The RF switch multiplexesthe signal with a square wave of 10 Hz that is synced to the lock-in amplifier on theoutput side before connecting to the AMC. The mechanics of the lock-in amplifierwill be explained in a later section.

The output side is responsible for detecting and digitizing the output signal. Italso works with the input side to further amplify the signal to large enough scales. ALow Noise Amplifier (LNA) receives the microwave signal and converts it to a voltagesignal that reflects the amplitude of its microwave input. The voltage signal from theLNA feeds into the lock-in amplifier, designed to read signals locked-in to frequencyof the RF switch. The lock-in amplifier outputs to the LabVIEW Data-Acquisition(DAQ) software for data analysis.

A diagnosis of the FTS was called into action when the lock-in amplifier failed to

23


Figure 4.2: Block diagram of the FTS. A Martin-Puplett interferometer encodes thefrequency information onto the amplitude of the signal and divides the FTS into aninput and output side.

detect microwave signals. Many components have interdependent input and outputrelations that confound the source of error. It was therefore crucial to design diagnosisthat test each suspect component in isolation using known functional equipment.These experiments could be split into the output and input sides.

4.2 Output Side Testing

4.2.1 Lock-in Amplifier

The lock-in amplifiers is a highly sensitive device with the ability to detect highlyminute and noisy signals. The only requirement is that users know beforehand thefrequency of the signal.

The frequency ”lock-in” feature is achieved by a electronic device called the phase-sensitive detector (PSD). As the RF switch multiplexes the input signal with a squarewave, it has a dominant frequency equal to the first harmonic of the square, denotedas ωr. The function generator that produces the square wave is synced to the lock-inamplifier which in turn commands a PSD to multiplex the incoming signal with a sinewave of frequency ωL. In theory, ωL should equal ωr, but may differ in phase. Theproduct of two sine waves with frequencies ωL and ωr , by trigonometric identities,is also the summation of two phase shifted sine (cosine) waves with frequencies equal

24


to the summation and difference of the two frequencies:

Vpsd = VsigVLsin(ωr t+ θsig)sin(ωL t+ θref )

=1

2VsigVLcos

([ωr − ωL

]t+ θsig − θref

)−1

2VsigVLcos

([ωr + ωL

]t+ θsig + θref

), (4.2)

where Vsig is the amplitude of the external signal, VL that of the reference signal,and θsig and θref the phase of the external and reference signals.

The proximity of ωL to ωr thus produces a DC addend. Using high roll-off low-pass filters, all input removed from the lock-in frequency are removed. The low-passfilters are RC filters with tunable time constants and roll off rates. The time constantis Tfc = 1/(2πftc), ftc being the −3 dB roll-off frequency. Beyond ftc, the filteredoutput rolls off at rates at multiples of 6 dB per octave as determined by the user.

The PSD-coupled and filtered signal, Vpsd = 12VLVsigcos

(θsig − θref

), is nonethe-

less a function of the phase difference. The dependence on phase is removed whenintroducing another PSD with a phase shift of 90 which produces a filtered outputof Vpsd2 = 1

2VLVsigsin

(θsig − θref

).

Calling the output of the first PSD X and the second Y , we may combine theseoutputs to obtain a phase independent output R just as one obtains the modulus ofa vector from combining its x and y components:

R =√(

X2 + Y 2)

=1

2VsigVL. (4.3)

25


Figure 4.3: SR830 Block Diagram. The external input is coupled to phase shiftedsine wave signals at the lock-in frequency by two-phase sensitive detectors (PSDs).The input is amplified twice – once at the AC gain before passing through the PSDsand low-pass filters, and again at the DC gain upon being filtered. Figure fromSR830 manual.

The maximum voltage of the external input to the SR830 lock-in amplifier isspecified by a tunable parameter called the sensitivity. A signal with amplitude ofthe maximum sensitivity is amplified to 10 V, so the lower sensitivity the larger theamplification. As decreasing the sensitivity increases the precision of the readout,the sensitivity should be set to as low as possible without causing an overload. Itcan be set to as low as 2 nV.

The signal undergoes a two-stage amplification process. At the first stage, itundergoes an AC gain that amplifies the input before transmitting through the PSDsand low-pass filters, amplifying both signal and noise. The second amplification takes

26


place at the DC gain after having non-lock-in frequency components removed.The two-stage amplification is a mechanism that contributes to the detection of

minute signals under large noise. A single AC gain becomes problematic for inputswith low signal-to-noise ratios (SNRs). Suppose a signal of 5 mV is tainted with60 dB of white noise as large as 5 V. A high AC gain will overload the amplifier withvoltage from noise; thereby it is necessary to introduce a DC gain that comes intoeffect after the noise is removed. The lower the SNR, the more amplification the DCtakes charge of.

Dynamic reserve is a parameter that controls the lowest allowable SNR. Noiselevel around the lock-in frequency ωr is limited by the roll-off rate of the low-passfilters, while noise at far-away frequencies is limited by internal noise. Numerically,the dynamic reserve is the largest possible noise-to-signal ratio (opposite of SNR)possible. The SR830 can have a reserve level as high as 100 dB.

Figure 4.4: Dynamic Reserve. The reserve is the largest noise-to-signal ratio in unitsof decibels. At frequencies nearby the lock-in frequency ωr the reserve is limited bythe roll-off rate of the low-pass filters, and at far-away frequencies limited by internalnoise of equipment. Figure from SR830 manual.

It may seem that the reserve should be set to as high as possible, but a highreserve presents a trade-off. At high dynamic reserves, the lock-in amplification isskewed to the DC gain in anticipation of large noise. Because the input of the DCgain is minimally amplified by the AC gain, the signal could small enough to becomparable to the internal noise of the DC amplifier. With a larger AC gain, theinternal noise of the DC amplifier is overshadowed by the input signal and so doesnot pose as a problem. Consequently, the reserve should be set the lowest valuetolerable by the SNR [13].

27


4.2.2 Testing with Thermal Source

As it was not immediately evident where the problem lay, the LNAs are tested inisolation first using a thermal source.

Assumptions need to be made on which pieces of equipment can fall under the”safe” category. To be considered functional, an item should either be diagnosed so,or are robust enough to be trusted without a full diagnosis. For instance the lock-inamplifier is a robust piece of equipment with overloading warnings and troubleshoot-ing features that safeguard it against malfunction. The thermal source likewise fallsinto this category for its simplicity – it is simply a radiator that emits blackbodyradiation (Fig. 4.5). If it glows it works.

Figure 4.5: Thermal Source. Radiation from aperture modulated by a chopper withfrequency synced to the lock-in amplifier.

The energy density per unit photon energy emitted by a blackbody source fallsunder the Planck spectrum:

µ(ε) =8π

(hc)3

ε3

eε/kT − 1, (4.4)

where T is the temperature in Kelvins, ε the photon energy, k the Boltzmann constantand h Planck’s constant.

One arrives at the total energy density by integrating the above profile over theentire energy range:

U

V=

8π5(kT )4

15(hc)3. (4.5)

28


To calculate the power emitted by the thermal source, we model it as a blackbodyinside a box with an aperture (Fig. 4.6). Photons that presently escape from anaperture of area A lay at a previous time in a hemispherical shell of thickness cdtand of radius R from the aperture. The angle a chunk of radiation makes with theperpendicular bisector of the aperture is denoted as θ which revolves from 0 to π/2.The polar angle φ, not shown in the figure, is the angle of revolution around theperpendicular bisector that ranges from 0 to 2π.

Figure 4.6: Modeling the Thermal Sources as Blackbody Radiation Escpaing Boxwith Aperture. Picture courtesy Daniel Schroeder.

In these coordinates, the volume of a infinitesimally small chunk of shell is

volume of chunk = (R dθ)× (R sinθ dφ)× (c dt).

The energy in said chunk is the volume multiplied by the total energy density (Eqn.4.5), so that

energy in chunk =U

Vc dt R2 sin θ dθ dφ.

We imagine the energy in each chunk to uniformly disperse in all directions, sothe energy that makes it out the aperture is the apparent area of the chunk as seenby an observer from the aperture over the total area of an imaginary sphere of radiusR:

probability of escape =A cosθ

4πR2

.

The total energy that escapes through the aperture within the time interval dt is

29


found by integrating θ and φ over the hemispherical shell:

total energy escaping from shell =

2pi∫0

dφ

2π∫0

dθA cos θ

4π

U

Vc dt sin θ

=A

4

U

Vc dt. (4.6)

The power emitted from the aperture is found by first dividing by the time intervaldt and substituting U

Vfor the expression in Equation 4.5. It has the simple expression:

P = σAT 4, (4.7)

where σ, the Stefa-Boltzmann constant, takes on the numeric value ofσ = 5.67 × 10−8Wm−2K−4 [14]. To find the power from a certain regime, theprofile in Equation 4.4 is integrated at the lower and upper bounds of the regime,and substituted into the energy density in Equation 4.6.

The thermal source is heated to 800 K, has an aperture size of 0.5 in. radius,and bypasses the Martin-Puplett interferometer to directly point at the LNA re-ceiver in order to reduce losses in transmission to a rate small enough to assumeno transmission loss. The transmitted power from the thermal source as calculatedfrom blackbody radiation is compared to the values received by the LNA, as shownin Table 4.1. That the experimental and expected values are on the same order ofmagnitude in this coarse test shows that the LNAs are functional.

Thermal Source Simulations and Experiment ComparisonThermal SourceInput (W)

Expected LNAOutput (V)

Experiment(V)

40 GHz 9.79 × 10−10 2.15 × 10−5 8.80 × 10−6

90 GHz 7.48 × 10−9 1.02 × 10−4 6.02 × 10−5

Table 4.1: Thermal Source Simulation and Experiment Comparison. The radiationfrom the thermal source at the 30− 50 GHz and 70− 90 GHz regimes is calculatedand compared to experimental measurements of the LNA. Because the experimentalresults are on the same order of magnitude as the expected results the LNAs areshown to be functional.

The voltages, though on the correct scale, jitter from time to time. An initialconjecture attributed the jitter to a noisy thermal source, motivating the author’s

30


research on thermal noise calculations. In his paper Thermal noise and correlationsin photon detection, Jonas Zmuidzinas formulates an equation for the thermal photonnoise, which at the Rayleigh-Jeans limit may be approximated as

σP =kT0√∆ντ

∆ν, (4.8)

where ∆ν is an infinitesimal frequency bandwidth to integrate over, and τ the inte-gration time [15]. Because noise at different frequencies are independent, the totalnoise over a certain regime can be found by integrating the σ2

P over the frequncyrange and taking the square root of the quadrature. At a integration time of 1 s inthe 30 − 50 Ghz range, the SNR of the thermal source is at an astoundingly largevalue of 60 dB. The high SNR eliminates thermal noise as the culprit behind thevoltage jitters.

An inspection of the lock-in amplifier revealed that it was set to high dynamicreserve when a low reserve is warranted for high fidelity thermal sources. A highreserve comes into use when dealing with extremely noisy inputs, but unless so,setting to high reserve levies a great deal of amplification on the DC gain, whichwhen dealing with feeble microwave signals can considerably contaminate it withthe DC amplifier’s internal noise. Configuring the lock-in to low reserve helped inreducing the voltage jitters.

The VCOs on the input side was found to be another contributor to the jitters.

4.3 Input Side Testing

4.4 Voltage Controlled Oscillators

The VCOs of model type HMC-C029 and HMC-C030 manufactured by Analog De-vices specify a output frequency that increases almost linearly with the tuning volt-age. Readings from the microwave counter (Fig. 4.7) , however, suggest otherwise.

31


Figure 4.7: Mismatched VCO Frequency Readings. Not only do the frequencies jitterin the mid tuning voltage range, they also follow a downward trend.

With much trial-and-error, the author discovered that the decoupler used tosplit the VCO output had a secondary output to low to be stably detected by themicrowave counter.

The observed Jitters from the VCOs were mitigated but not fully eliminated aftersidestepping the decoupler. This remaining jitter, after resolving transmission issues,reflects reflects the VCOs intrinsic noise that turned out to be cased by overheating.Jitters were resolving after installing RF components onto a heatsunk platform (Fig.4.8).

32


Figure 4.8: Heatsunk platform for RF components. Installed components includethe VCOs, decoupler and RF switch.

Two diagnoses independently verify the linear response VCO output to the tun-ing voltage as specified by the data sheets. The first test records on-screen readingsfrom the microwave counter (Fig. 4.9a). A second test Fourier Transform the in-terferogram of the FTS at a single tuning voltage. From the frequency spectrum(Fig. 4.9b), we not only see an increase dominant frequency with respect to thetuning voltage but also incidentally an increase in frequency amplitude. A detaileddiscussion of FTS experimentation techniques is covered in the next chapter.

33


(a) VCO Frequency Readings from Mi-crowave Counter

(b) Frequency Spectrum of C029 VCO atconstant tuning voltages

Figure 4.9: Two diagnoses of VCO output both confirm the linearity of frequencywith respect to tuning voltage as specified by data sheets.

4.4.1 RF Switch

The HMC-C011 RF Switch, controlled by a square wave generator, is tested for itsoutput frequency and amplitude.

The margin for error for frequency is governed by the roll-off frequency ftc of thelow-pass filters on the lock-in. It is a parameter interchangeable with Tfc = ffc×2π.The larger the time constant the smaller the noise bandwidth, and the cleaner themeasurements. On the flip side, low-pass filters exhibit signals delays proportional toTfc, so as the linear stage moves at a rate of 1 mm s−1, Tfc must be small enough tokeep up with the changes in the signal. Empirical observations of the interferogramshow that a time constant of 100 ms is sufficient for our purposes. This time constantrenders a −3 dB roll-off frequency of 1.59 Hz, the maximum deviation from the lock-in frequency before the signal becomes largely attenuated. The RF switch outputis made detectable by a microwave detector hooked to an oscilloscope. The testindicates a output frequency well within the confines when tuned at 10 Hz.

34


Figure 4.10: RF Switch Test. Switch output is of correct frequency but insufficientamplitudes.

The same test additionally measured an output amplitude that was orders ofmagnitude smaller than the switch’s specifications (Fig. 4.10). A replacement of theswitch brought the FTS back into good shape.

35

Chapter 5

Experimentation

5.1 Methodology

When performing Fourier Transforms on disrectized interferograms, one needs to takeinto account the finite frequency range and frequency resolution. These parametersare determined by the spatial resolution and spatial range in a criss-cross fashion(Fig. 5.1) in a quantifiable way.

Figure 5.1: Interdependence Between Spatial and Frequency Domains.

Before a discussion on how the spatial resolution determines the frequency range,it is prudent to first clarify the meaning of spatial frequency. Equation 4.1 rendersthe output amplitude of a monochromatic source as a function of the phase difference∆ of light, expressible from simple geometry in terms of the spatial path difference

36

Chapter 5. Experimentation

between the mirrors ∆d and the wavelength of light λ:

∆ = 2× 2π × d

λ.

The factor 2 in the front accounts the light to travel to and from the mirror whicheffectively doubles the path length.

The spatial period Tspace is defined as the path difference ∆d that yields a phasedifference ∆ of 2π. The spatial frequency fspace, the reciprocal of Tspace, relates tothe frequency of light flight by the equation

1

Tspace= fspace =

2

λ=

2

cflight.

The spatial frequency may be thereby converted to the true frequency of light at aconversion ratio of c/2.

The Nyquist frequency, or the frequency corresponding to one half of the spatialresolution, gives the maximum resolvable frequency for discrete samples. The DAQsamples the data very fast, but at slightly irregular rates. Only by binning thesamples at regular intervals can one perform a Fast Fourier Transform. The rawdata undergoes additionally discretization as we average the data at each bin ontoa single point. Choosing a bin size of 0.5 mm−1, the maximum resolvable lightfrequency is given by

fmaxlight =1

2× c

2× bin size = 150 GHz.

Section 3.1.2 demonstrates that the larger the spatial range the sharper the fre-quency resolution (∆flight). The linear stage that mounts the moving mirror coversa range of 120 mm. At a spatial range of 120 mm covered by the linear stages thatmounts the moving mirror, the frequency resolution, formally defined as the spacingbetween frequency samples, takes on the value

∆flight =fmaxlight

Spatial Range/Bin Size=

c

4

1

Spatial Range= 0.63 GHz.

The transmission rate, an important property of filter recoverable from the fre-quency spectrum, is regarded as the ratio of frequency amplitudes with and withoutapplying a filter:

T =AFilterANoFilter

. (5.1)

37


The random error accounted by error propagation takes the form

σ2Trans = T 2 ×

(σ2NoFilter

NA2NoFilter

+σ2Filter

NA2Filter

), (5.2)

where N is the number of runs. Random errors can be confined to a few percentwhen averaged over 15 runs.

5.2 Results

The filters under test at the 40 GHz and 90 GHz regimes are listed in Table 5.1. Fullresults are displayed in the Appendix.

List of Filters

150 µm Metal Mesh Filter300 µm Metal Mesh Filter400 µm Metal Mesh Filter

Si Powder Filter (no anti-reflective coating)Polyethylene (PE) window

Table 5.1: List of Filters Tested at 40 GHz and 90 GHz regimes.

5.2.1 Sample Plots

Figure 5.2 shows a sample interferogram and its frequency spectrum of the 150 µmmetal mesh filter tested at 90 GHz. The green bands in the frequency spectrummark the highest intensity regions that yield the most meaningful measurements.

38


(a) Raw Interferogram (b) Frequency Spectrum

Figure 5.2: Sample Interferogram and Frequency Spectrum of 150 µm Metal MeshFilter at 90 GHz.

Transmission rates with 1 − σ error bars are plotted in Figure 5.3. Measure-ments outside the green bands are discarded as they lie beyond the dominant outputregimes of the AMC. The sample filter displays > 80% in-band transmission andtotal transmission rate > 90% rate. These numbers fluctuate across various filtersand frequencies, but are all surpass roughly 60% at the in-band regions and havetotal transmission capping 79%.

Figure 5.3: Sample Transmission Plot. The sample displays > 80% transmission atin-band regions and > 90% overall transmission.

Occasionally the in-band transmission rates would exceed 100% at levels muchhigher than the error bar span (Fig. 5.4). While these occurrences are rare, the may

39


be the result of systematic errors that must be closer scrutinized. The readings areoccasionally observed to drift at very rates so slow that they are observed only bycomparing the DC offset of the interferogram at different runs. These drifts may alsoalter the frequency spectrum.

Figure 5.4: Transmission Rate with Outlier for 400 µm Metal Mesh Filter at 90 GHz.Outlier that overwhelmingly exceeds 100% suggests possible systematic errors thatloom large.

5.2.2 Verification of Si-powder Filter Parameters

Of particular interest is the experimental verification of parameters of the Si-Powderfilter developed by former lab mate Fletcher Boone, comprised of silicon powders dis-tributed in a polymethylpentene (PMP) substrate, designed to transmit microwavesignals but attenuate infrared. It’s attenuation factor γ is governed by the formula

γ = Nπr2Qsca, (5.3)

where N is the scatterer concentration, r the radius of Si pebbles, and Qsca the Miescattering efficiency that is a function of scatterer concentration, radius, wavelengthand refractive index of the PMP substrate and Si [9].

Most of the parameters such as the refractive index of PMP and Si are wellunderstood, leaving the scatter concentration and radius to be verified. Modelingthe Si-powder as an optical film, its attenuation factor as a function of N and rcan be substituted into Equation 3.7. Though the formula for Mie scattering is veryinvolved, a general trend emerges as one sweeps these two parameters – scatterer

40


concentration N governs the cut-off frequency and phase shift of oscillations andscatter radius r the amplitude of oscillations. Shown in Figure 5.5, the parametersare best matched to experimental results through inspection. For a 9 mm thick filterthe values take on estimate values of N = 2 × 106 m−3 and r = 4 × 10−5 m.

(a) Si-powder Filter Transmission Simu-lation with Best-Fit Parameters (b)

Figure 5.5: Simulated verses Experimental Transmission Rate for Si-powder Filter.Experimental transmissions shows oscillations that from close to unity to roughly85% with troughs at roughly 25 GHz and 37 GHz and peaks at 30 GHz and 43 GHz.N and r are chosen to best fit simulation to experiment.

41

Chapter 6

Conclusion

The transmission rate of various IR-blocking filters were tested at microwave regimesresembling the CLASS in-band range. The design criteria for IR-blocking filters areto reduce IR by a factor 400, 000 while minimizing in-band losses. The FTS, withthe current data binning size, can only sample to a maximum frequency of 150 GHzthat falls under the IR range. Even were it able to sample at a higher frequency,the error bars are likely to be too large to determine IR transmission rates at thedesired scale. Under the assumption that IR can be eliminated, the focus of theexperiment was to test CLASS in-band transmission. Among the 5 filters tested atthe two regimes, the 150 µm metal mesh filter displays the most promising results.Its total transmissions rate under respectively the 40 GHz and 90 GHz tests are91.93% and 97.65%. At the in-band range of the AMC source, its transmission at asingle frequency consistently stays above 75%. These rates are believable in that allin-band data points with error bars lie below or at most cross the 100% transmissionrate. Rare instances that show data points overwhelming exceeding unity suggestsystematic errors to be further scrutinized to further enhance the credibility of thedata.

42

Appendix A

Full Measurements

(a) 150 µm Metal Mesh at 40 GHz (b) 150 µm Metal Mesh at 90 GHz

(c) 300 µm Metal Mesh at 40 GHz (d) 300 µm Metal Mesh at 90 GHz

43

Appendix A. Full Measurements

(e) 400 µm Metal Mesh at 40 GHz (f) 400 µm Metal Mesh at 90 GHz

(g) Si-powder at 40 GHz (h) Si-powder at 90 GHz

(i) PE Window at 40 GHz

Figure A.1: Plots of All Filter Measurements

44

Bibliography

[1] Bennett, C. L.; et al. “Nine-year Wilkinson Microwave Anisotropy Probe(WMAP) observations: Final maps and results.” The Astrophysical Jour-nal Supplement Series 208 (2): 20. 2013. arXiv:1212.5225. doi:10.1088/0067-0049/208/2/20.

[2] Barbara Ryden. Introduction to Cosmology (First Edition). Boston, Addison-Wesley (2002).

[3] Fritz Zwicky. “Die Rotverschiebung von extragalaktischen Nebeln.” HelveticaPhysica Acta, Vol. 6, p. 110-127.

[4] Albert Einstein. Kosmologische Betrachtungen zur allgemeinen Relativitatstheo-rie (Cosmological Considerations in the General Theory of Relativity). KoniglichPreußische Akademie der Wissenschaften, Sitzungsberichte (Berlin): 142–152.1917.

[5] J. Bock et al.. “Task force on cosmic microwave background.” eprint arXiv:astro-ph/0604101. 2006.

[6] Wayne Hu. Intermediate Guide to the Acoustic Peaks and Polarization. Written2001. http : //background.uchicago.edu/ whu/intermediate/intermediate.html

[7] Thomas Essinger-Hileman; et al. “CLASS: The Cosmology Large Angular ScaleSurveyor.” Proceedings of the SPIE, Volume 9153, id. 91531I 23 pp. 23. 2014.DOI: 10.1117/12.2056701

[8] David Chuss; et al. “The Cosmology Large Angular Scale Surveyor(CLASS) Tele-scope Architecture.” 2014.http : //ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20140007513.pdf

[9] Boone, F. I.; 2014. “Silicon Powder Filters for Large-Aperture Cryogenic Re-ceivers.” Bachelor of Sciences Dissertation. Johns Hopkins University.

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[10] Martin, D.H. and Puplett E. “Polarised Interferometric spectrometry for themillimetre and submillimetre spectrum.” Infrared Physics, 1969, Vol. 10, pp.105-109. Great Britian, Pergamon Press.

[11] “Fourier Transform and Apodization.”http : //www.shimadzu.com/an/ftir/support/tips/letter15/apodization.html

[12] Eugene Hecht. Optics, Fourth Edition. New Jersey, Pearson Education (2004).Chapter 9.7.1.

[13] “About Lock-In Amplifiers.” Stanford Research Systems.http : //www.thinksrs.com/downloads/PDFs/ApplicationNotes/AboutLIAs.pdf.

[14] Daniel Schroeder. An Introduction to Thermal Physics (First Edition). NewJersey, Pearson (1999). Chapter 7.4.

[15] Jonas Zmuidzinas. “Thermal noise and correlations in photon detection.” ApplOpt. 2003 Sep 1;42(25):4989-5008.

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