an Apparatus for Detection of Nitric Oxide
through Faraday Modulation Spectroscopy
Master Thesis in Engineering Physics
Faraday modulation spectroscopy (FAMOS) is a technique for detection of paramagnetic molecules. By applying a magnetic field over a gaseous sam-ple, the presence of paramagnetic species will rotate the polarization plane of light, addressing a transition in such a species. By placing a gas containing paramagnetic molecules between almost crossed polarizers and modulating the magnetic field, the intensity of the transmitted light will consequently be modulated. Since the rotation of the polarization plane of light is propor-tional to the concentration of species, this technique can be used for quanti-tative analysis of paramagnetic molecules.
Since FAMOS is solely sensitive to paramagnetic molecules it is superior to many other types of laser-based detection techniques, drastically decreas-ing various types of noise, background signals, as well as signals from other molecules; e.g. flicker noise, etalon effects and signals from water and CO2 molecules.
An experimental setup for detection of nitric oxide (NO) by FAMOS has been developed and optimized. This system is based on a quantum cascade laser emitting light at 5.331 μm, addressing the—for FAMOS—most sensi-tive transition in NO, Q3/2(3/2). Optimized parameters include a pressure of 60 mbar, a magnetic field of 190 G and a polarizer uncrossing angle of 0.75°.
In its present configuration, this system has demonstrated a detection of NO down to 200 ppb for a detection rate of 10 measurements per second. It is very possible that the limit of detection is even lower than this number since this lowest acquirable concentration is limited by the specifications of the gas mixer. A standard deviation between subsequent measurements, of 15 s time separation, is calculated to 30 ppb. However, this is far from the expected ultimate detection limit of this system and this technique in general.
Gasen kvävemonoxid är en restprodukt vid förbränning av fossila bränslen, men den återfinns också i människokroppen som en signalsubstans. I stora mängder är den giftig då den interagerar med cellandningen och påverkar arvsmassan. I kombination med syre övergår den sakta till gasen kvävedioxid som i sin tur övergår till salpetersyra. Detta innebär att det är viktigt att kunna detektera denna gas i miljöer där risk för spridning kan anses vara stor.
Kvävemonoxid används inom sjukvården som inhalationsgas för tidigt födda barn. Den förekommer även i ökad koncentration i utandningsluft hos människor med astma, Alzheimers eller Parkinsons sjukdom. Därmed finns det också ett intresse av att kunna mäta denna noggrant inom sjukvården, för att ge rätt dosering av gasen samt för att enklare kunna ställa korrekta diagnoser.
Att mäta koncentration av kvävemonoxid görs ofta på kemiska vägar. Då detta är tidskrävande finns det anledning att utveckla nya och snabbare sätt att detektera denna gas. En effektiv teknik för att mäta kvävemonoxid är Faraday-moduleringsspektroskopi, FAMOS. Denna teknik bygger på att man med ett magnetfält, genom den så kallade Faradayeffekten, utnyttjar somliga fysikaliska egenskaper hos just kvävemonoxid, för att vidare be-stämma dess mängd.
FAMOS möjliggör mätningar i realtid och har en låg minsta detektions-gräns i jämförelse med andra laserbaserade tekniker. Framtidsutsikterna för denna teknik är mycket goda och möjligheterna utökas allt eftersom optisk utrustning förbättras.
Abbreviations
2f-WMS . . . Second harmonic WMS AC . . . Alternating current CaF2. . . Calcium Fluoride
DAS . . . Direct absorption spectroscopy DC . . . Direct current
FAMOS . . . Faraday modulation spectroscopy FRS . . . Faraday rotation spectroscopy FWHM . . . Full width at half max
HITRAN . . . High resolution transmission database HWHM . . . Half width at half max
IR . . . Infra red L . . . LHCP
LHCP . . . Left-handed circularly polarized light MgF2. . . Magnesium Fluoride
MRS . . . Magnetic resonance spectroscopy N2. . . Nitrogen gas
NEP . . . Noise equivalent power NO . . . Nitric oxide
ppb . . . Parts per billion ppm . . . Parts per million QCL . . . Quantum cascade laser R . . . RHCP
RHCP . . . Right-handed circularly polarized light RMS . . . Root mean square
SNR . . . Signal-to-noise ratio SOT . . . Sample optical thickness
TDLAS . . . Tunable diode laser absorption spectroscopy TEC . . . Temperature controller
WM-TDLAS . . Wavelength modulated TDLAS WMS . . . Wavelength modulated spectroscopy ZMS . . . Zeeman modulation spectroscopy
Symbols and functions
𝛼 . . . Sample optical thickness, SOT 𝛼0. . . On-resonance SOT
𝜒 . . . Area normalized lineshape function 𝜒G. . . Gaussian lineshape function 𝜒L. . . Lorentzian lineshape function
𝜒V. . . Voigt lineshape function 𝜒abs. . . Attenuative lineshape function 𝜒disp. . . Dispersive lineshape function
̂
𝜒abs. . . Peak-normalized attenuative lineshape fcn.
̂
𝜒disp. . . Peak-normalized dispersive lineshape fcn.
̂ 𝜒Ldisp. . . ̂𝜒dispof LHCP ̂ 𝜒Rdisp. . . ̂𝜒dispof RHCP 𝛿 . . . Attenuation 𝛿𝜈0. . . Lorentzian HWHM
𝛿𝜈D. . . Doppler width, Gaussian FWHM Δ ̂𝜒abs. . . Difference between ̂𝜒abs
L and ̂𝜒 abs R
Δ ̂𝜒disp. . . Difference between ̂𝜒disp L and ̂𝜒
disp R
Δ ̂𝜒abs
1 . . . First Fourier coefficients of Δ ̂𝜒 abs
Δ ̂𝜒1disp. . . First Fourier coefficients of Δ ̂𝜒disp
Δ𝛿 . . . Difference in attenuation Δ𝜙 . . . Difference in dispersion ΔΦ . . . Uncrossing angle
Δ𝐸 . . . Difference in energy between states Δ𝑘 . . . Difference of 𝑘 between the two frequencies Δ𝑡 . . . Decay time of a change of state
𝜖0. . . Vacuum permittivity
𝜀 . . . Unbalancing ter between LHCP and RHCP
̄
𝜀 . . . Unbalancing ter between LHCP and RHCP 𝜂 . . . Instrumentation factor
𝛾c. . . Particle collision rate
𝜇B. . . Bohr magneton 𝜈 . . . Frequency parameter 𝜈0. . . Frequency of transition 𝜈d. . . Detuning frequency 𝜈B. . . Magnetic detuning frequency
̄
𝜈 . . . Modified detuning frequency
̄
𝜈L. . . Modified detuning frequency of LHCP
̄
𝜈R. . . Modified detuning frequency of RHCP 𝜔 . . . Radial frequency
𝜙 . . . Dispersion
𝐵 . . . Magnetic field
𝐵0. . . Magnitude of magnetic field
𝑐 . . . Speed of light
𝒞 . . . Transforming factor of interacting fields const . . . Constant
c.c. . . Complex conjugate 𝐷∗. . . Detector detectivity
̂𝐞 . . . Electric field, unit vector erfc . . . Complex error function 𝐄 . . . Electric field vector
𝐄∗. . . Complex conjugate of electric field vector 𝐄L. . . LHCP electric field vector
𝐄R. . . RHCP electric field vector 𝐄t. . . Transmitted electric field vector 𝐄x. . . Electric field vector, linearly x-polarized 𝐄y. . . Electric field vector, linearly y-polarized
̃
𝐄x. . . Modified electric field, x-polarized
̃
𝐄y. . . Modified electric field, y-polarized 𝐄∗. . . Complex conjugate of electric field vector
𝐸0. . . Electric field magnitude 𝐸photon. . . Photon energy
𝑔′. . . Gyromagnetic ration of upper state
𝑔″. . . Gyromagnetic ration of lower state
ℎ . . . Planck constant ℏ . . . Reduced Planck constant
𝐼0. . . Magnitude of transmitted laser intensity 𝐼t. . . .Transmitted laser intensity
𝐽″. . . Lower state quant. num. of tot. ang. mom.
𝑘+. . . Mean wavevector of 𝑘 Land 𝑘L
𝑘−. . . Half the distance between 𝑘 Land 𝑘L
𝑘0. . . Wavevector without light-matter interaction
𝑘B. . . Boltzmann constant 𝑘L. . . Wavevector of LHCP
𝑘R. . . Wavevector of RHCP
𝐿 . . . Distance of particle-light interaction 𝑚X. . . Particle mass
𝑀′
F. . . Index of hyperfine splitting of upper state
𝑀″
F . . . Index of hyperfine splitting of lower state
𝑁 . . . Number of particles 𝑝 . . . Pressure
𝑃t. . . Transmitted laser power 𝑃L. . . Laser power
𝑝part. . . .Partial pressure of absorber ℛ . . . Detector responsivity 𝑆 . . . .DAS signal ̄ 𝑆 . . . .Normalized signal 𝑆F. . . FAMOS signal 𝑡 . . . Time parameter 𝒯 . . . Transmission factor 𝑇 . . . Temperature 𝑢 . . . Integration parameter 𝑣part. . . Particle speed 𝑉 . . . Volume
1 Introduction 1
2 Direct Absorption Spectroscopy 1
2.1 Beer-Lambert Law . . . 2 2.2 Rotation-Vibration Spectrum . . . 3 2.3 Lineshape Functions . . . 4 2.4 Detection of Light . . . 11 3 Modulation 13 3.1 Wavelength Modulation . . . 13 3.2 Faraday Modulation . . . 16
4 Design of FAMOS Apparatus 24 4.1 Setup . . . 24 4.2 FAMOS Signal . . . 28 4.3 Optimization . . . 30 5 Results 32 5.1 FAMOS Response . . . 32 5.2 Error Analysis . . . 34
1 Introduction
Nitric oxide, NO, sometimes called nitrogen oxide or nitrogen monoxide, is a poisonous
gas and a free radical. It is therefore both dangerous, highly reactive and of importance to be able to detect in various environments. NO is a very small molecule that is found in many different places and circumstances. It occurs in nature, as a signal substance in mammals, therefore low concentrations are naturally present in our bodies. Beyond this, increased amounts of NO have been noted [1–3] as results of airway inflammation and of the medical conditions asthma and; Alzheimer’s and Parkinson’s disease. There are also other reasons why one wants to measure the concentration of this molecule, including environmental applications [4].
There are many ways to measure and detect NO where some of these are through chemical reactions [5]. What these methods have in common is that they are not operating in real time. By using laser-based techniques the gas concentration may be monitored as time progress which may be hundreds of measurements per second. Compared to chemical ways of detection, one has to wait for a laboratory analysis in order to get the result of one measurement.
The purpose of this thesis is to construct a setup where measurements of NO may be performed by use of laser spectroscopy and a method called Faraday modulation
spec-troscopy, FAMOS. This method is modulation based and, contrary to many other
tech-niques where properties of the laser light are modulated, transition properties of molecules are modulated instead. For a given geometrical setup, a magnetic field rotates the polariza-tion of laser light that propagates through a paramagnetic sample, which may be detected by a polarizer and a photodiode.
Even though there are other laser-based spectroscopic techniques at hand, this one is very well suited for the purpose of measuring NO as it effectively minimizes the influence of other gases, which are not paramagnetic. It also allows a local analysis, i.e. detection at the location of the magnetic field, whereby other gases that are in the path of the laser light are left out. This is in contrast to other techniques, primarily those involving modulation of the laser frequency, for which the resulting signal depends on all different kinds of particles between the diode and the detector.
2 Direct Absorption Spectroscopy
In direct absorption spectroscopy (DAS) laser light with a swept wavelength is sent through a gaseous sample and then analyzed by a photo detector. Some wavelengths will be ab-sorbed by the gas depending on the internal molecular structure of its components. This phenomenon leads to a decreased intensity of the analyzed laser light in certain regimes of the tuned wavelength. The intensity as a function of wavelength is characteristic for the specific substance and provides a unique fingerprint corresponding to the gas’s compo-nents. In fig. 2.1 it is shown how an implementation of DAS may look like. The accuracy of this technique is quite limited and much better results may be acquired by rather simple modifications.
Due to the many appealing properties of diode lasers, a commonly used laser-based technique for measuring the sample concentration is tunable diode laser absorption
Gas cell Laser Detector Sweep Data acquisition
Figure 2.1: A schematic setup for detecting a sample sub-stance by direct absorption spectroscopy.
in a triangular path over a specific transition. We refer to this tuning of wavelength as the
sweep. The transmitted laser intensity is then acquired as a function of frequency and by
data processing, the concentration can be assessed. For every new sweep that is performed a new set may be recorded.
2.1 Beer-Lambert Law
The event of photon absorption is probabilistic and its probability is approaching 1 when the distance of interaction goes to infinity1. The laser intensity is decreased exponentially over the propagated distance which results in the Beer-Lambert law [6] which in spectro-scopic applications is described by
𝐼t(𝜈) = 𝐼0(𝜈)𝑒−𝛼(𝜈). (2.1)
Here, 𝐼t(𝜈) is the transmitted light intensity as a function of frequency and 𝐼0(𝜈) is the intensity of the incident beam. The absorbance or the sample optical thickness (SOT), denoted by 𝛼(𝜈), is a substance specific measure [7] of the ability of a medium to interact with light and absorb photons at a given frequency, 𝜈.
Even though the SOT may be estimated by a theoretical approach it is in spectroscopic applications often just measured and compared to experimentally obtained data. It is safe to assume that the probability of light-molecule interactions increases linearly with an increased number of molecules, so we can separate this number from other factors that make up the SOT. The SOT can thereby be written
𝛼(𝜈d) = 𝑝part𝒮 𝜒(𝜈d)𝐿 , (2.2)
where 𝑝partis the partial pressure of the absorber, 𝒮 is the integrated linestrength of the energy transition, 𝜒(𝜈d) is an area normalized lineshape function, while 𝐿 is the distance of
1This is true given that the laser frequency is in resonance with the molecular transition. This is explained
interaction which for many spectroscopic applications is the length of the gas cell. More-over, 𝜈dis the detuning frequency given by
𝜈d= 𝜈 − 𝜈0. (2.3)
This implies that 𝜒(𝜈d) is centered about the frequency 𝜈0, as will be further explained in section 2.3. 𝛼 is dimensionless as eq. (2.1) requires and the partial pressure is the total pressure times the fraction, the amount of the interacting gas is to the total amount of gas inside the cell. The integrated linestrength, is also called the spectral linestrength, and will be discussed further in section 2.2. It is given in different units depending on the application. However, from eq. (2.2) we calculate its dimension to be [cm−2atm−1].
It is of interest to solve eq. (2.1) for the SOT since it is this quantity that contains the information about the sample. Doing so, yields
𝛼(𝜈d) = − ln 𝐼t(𝜈d) 𝐼0(𝜈d)
. (2.4)
If we assume the medium to be optically thin, i.e. a small total absorption at the end of the light-molecule interaction, 𝐼t(𝜈d)/𝐼0(𝜈d) ≪ 1 and we may series expand eq. (2.1) into
𝐼t(𝜈d) = 𝐼0(𝜈d) [1 − 𝛼(𝜈d)] . (2.5) We see that the transmitted intensity is approximated to a linear function of the SOT, which facilitates calculations when experimentally relating partial pressure of an analyte with the measured light intensity. By integrating the intensity over the beam cross section, we get a similar expression for the transmitted power,
𝑃t(𝜈d) = 𝑃L(𝜈d) [1 − 𝛼(𝜈d)] , (2.6) where 𝑃Lis the laser power impinging on the sample.
2.2 Rotation-Vibration Spectrum
On a microscopic level there are interactions between photons and molecules. The atoms, which a molecule is made of, rotate and vibrate relative to each other and according to quantum mechanics these particles can only do so in certain discrete states. We say that they are quantized.
It is known that photons of different frequency, 𝜈, have different energy,
𝐸photon= ℎ𝜈 , (2.7)
Figure 2.2: Rovibra-tional spectrum of nitric oxide (NO). From left to right the P-, Q- and R-branches are visible. The data has been taken from the HITRAN 2008 database.
of molecules have different energy levels since their internal particles differ in number and structure. Fundamentally, this is what distinguishes different molecules (and atoms) and it is the origin to their unique fingerprints.
When performing laser-based spectroscopy there are usually two approaches depend-ing on what you are lookdepend-ing for. If you want to characterize an unknown substance you look at a spectrum at wide range of wavelengths which contains several of these energy transitions and try to fit it to curves of already known substances. This is called a
rotation-vibration or rorotation-vibration spectrum. See fig. 2.2 for such a spectrum for NO in the range of
1700 cm−1to 2000 cm−1and fig. 2.3 for a closeup on its Q-transitions.
The integrated linestrength, 𝒮 , is a quantity that describes the interaction strength as a function of wavenumber. The HITRAN 2008 database contains numerical values for linestrengths of different transitions and the dimension of 𝒮 provided by HITRAN is [cm−1/(molecule × cm−2)] which in words is the probability of a molecular transition per the ratio of molecular cross-sectional area to the beam cross-sectional area, and per distance unit. This is different from the dimension as it was previously described, but it is of low importance since the spectral shape remains the same.
If a substance is known and its concentration is to be measured, one normally selects a transition that typically is strong and, in addition, is very characteristic for the specific species. The wavenumber range of the sweeping is then shortened until the point that only this transition is targeted by the laser light. This, one, transition to be measured is guaranteed to exist if we know that the substance is present. As eq. (2.2) and (2.5) states, the intensity is related to the partial pressure of the analyte and so the concentration of analyte is found.
2.3 Lineshape Functions
By sweeping the laser’s frequency over a certain energy transition, we realize that it, in fact, is not an infinitesimally narrow line which one might have expected from the deriva-tion of its existence in secderiva-tion 2.2. It is broadened in wavenumber space due to some broadening mechanisms and three of these mechanisms are considered for DAS, among other spectroscopic implementations, and they will be further examined in this section.
Figure 2.3: Rovibra-tional spectrum of nitric oxide (NO). The Q-branch is here zoomed in and its peaks are named XΩ(𝐽″) where
X is the corresponding branch, 𝐽″ is the lower
state angular momentum and Ω is the projection of angular momentum along the internuclear axis. The data has been taken from the HITRAN 2008 database.
The height of the lineshape is given in absorption units and these lineshape functions are normalized in the sense that if you integrate over a spectral transition, the contribution from the lineshape will equal 1 and be dimensionless. The linestrength at a transition has a known value and if we model the broadening, integrating over the lineshape must equal 1 and preserve the linestrength dimension. This means that, since the frequency is given in wavenumbers [cm−1], its integration is dimensionless and the dimension of an absorption unit must be centimeters [cm].
2.3.1 Natural Broadening
This type of broadening occurs due to the limited decay time of an excited state. For shorter decay times the energy uncertainty increases according to the uncertainty principle [8], i.e. as
Δ𝐸Δ𝑡 ≥ ℏ
2. (2.8)
Here, Δ𝐸 is the energy uncertainty, Δ𝑡 is the time difference, or decay time, and ℏ is the reduced Planck constant. This energy difference is very small so if natural broadening is the limiting factor for the width of a measured transition line, it is not possible to reduce this width any further. Hence, there cannot exist infinitesimally narrow transition lines.
The appearance of natural broadening takes the shape of the symmetrical Lorentzian lineshape profile [9, p. 112], 𝜒L(𝜈𝑑). Its form is shown in fig. 2.4(a) and it is given by
𝜒L(𝜈) = 𝛿𝜈0/𝜋 (𝜈 − 𝜈0)2+ 𝛿𝜈2
0
. (2.9)
where 𝜈0is the center frequency and 𝛿𝜈0is the half-width-half-maximum (HWHM), that is half the width of the bell shaped function at half its maximum and is for natural broadening given by
𝛿𝜈0= 1
4𝜋 (𝐴1+ 𝐴2) , (2.10)
these values can be found, for the transitions of interest in this thesis, in the HITRAN 2008 database.
2.3.2 Collision Broadening
An absorbing molecule that is exposed to a relatively high rate of collisions gets its spectral lines distorted because of interactions between the respective particles’ electronic field po-tentials. A medium of high pressure equals high density and an increased rate of molecular collisions. Therefore, another name for this type of broadening is pressure broadening, which might be more intuitive since the amount of this broadening is proportional to pres-sure.
The lineshape of pressure broadening is Lorentzian, just as the natural broadening is [9, p. 99], but with a different HWHM, 𝛿𝜈0given by
𝛿𝜈0= 𝛾c
2𝜋. (2.11)
In this equation, 𝛾cis the collision rate which is molecule specific and it can more or less be modeled from the structure and cross-sectional diameter of the absorber’s components, the partial pressures of the gases together with the temperature. Most often, however, it is assessed by measurements or is obtained from databases, such as HITRAN 2008, which will be done in this thesis.
2.3.3 Doppler Broadening
This type of broadening originates from the thermal motion of the molecules. By their movement along the propagation of light, the velocity of a particle in relation to the mean velocity of the sample gas introduces differing transition energies between molecules of different thermal velocity. This motion changes the frequency, and so the wavenumber, of the photon in the frame of the particle, which is the frame in which the absorption occurs. The individual particles’ velocities, projected on the axis of light propagation, results in a broadening of the absorption lineshape, even though their mean velocity equals zero. The frequency as is perceived by the absorber, 𝜈, is described by the Doppler shift equation,
𝜈 = 𝜈0(1 − 𝑣part
𝑐 ), (2.12)
where 𝜈0is the laser frequency in the rest frame, 𝑣partis the particle speed along the prop-agation of light and 𝑐 is the speed of light. The distribution of particle velocities due to the random nature of thermal movement is expressed by the one-dimensional Maxwell-Boltzmann distribution, given by
d𝑓 (𝑣part) =( 𝑚X 2𝜋𝑘B𝑇 ) 1/2 exp ⎛ ⎜ ⎜ ⎝ −𝑚X𝑣2part 2𝑘B𝑇 ⎞ ⎟ ⎟ ⎠ d𝑣part. (2.13)
Detuning frequency [cm−1] -0.01 0 0.01 A b so rp ti o n u n it s [c m ] 0 50 100 150 200 250 20 mbar 40 mbar 60 mbar 80 mbar 100 mbar
(a) Lorentzian lineshape functions calculated for NO diluted in air, at a transition at 1875.81 cm−1. Detuning frequency [cm−1] -0.01 0 0.01 A b so rp ti o n u n it s [c m ] 0 50 100 150 200 250 300 K 400 K 500 K 600 K 700 K
(b) Gaussian lineshape functions calculated for NO at a transi-tion at 1875.81 cm−1.
Figure 2.4: Area normalized lineshape functions. (a) is the shape of transition broadening due to the natural and colli-sional parts. A lower pressure results in a narrower width. (b) is the broadening that arises due to Doppler effect. A lower temperature results in a narrower width.
[9, p. 105] for Doppler broadening. We call this the Gaussian lineshape,
𝜒𝐺(𝜈) = 𝛿𝜈1 D( 4 ln 2 𝜋 ) 1/2 exp ( −4(𝜈 − 𝜈0)2ln 2 𝛿𝜈2D ) (2.14)
where 𝜈0is the center frequency and
𝛿𝜈D= 2𝜈0 𝑐 ( 2𝑘B𝑇 𝑚X ln 2 ) 1/2 (2.15)
is the Doppler width. The Doppler width is normally taken as the full-width-half-maximum (FWHM) for the Gaussian profile. That is the width of the bell shaped function at half its maximum value, or twice the HWHM. Its form is shown in fig. 2.4(b).
2.3.4 Voigt Profile
The Lorentzian and the Gaussian lineshape profiles dominate at different regimes of pres-sure. For low pressures the Gaussian profile is dominant, and for high pressures the Lorentzian profile dominates. When the pressure is in between, the profiles are com-bined into a common lineshape profile, the Voigt profile. Even though many applications are carried out in regions that safely can be expressed by the Lorentzian or the Gaussian lineshape functions, the Voigt lineshape is often used instead.
This lineshape is found by calculating the convolution of the Lorentzian and the Gaussian lineshapes. The Lorentzian lineshape function eq. (2.9) is modified to take the Doppler shift into account, that is
𝜒L(𝜈, 𝑣part) = 𝛿𝜈0/𝜋
(𝜈0− 𝜈 − 𝑣part𝜈/𝑐)2+ 𝛿𝜈20
which is on resonance when eq. (2.12) is fulfilled. What this convolution describes in more practical terms is an integration over all velocities of this function times the velocity distribution eq. (2.13). 𝜒Vis the Voigt lineshape function given by
𝜒V(𝜈) = ∫ ∞ −∞ d𝑣part𝜒L(𝜈, 𝑣part) ( 𝑚X 2𝜋𝑘B𝑇 ) 1/2 exp⎛⎜ ⎜ ⎝ −𝑚X𝑣2part 2𝑘B𝑇 ⎞ ⎟ ⎟ ⎠ . (2.17)
If we are on resonance, this equation may conveniently be expressed [9, p. 109] using the
complementary error function, erfc(𝑏),
𝜒V(𝜈 = 𝜈0) = ( 4 ln 2 𝜋 ) 1/2 1 𝛿𝜈D 𝑒 𝑏2erfc(𝑏) , (2.18) where erfc(𝑏) = 2 𝜋1/2 ∫ ∞ 𝑏 d𝑢 𝑒−𝑢2 (2.19) and 𝑏 = √4 ln 2 𝛿𝜈0 𝛿𝜈D. (2.20)
The above lineshape is good for describing the on-resonance absorption. However, when this is not the case, the complexity of the Voigt profile [9, p. 111] is slightly in-creased; it can be written
𝜒V(𝜈) = 1 𝜋3/2 𝑏2 𝛿𝜈0∫ ∞ −∞ 𝑒−𝑦2 (𝑥 + 𝑦)2+ 𝑏2d𝑦 = 1 𝜋3/2 𝑏2 𝛿𝜈0 𝜋 𝑏 Re [w(𝑥 + 𝑖𝑏)] . (2.21)
w is here called the error function of complex argument and numerical values of it may be found in mathematical handbooks or by using certain computer implementations for software such as MATLAB.2𝑥 is a detuning parameter that is a measure of how close to resonance the incident light is, given by
𝑥 = √4 ln 2 (
𝜈0− 𝜈 𝛿𝜈D )
(2.22)
It thus equals zero when there is perfect resonance. The broadening of a Voigt profile is given by the parameter 𝑏, and so we will refer to this value when discussing the practical influence of pressure. We call it the Voigt parameter.
2.3.5 Absorption and Dispersion Lineshapes
As the Beer-Lambert law states, the laser intensity is exponentially decreased inside a gaseous medium. As we will see, also the phase is retarded here and it is convenient to correlate these behaviors to the Voigt profile. Here we will examine how a linearly
2ERFZ(z) is a MATLAB function for calculation of the error function and it has been expanded to cover
polarized electric field behaves and to begin with, the electric field is expressed as a plane-polarized monochromatic wave.
𝐄(𝑧, 𝑡) = 𝐸0 ̂𝐞 cos (𝑘𝑧 − 𝜔𝑡)
= 𝐸0 2 ̂𝐞 𝑒
𝑖𝑘𝑧−𝜔𝑡+ c.c. (2.23)
𝐸0is the electric field amplitude, ̂𝐞 is the polarization vector, 𝑘 is the wavenumber, 𝑧 is the spatial parameter, 𝜔 is the angular frequency and 𝑡 is the time parameter. For vacuum, the intensity is related to the electric field according to
𝐼0= 2𝑐𝜖0𝐄(𝑧, 𝑡)𝐄∗(𝑧, 𝑡) = 12𝑐𝜖0𝐸02, (2.24) where 𝑐 is the speed of light and 𝜖0is the vacuum permittivity.
Both the absorption and the dispersion changes exponentially along the propagation and so it is common practice to expand the wavenumber, 𝑘, into the complex plane where the absorption is described by its imaginary part and the dispersion by its real part. We now call it the wavevector and its amplitude and phase changes from medium to medium. If we consider a cell containing some specific gas, we get one wavevector on the outside, 𝑘0, where no absorbers are present and it is therefore is real valued; and one wavevector on the inside, 𝑘. This latter wavevector depends on the frequency, as light of certain wavenumbers is absorbed and light of others is not. We are interested in the transmitted electric field when the plane wave has exited the gas cell, i.e. the transmitted field, 𝐸t, when 𝑧 > 𝐿 where 𝐿 is the length of the cell. Hence we modify eq. (2.23) accordingly.
𝐄t(𝑧, 𝑡, 𝜈) = 𝐸0 2 ̂𝐞 𝑒 𝑖[𝑘0(𝑧−𝐿)+𝑘(𝜈)𝐿−𝜔𝑡]+ c.c. = 𝐸0 2 ̂𝐞 𝑒 𝑖[(𝑘(𝜈)−𝑘0)𝐿+𝑘0𝑧−𝜔𝑡]+ c.c. (2.25)
We see that 𝐸tcan be seen as a product of some frequency dependent complex number, 𝒞 (𝜈), and the non-absorbed electrical plane wave eq. (2.23). We write
𝐄t(𝑧, 𝑡, 𝜈) = 𝒞 (𝜈)𝐄(𝑧, 𝑡) . (2.26)
This multiplying complex factor, which can be interpreted as a complex transmission func-tion, can be written as
𝒞 (𝜈) = 𝑒𝑖[𝑘(𝜈)−𝑘0]𝐿= 𝑒𝑖𝜙(𝜈)−𝛿(𝜈), (2.27) where 𝛿(𝜈) and 𝜙(𝜈) are the attenuation and the phase shift of the electrical field, respec-tively. Under the assumption that 𝑘0is real, these can be expressed as
𝛿(𝜈) = Im [𝑘(𝜈)] 𝐿
𝜙(𝜈) = {Re [𝑘(𝜈)] − 𝑘0} 𝐿 .
(2.28)
They are thus both related to the wavevector, 𝑘(𝜈) by their imaginary and real parts, re-spectively.
By use of the relation between intensity and electric field in eq. (2.24) it is possible to conclude that the intensity of the transmitted light only depends on the absorbing proper-ties of the medium, namely
𝐼t(𝜈) = 2𝑐𝜖0𝐄t(𝑧, 𝑡, 𝜈)𝐄∗t(𝑧, 𝑡, 𝜈) = 2𝑐𝜖0𝒞 𝒞∗𝐄(𝑧, 𝑡)𝐄(𝑧, 𝑡)∗ = 𝐼0𝑒−2𝛿(𝜈).
From the Beer-Lambert law eq. (2.1) we can relate the SOT, 𝛼(𝜈), to the attenuation, 𝛿(𝜈) as
𝛼(𝜈) = 2𝛿(𝜈) . (2.30)
Since all gaseous media affect light by both attenuation and dispersion, the lineshape functions describing the light-matter interaction are in general complex. It is therefore convenient to split the real and imaginary parts of the complex lineshape function into separate lineshape functions, corresponding to the dispersion and the attenuation of the light, respectively. The SOT can then be expressed as
𝛼(𝜈) = 𝑝part𝒮 𝐿𝜒abs(𝜈) = 𝑝part𝒮 𝐿𝜒abs(𝜈0) 𝜒 abs(𝜈) 𝜒abs(𝜈 0) = 𝛼0𝜒̂abs(𝜈) , (2.31)
where 𝜒abs(𝜈0) corresponds to the imaginary part of the complex wavevector (and thereby the imaginary part of the complex lineshape function) and where we have introduced 𝛼0 as the on-resonance sample optical thickness, which is the SOT at the transition resonance 𝜈0, and ̂𝜒abs(𝜈) as the peak-normalized absorption lineshape function. By identification, it is thus possible to conclude that 𝛼0is given by
𝛼0= 𝛼(𝜈0)
= 𝑝part𝒮 𝐿𝜒abs(𝜈0),
(2.32)
where 𝜒abs(𝜈0) is the absorption lineshape of the transition evaluated on resonance. The peak-normalized attenuation lineshape function is then defined as
̂ 𝜒abs(𝜈) = 𝜒 abs(𝜈) 𝜒abs(𝜈 0) . (2.33)
By substitution into eq. (2.30), we get
𝛿(𝜈) = 𝛼0 2𝜒̂
abs(𝜈) , (2.34)
and note that the frequency dependence of the transmitted intensity comes from the atten-uation of the medium solely. This result shows that it is possible to rewrite the absorption lineshape in terms of the attenuation and the on-resonance SOT as
̂ 𝜒abs(𝜈) = 2𝛿(𝜈) 𝛼0 , (2.35) where 𝛼0= 𝛼(𝜈) ̂ 𝜒abs(𝜈) . (2.36)
In a similar way as for attenuation, the dispersion amplitude of the laser light changes exponentially. Here, the phase of the electric field changes rather than the amplitude (or the intensity). This dispersion of light corresponds to the real part of the wavevector and the complex lineshape function. Quantitatively, this property is what we call the refractive index of the medium. A lineshape function corresponding to dispersion can be defined as
̂
𝜒disp(𝜈) = 2𝜙(𝜈) 𝛼0
and correspondingly
𝜙(𝜈) = 𝛼0 2 𝜒̂
disp(𝜈) . (2.38)
In section 3.2 we will evaluate the consequence of applying an external magnetic field in terms of these lineshape functions and, as we will see, for Faraday modulation: only the latter lineshape function, 𝜒disp(𝜈), will affect the final signal which is the reason for the use of this particular separation.
2.4 Detection of Light
The detection of transmitted light is typically performed by a photo-voltaic detector as it has a better response time than the alternatives. For an even better response time it may be operated under a reverse bias voltage where the response time goes from 80 ns to 0.7 ns effectively3, but for DAS and FAMOS the response time is not a limiting factor as measurements are carried out in in speeds of 0.1 s and 0.1 ms respectively.
2.4.1 Detector signal
The detector signal is related, but not equal, to 𝑃t(𝜈), eq. (2.6). Other factors have to be accounted as well, such as the loss of intensity due to reflections and absorptions by the optics, as well as a limited detection area of the detector. This can be measured by running the laser through the system without any present absorber and then measure the losses. We name this fraction that is transmitted through the system, the transmission factor, 𝒯 . There is also a factor called the instrumentation factor, 𝜂, which is introduced by amplifiers and other electronic equipment. For a given spectroscopic setup, the instrumentation factor is often constant and it is not meaningful to consider it any further as it effectively multiplies a small signal equally as much as a large signal. However, as will be shown in section 4.2, 𝜂 has to be considered as it may be required to change it from one measurement to another. All this implies that for small SOT, for which it is appropriate to assume exp [−𝛼(𝜈)] ≈ 1 − 𝛼(𝜈), the signal is written
𝑆(𝜈) = 𝜂𝒯 𝑃L[1 − 𝛼(𝜈)] . (2.39)
When analyzing the signal quality it is sometimes more illustrative to calculate the
normalized signal,𝑆(𝜈). This entity always has the amplitude 1 as it effectively is thē signal divided by its own amplitude. This property is given by
̄
𝑆(𝜈) = 𝑆(𝜈)
max [𝑆] − min [𝑆]. (2.40)
2.4.2 Flicker noise
For DAS, the most dominant noise is flicker noise, which sometimes is called pink noise or 1/f-noise because of its frequency dependence. It occurs in almost all electric circuits and it is large for low frequency measurements and small for higher frequency measurements. For flicker noise, the change of noise equivalent power (NEP) per octave is constant [10]. This is a constant change in noise amplitude when doubling the frequency. It means that
Figure 2.5: Signal, experimentally collected by aligning a quantum cascade laser operating at 5.331 μm onto a mid-IR detector. The data set of the two plots are the same where, for the second plot, fast Fourier transform breaks the devia-tions down into its frequency components and flicker noise is visualized as linearly decreasing as a function of frequency (but on a logarithmic scale).
every octave carries approximately the same amount of energy, and also the same amount of flicker noise as a consequence of Parseval’s theorem.
For a regular implementation of DAS the frequency of measurements is normally low enough to produce much more pink than white noise, which is of a random nature. It is also possible to detect the main hum which is the frequency of the main power-line. To increase the signal-to-noise ratio (SNR) the reduction of flicker noise should be prioritized while keeping away from the first few harmonics of the main hum. We will need to measure fast, and the faster we measure the lesser flicker noise we detect. This can be achieved by use of modulation.
but also white. In the second graph we have the discrete Fourier transform of the signal. We see that the voltage, as a function of the frequency, decays exponentially towards zero. Around 1 MHz the linearity tends to vanish, this is the point where white noise overtakes the pink.
3 Modulation
We want to minimize the interference between low frequency noise and the signal. The first action to take is to move the measurement rate up in the frequency spectrum and from that point separate the signal from noise of lower frequency.
3.1 Wavelength Modulation
In the technique called wavelength modulation the wavelength is modulated with a given frequency while the signal is demodulated at a harmonic of the same frequency. This increases the sensitivity of TDLAS remarkably, we call it WM-TDLAS wavelength
mod-ulated tunable diode laser absorption spectroscopy. The frequency of this modulation is
usually many orders of magnitudes higher than that of the sweep, preferably in kilohertz up to megahertz range, ideally at the frequency where white noise prevails, see fig. 2.5. The amplitude of the modulation is in the order of the width of the addressed transition, and despite its name, wavelength modulation is most often described as a modulation of the frequency of the light. We are interested in moving back and forth over the peak of the lineshape and every time we do, we get an intensity curve, see fig. 3.1(a), which can be related to a lineshape of a specific energy transition.
Now assume that we have some noise with frequency several times lower than that of the modulation. Instead of distorting the signal, this noise would rather add an offset to it and also a small “tilt”, that is one side of the signal curve would raise a little relative to the other side. We will see that its shape is somewhat preserved which it would not be if this modulation frequency was not high compared to the frequency of the noise. To get rid of this offset which still is added to the intensity we multiply it to the modulation function and noise of lower frequency than this modulation function is averaged out. This procedure is performed by a lock-in amplifier and the flicker noise, which is more dominant at lower frequencies, is thereby greatly reduced. The detector signal is input to the lock-in amplifier together with the modulation signal as reference. The output of this lock-in amplifier is the wavelength modulated signal. Every time a period of modulation is convoluted with the reference function, which may be several thousands of times per second depending on the modulation frequency, information about the SOT may be analyzed.
As we see in fig. 3.1, we get different demodulated signal (intensity) depending on what phase shift there is between the detector signal and the reference function. Two of these phases contain the same information about the sample (but are of opposite sign) and two of the phases will cancel the demodulated signal. One of these phases that contain the most information is set to the lock-in amplifier by the operator4and is then extracted by the lock-in amplifier.
4Some lock-in amplifiers provide the option to use two input channels, instead of one, and the root mean
(a) Phase 0 (b) Phase 𝜋/2
(c) Phase 𝜋 (d) Phase 3𝜋/2 Figure 3.1: Illustrative scheme of the effect of a modula-tion at a given detuning from a transimodula-tion for four phases of modulation (0, 𝜋/2, 𝜋 and 3𝜋/2) of the SOT and the trans-mitted intensity. For the lineshape, the vertical axis is fre-quency, while the horizontal axis represents absorption. For the modulation, the sample optical thickness and the inten-sity, the vertical axes are arbitrary while the horizontal axis is time.
We sweep over the lineshape function to make greater sense out of the modulated signal and in fig. 3.2 we see how the modulated signal appears as we cover different wavenum-ber ranges over the lineshape. In this figure we see that when the modulation amplitude is centered over the peak of the lineshape, the frequency of the intensity is double the mod-ulation frequency, this is the second harmonic of the modmod-ulation and it is the reason why this technique sometimes is referred to as 2f-WMS.
(a) Negative detuning (b) On-resonance
(c) Positive detuning
Figure 3.2: Illustrative scheme of the effect of a modula-tion with a large amplitude for: a negative detuning (a), on-resonance (b) and positive detuning (c); on the SOT and the transmitted intensity. For the lineshape, the vertical axis is frequency, while the horizontal axis represents absorption. For the modulation, the sample optical thickness and the in-tensity the vertical axes are arbitrary while the horizontal axis is time.
Fourier coefficients as the second coefficient corresponds to the even component of the product between the detector signal and the modulation function, because the lineshape is almost symmetrical5 around the peak. The averaged product is positive for an equal phase of signal and reference, fig. 3.3(a), it is negative when they are completely out of phase, fig. 3.3(c), we get a negative product. At the turning point, fig. 3.3(b), we see that the second harmonic, instead of the first, is present. Since the lineshape is symmetric, the modulated signal will be anti-symmetric as the signal is negative on one side of the transition and positive on the other.
A problem with wavelength modulation is that we not only modulate the interaction between the laser and sample molecules but also the interaction between the laser and surrounding molecules that are of no interest for us. Usually one has a setup where a sample gas is placed inside a gas cell and laser light is led through it, onto a detector. By modulating the frequency of the light itself, the response from all molecules between the diode and the detector are subject to measurement. We need some way to modulate only the contents of the gas cell in order to bypass the excessive signal that is picked up from the
5Practically, the acquired lineshape is not perfectly symmetrical due to variations in laser intensity over the
(a) Negative detuning (b) On-resonance
(c) Positive detuning (d) Off-resonance Figure 3.3: Illustrative scheme of the effect of a modula-tion with a small amplitude for: a negative detuning (a), on-resonance (b), positive detuning (c) and off-resonance (d); on the SOT and the transmitted intensity. For the line-shape, the vertical axis is frequency, while the horizontal axis represents absorption. For the modulation, the sample optical thickness and the intensity the vertical axes are arbi-trary while the horizontal axis is time.
surroundings. By finding a property of the sample to modulate instead of the wavelength of the laser, a much cleaner signal may be obtained. The laser amplitude noise would also be reduced as the high frequency modulation of the laser would be removed.
Moreover, there is also a phenomenon called etalon effects present in WMS. It is intro-duced as resonances between the optical components, such as mirrors and beam-splitters. Basically, if the distance between two reflective surfaces is an even number of wavelengths, the signal intensifies. The effect of etalons looks similar to actual lineshapes when the sig-nal is processed. Then, if the wavelength is modulated, a WMS sigsig-nal will appear when the specific wavelength is matched by the modulator. It would be good if also this effect could be eliminated.
3.2 Faraday Modulation
Even though wavelength modulation is a very powerful technique, which may be used on a wide variety of different compound gases, there are situations where one can achieve even better results by using other techniques. One of these is Faraday modulation
3.2.1 Faraday effect
In the year 1845, scientist Michael Faraday discovered the interaction between magnetism and light. This was done before James Clerk Maxwell had invented the modern framework for macroscopic description of electromagnetic fields.
Faraday discovered what is now called the Faraday effect. That is, in a magnetic field directed along the propagation of light, left-handed circularly polarized light (LHCP) and
right-handed circularly polarized light (RHCP) are dispersed differently by some
molecu-lar substances. That is, they experience different refractive indices. More generally, they will experience different wavevectors, 𝑘(𝜈), for which the real component is the refractive index. This can be seen as the two circularly polarized light components are phase shifted differently. The larger the amplitude of the magnetic field gets, the larger the experienced phase shift between RHCP and LHCP gets. Now, if the magnetic field is of opposite di-rection to the propagation of light, the refractive indices appear to be sign shifted—the part of the light which before was experiencing a higher refractive index is now, instead, experiencing a lower refractive index and vice versa. Hence, by modulating the magnetic field, the response of the molecules will be modulated. This is FAMOS.
3.2.2 General principle of FAMOS
Linearly polarized light may be written as a superposition of RHCP and LHCP. As such light is both subject to an external magnetic field and passes a paramagnetic substance, the angle of polarization is slightly rotated. This rotation thus depends on the dispersive component of the wavevector, and not the absorptive. In order to analyze the resulting angular shift, two linear polarizers are used, one before the sample cell and one after.
The two polarizers are placed with a relative angle that is close to, but not exactly, 90° so that light is almost blocked. When a rotation of the polarization angle occurs in between the polarizers, the intensity of the transmitted light will change. The rotation is very small and for small angles it is assumed that the intensity change is linear with the rotation. By modulating the magnetic field, also the transmitted light will be modulated. After demodulation of the transmitted light at the first harmonic, the resulting height of the peak is almost directly proportional to the number of light-molecule interactions, and so the concentration is found. Therefore, the FAMOS signal can be interpreted as a consequence of the dispersive properties of the interacting molecules.
When the transitions between the molecular states occur, there are both absorption and dispersion but it is only the dispersive properties that affect the polarization. That being said, there is still a change of absorption when FAMOS is performed and this change of signal due to absorption is thus very weak and does not affect the final intensity as much as the shift of polarization angle does.
3.2.3 Relation between dispersion and magnetic field
The total difference between the LHCP and RHCP wavevectors is expressed by the absorp-tion and dispersion lineshape funcabsorp-tions eq. (2.35) and (2.37). Since lineshape funcabsorp-tions are depending on the frequency of the light as well as the center frequency of the transition, we may write these as functions of the detuning frequency, 𝜈d, eq. (2.3) instead.
fre-quency due to the presence of an external magnetic field [11, 12]. It is given by 𝜈B = (𝑀F′𝑔′− 𝑀F″𝑔″) 𝜇B𝐁 .
𝑀Fis here the magnetic quantum number of the states addressed, here denoted F. 𝑔′and 𝑔″are the gyromagnetic ratios of the two states between which the transition takes place, single prime is upper state and double prime is lower state. 𝜇Bis the Bohr magneton and 𝐁 is an applied magnetic field that is either parallel or anti-parallel to the laser light. For a given rovibrational transition, the 𝑔-factors, the set of 𝑀F quantum numbers and the Bohr magneton are all constant. It suffices therefore to say that this phase shift is linearly dependent on the magnetic field, 𝐁, which may be changed over time. This implies that we can write
𝜈B(𝑡) = const × 𝐵(𝑡) . (3.1)
The effective frequency, or the modified detuning frequency, ̄𝜈(𝑡), for the transition when taking the shift due to a magnetic field into account, can thereby be written
̄
𝜈(𝑡) = 𝜈d− 𝜈B(𝑡) . (3.2)
In section 2.3.5 we saw that the dispersive and attenuating properties of a medium at a transition may be expressed by their own separate lineshape functions. For convenience eq. (2.38) is solved for the phase shift using the above modifications of the phase shift for LHCP (L) and RHCP (R). This gives
𝜙( ̄𝜈L/R) = 𝛼0
2𝜒̂ disp( ̄𝜈
L/R) . (3.3)
The difference in phase shift between the LHCP and RHCP can then be written
Δ𝜙( ̄𝜈L/R) = 𝛼0 2Δ ̂𝜒 disp( ̄𝜈 L/R) , (3.4) where
Δ ̂𝜒disp( ̄𝜈L/R) = ̂𝜒disp( ̄𝜈L) − ̂𝜒disp( ̄𝜈R) , (3.5) is the difference in peak-normalized dispersion lineshapes between LHCP and RHCP.
There is an equation similar to eq. (3.4) that corresponds to the attenuation but as the method FAMOS relies on a dispersive shift rather than a total decrease of intensity, it has a smaller impact on the FAMOS signal. This is given by
Δ𝛿( ̄𝜈L/R) = 𝛼0
2Δ ̂𝜒 abs( ̄𝜈
L/R) . (3.6)
where Δ ̂𝜒abs( ̄𝜈L/R) is the corresponding difference in peak-normalized attenuation line-shapes between LHCP and RHCP. It is written
Δ ̂𝜒abs( ̄𝜈L/R) = ̂𝜒abs( ̄𝜈L) − ̂𝜒abs( ̄𝜈R) . (3.7)
The propagating plane wave electric field of light polarized in fixed cartesian coordi-nates, in 𝑥- and 𝑦-directions, is written
The fields for RHCP and LHCP can be written 𝐄L(𝑘L( ̄𝜈)) = 𝐸0 √2 ( 1 𝑖)𝑒𝑖(𝑘L ( ̄𝜈)𝑧−𝜔𝑡) 𝐄R(𝑘R( ̄𝜈)) = 𝐸0 √2 ( 1 −𝑖)𝑒𝑖(𝑘R ( ̄𝜈)𝑧−𝜔𝑡). (3.9)
For the case when 𝑘R( ̄𝜈) = 𝑘L( ̄𝜈) = 𝑘( ̄𝜈), linear combinations of the two circularly polar-ized electric fields reproduce the linearly polarpolar-ized fields, namely
𝐄x= 1 √2(𝐄L (𝑘( ̄𝜈)) + 𝐄R(𝑘( ̄𝜈))) 𝐄y= −𝑖 √2(𝐄L (𝑘( ̄𝜈)) − 𝐄R(𝑘( ̄𝜈))) . (3.10)
However, under the exposure to an external magnetic field, the two wavevectors, for circu-larly polarized light, are shifted so that 𝑘R≠ 𝑘L. For future use, it is suitable to decompose these two 𝑘-vectors into two other ones, 𝑘+and 𝑘−, as
𝑘L( ̄𝜈) = 𝑘++ 𝑘−
𝑘R( ̄𝜈) = 𝑘+− 𝑘−, (3.11)
where the frequency dependence, for readability, is not written out. 𝑘+and 𝑘−are given by 𝑘+= 𝑘L( ̄𝜈) + 𝑘R( ̄𝜈) 2 𝑘−= 𝑘L( ̄𝜈) − 𝑘R( ̄𝜈) 2 . (3.12)
The modified electric fields, ̃𝐄xand ̃𝐄y, are the transformations of 𝐄x and 𝐄ywhen exposed to an external magnetic field due to the Faraday effect. They are found by substi-tuting eq. (3.11) into eq. (3.9) and (3.10). The modified electric fields read
Laser
Polarizer Polarizer
Gas cell and electromagnet
Detector
Figure 3.4: Principle of how the polarization vector of lin-early polarized light is rotated by an electromagnet when exciting paramagnetic molecules, and how FAMOS makes use of this phenomenon.
and 𝐄y B-field −−−−→ ̃𝐄y= −𝑖 √2(𝐄L(𝑘 ++ 𝑘−) − 𝐄 R(𝑘+− 𝑘−)) = −𝑖𝐸0 2 [( 1 𝑖)𝑒𝑖(𝑘 ++𝑘− )𝑧− ( 1 −𝑖)𝑒𝑖(𝑘 +−𝑘− )𝑧 ]𝑒 −𝑖𝜔𝑡 = 𝐸0 2 ( −𝑖𝑒𝑖𝑘−𝑧+ 𝑖𝑒−𝑖𝑘−𝑧 𝑒𝑖𝑘−𝑧+ 𝑒−𝑖𝑘−𝑧 )𝑒 𝑖(𝑘+𝑧−𝜔𝑡) = 𝐸0(sin (𝑘 −𝑧) cos (𝑘−𝑧))𝑒𝑖(𝑘 +𝑧−𝜔𝑡) = [sin (𝑘−𝑧) 𝐄x+ cos (𝑘−𝑧) 𝐄y] 𝑒𝑖(𝑘 +−𝑘( ̄𝜈))𝑧 . (3.14)
We see that the magnetic field generates a rotation of the electric field around the propa-gation of light an amount of 𝑧𝑘−. It also provides an overall phase shift of (𝑘+− 𝑘( ̄𝜈))𝑧. From eq. (3.12) it is noted that 𝑘+may be interpreted as the mean wavevector of the LHCP and RHCP. Furthermore, the rotation angle 𝑘−𝑧, which is related to the difference between LHCP and RHCP, can be expressed as
𝑘−𝑧 = Δ𝑘( ̄𝜈L/R)
2 𝑧 , (3.15)
where Δ𝑘( ̄𝜈L/R) is given by 𝑘( ̄𝜈L) − 𝑘( ̄𝜈R). A rotation of the electric field of light is the same as a rotation of polarization by definition. Since the rotation of the polarization takes place between two almost perpendicularly oriented polarizers, it gives rise to an alteration of the transmitted amount of light. In fig. 3.4 it is shown how the rotation of the linear components of the electric field is related to the detected laser intensity in FAMOS.
The rotation angle Δ𝑘( ̄𝜈L/R)𝑧 / 2 is a result of the Faraday effect and it is of interest to relate Δ𝑘( ̄𝜈L/R) to Δ𝜙( ̄𝜈L/R). From eq. (2.28) we know the relation between the wavevector and the phase shift. Along with eq. (3.4), the wavevector is written
This equation is substituted into eq. (3.15) to obtain the rotation. It is known that a trigono-metric function with an imaginary argument is purely imaginary itself. This means that imaginary parts of Δ𝑘( ̄𝜈L/R) are directly translated into a purely imaginary electric field in eq. (3.13) and (3.14) which, in turn, implies that the Re-operator may be removed.
For 𝑧 = 𝐿, the rotation of polarization is calculated to be
𝑘−𝑧 = Δ𝑘( ̄𝜈L/R)𝐿 2 = 𝛼0 4Δ ̂𝜒 disp( ̄𝜈 L/R) . (3.16)
Since 𝑘−𝑧 is the amount of rotation and Δ ̂𝜒disp( ̄𝜈L/R) changes for different magnetic fields, it is clear that the rotation of polarization depends on the applied magnetic field as well as the SOT. The rotated light may pass the second polarizer to a different degree than what non-rotated light will. The higher the concentration of sample gas we have, the more the polarization of the light is rotated, and the larger fraction of laser light is transmitted through the second polarizer6and may impinge upon a detector.
3.2.4 Polarizer angles
To achieve a high signal-to-noise ratio, it is intuitive to let the modified detuning frequency result in a blocking of all light at one turning point of the modulation period. This lets us have as high transmission as possible at the other modulation turning point, and the difference in transmitted light increases between the two turning points. This means that we cannot have exactly 90∘angle between the two polarizers, we must rotate one polarizer a small angle which corresponds to half the peak-to-peak rotation of the linear polarization due to the applied B-field. This angle may be expressed analytically [12, 13] but in this thesis we will settle with the optimization of this angle experimentally. The small angle, by which the second polarizer deviates from 90° from the first, is named the uncrossing
angle. It is given in degrees and it is defined as
ΔΦ = Φ1− Φ2+ 90 , (3.17)
where Φ1and Φ2are the angles of the two polarizers, relative to an angle that only would transmit vertically polarized light, respectively.
For ΔΦ = 0 there would not be any transmitted light if the polarizers would be perfect, (i.e. having an extinction ratio of zero). For practical systems, where there exist small polarizer imperfections, the polarization becomes slightly elliptical rather than perfectly linear after the first polarizer. Depending on the extinction ratios of the polarizers, this may become a problem. After the second polarizer, including small polarizer imperfections and assuming small absorption, the final transmitted laser intensity is calculated by Jones’ matrices. It can be written [11, 12, 14], after series expansion, as
𝐼t≈ 𝐼0
2 [1 − cos (2ΔΦ) + sin (2ΔΦ) Δ𝜙 + 𝜀Δ𝛿] , (3.18) where Δ𝜙 is given by eq. (3.4) and Δ𝛿 is given by eq. (3.6). 𝜀 is a small unbalancing term between the two circularly polarized components of light. This implies that light after the first polarizer, whose polarization plane is along the 𝑥-axis, can be written as
𝐄 (𝑘( ̄𝜈)) = 𝐸0 √2 (
1
𝑖𝜀)𝑒𝑖(𝑘( ̄𝜈)𝑧−𝜔𝑡). (3.19)
6This is provided that the second polarizer is oriented in a way such that the polarization is rotated away from
Both Δ𝛿 and 𝜀 are small in comparison to the other terms which makes eq. (3.18) somewhat less complicated.
3.2.5 Modulation and demodulation
For Faraday modulation spectroscopy, the magnetic field is modulated. This process af-fects the demodulated signal and its shape is therefore different from the DAS technique. The first harmonic, 1f, is extracted by the use of a lock-in amplifier so that some flicker noise is eliminated. The magnetic field is described by
𝐵(𝑡) = 𝐵0cos (𝜔𝑡) . (3.20)
This technique is quite similar to the technique of WMS with the exception that for WMS the second harmonic is extracted instead of the first. Since the modified detun-ing frequency for LHCP and RHCP, ̄𝜈L/R, depends on the magnetic field, we have ̄𝜈L/R =
̄
𝜈L/R(cos(𝜔𝑡)) but it is not written out for simplicity. We also write
Δ𝜙 [ ̄𝜈L/R(cos(𝜔𝑡))] Δ𝛿 [ ̄𝜈L/R(cos(𝜔𝑡))] as Δ𝜙( ̄𝜈L/R) Δ𝛿( ̄𝜈L/R) and Δ ̂𝜒disp[ ̄𝜈L/R(cos(𝜔𝑡))] Δ ̂𝜒abs[ ̄𝜈L/R(cos(𝜔𝑡))] as Δ ̂𝜒 disp( ̄𝜈 L/R) Δ ̂𝜒abs( ̄𝜈L/R) for the same reason.
The demodulated signal is formulated as the averaged product between the transmitted intensity, 𝐼t, and the modulation function, cos (𝜔𝑡). There are similarities between the FAMOS signal, 𝑆F, and the DAS signal, eq. (2.39) where the instrumentation factor 𝜂 persists but the transmission factor, 𝑇 , is non-present. The uncrossing angle is moved out for convenience and so the unbalancing term needs to be adjusted accordingly.
For a modulated case, the terms of the signal that are not time dependent average out to zero because of the periodic integration and are therefore left out. We write the FAMOS signal as 𝑆F= 𝜂 𝜏 𝜏 ∫ 0 𝐼t(𝑡) cos (𝜔𝑡) d𝑡 = 𝜂𝐼0sin (2ΔΦ) 2 ⎧ ⎪ ⎨ ⎪ ⎩ 1 𝜏 𝜏 ∫ 0 Δ𝜙( ̄𝜈L/R) cos (𝜔𝑡) d𝑡 + ̄𝜀1 𝜏 𝜏 ∫ 0 Δ𝛿( ̄𝜈L/R) cos (𝜔𝑡) d𝑡 ⎫ ⎪ ⎬ ⎪ ⎭ (3.21) = 𝜂𝛼0𝐼0sin (2ΔΦ) 4 ⎧ ⎪ ⎨ ⎪ ⎩ 1 𝜏 𝜏 ∫ 0 Δ ̂𝜒disp( ̄𝜈L/R) cos (𝜔𝑡) d𝑡 + ̄𝜀1 𝜏 𝜏 ∫ 0 Δ ̂𝜒abs( ̄𝜈L/R) cos (𝜔𝑡) d𝑡 ⎫ ⎪ ⎬ ⎪ ⎭ ,
where we have introduced the modified unbalancing term ̄𝜀, which is given by
̄
𝜀 = 𝜀
Modified detuning frequency, ¯ν P h a se sh if t [a .u .] -0.5 -0.25 0 0.25 0.5 0.75 φ R φL
(a) Separation of phases be-tween LHCP and RHCP.
Modified detuning frequency, ¯ν
P h a se sh if t [a .u .] -0.5 -0.25 0 0.25 0.5 0.75 φ R −φL φR− φL
(b) Difference between phases in an external magnetic field of optimal magnitude.
Modified detuning frequency, ¯ν
P h a se sh if t [a .u .] -0.5 -0.25 0 0.25 0.5 0.75 φR −φL φR− φL
(c) Difference between phases in an external magnetic field of too weak magnitude.
Modified detuning frequency, ¯ν
P h a se sh if t [a .u .] -0.5 -0.25 0 0.25 0.5 0.75 φR −φL φR− φL
(d) Difference between phases in an external magnetic field of too strong magnitude. Figure 3.5: Separation and differences between the first harmonic phases of RHCP and LHCP light in a modulated magnetic field parallel to the optical path. Figure (b-d) shows the resulting difference between the phases for dif-ferent strength of the magnetic field.
In the last line of eq. (3.21), the eqs. (3.4) and (3.6) are used to express the signal in terms of the dispersion and attenuation lineshape functions. This equation is decomposed into two terms that are specific for the first harmonic demodulation. These contain two integrands that can be expressed as
1 𝜏 𝜏 ∫ 0 Δ ̂𝜒disp( ̄𝜈L/R) cos (𝜔𝑡) d𝑡 = 1 𝜏 𝜏 ∫ 0
[ ̂𝜒disp( ̄𝜈L) − ̂𝜒disp( ̄𝜈R)] cos (𝜔𝑡) d𝑡 = ̂𝜒1disp( ̄𝜈L) − ̂𝜒1disp( ̄𝜈R)
= Δ ̂𝜒1disp( ̄𝜈L/R)
and 1 𝜏 𝜏 ∫ 0 Δ ̂𝜒abs( ̄𝜈L/R) cos (𝜔𝑡) d𝑡 =1𝜏 𝜏 ∫ 0
[ ̂𝜒abs( ̄𝜈L) − ̂𝜒abs( ̄𝜈R)] cos (𝜔𝑡) d𝑡 = ̂𝜒1abs( ̄𝜈L) − ̂𝜒1abs( ̄𝜈R)
= Δ ̂𝜒1abs.
(3.24)
̂
𝜒1disp/abs( ̄𝜈L) are the first Fourier coefficients of the modulated lineshape functions. Like-wise, the Δ ̂𝜒1absare the differences between each pair of Fourier coefficients. This lets us write the FAMOS signal as
𝑆F=
𝜂𝛼0𝐼0𝑠𝑖𝑛 (2ΔΦ)
4 [Δ ̂𝜒
disp
1 ( ̄𝜈L/R) + ̄𝜀Δ ̂𝜒1abs( ̄𝜈L/R)] . (3.25) By use of the WWA-algorithm [15], using transition parameters for a Q-branch Voigt profile of NO at room temperature, examples of the dominating term,
Δ𝜙1= 𝜙1,L− 𝜙1,R = 𝛼0 2 [𝜒̂ disp 1 ( ̄𝜈L) − ̂𝜒 disp 1 ( ̄𝜈R)] = 𝛼0 2 Δ ̂𝜒 disp 1 ( ̄𝜈L/R) ,
are commonly calculated numerically. These are plotted in fig. 3.5. As it is seen from this figure, there is an optimal magnetic field for an implementation of FAMOS and it is also noted that a stronger B-field does not always imply a stronger signal.
The reason for using the first harmonic for FAMOS, rather than the second, which is used in WMS, originates from the symmetrical intensity dependence around 𝜈0for WMS, see fig. 3.2(b). For FAMOS the passing intensity depends on an anti-symmetrical phase shift around 𝜈0for the two circular polarization components, see fig. 3.5(a), therefore the harmonic which contains the signal is odd rather than even.
4 Design of FAMOS Apparatus
The main goal of this thesis work is to construct and optimize a FAMOS system that is capable of measuring the concentration of nitric oxide, NO. Based on the theoretic foundation given in the previous sections, a basic setup has been constructed.
4.1 Setup
Out In Temperature Amplifier Function Laser Lock-in amplifier Data acquisition Detector Polarizer Polarizer
Gas cell and electromagnet
controller Temperature controller synthesizer Pre-amplifier Trans-impedance amplifier
Figure 4.1: A schematic model of a Faraday modulation spectroscopy setup.
the detector signal, which is amplified by a pre-amplifier. The FAMOS signal, 𝑆𝐹, is ac-quired from the lock-in amplifier and by data processing, quantitative information about the sample concentration is assessed, as the amplitude of the signal curve.
4.1.1 Laser, laser driver and function synthesizer
A Fabry Perot quantum cascade laser, QCL, is mounted in a LDMC20 c-mount and fo-cused by an aspheric lens, Black Diamond, both constructed by Thorlabs. To address the most suitable transition, Q3/2(3/2) in NO, the laser is operating at approximately 5.331 μm, which corresponds to the wavenumber 1875.81 cm−1. The lens has an anti-reflection coat-ing that has a reflectance of 4.9% at this frequency (accordcoat-ing to specification provided by the manufacturer).
The laser driver, Thorlabs LDC4002, is a combined current driving unit and a temper-ature controller, which is abbreviated TEC. It can provide a maximum current of 2 A and it has a current accuracy of 1.4 mA. The integrated TEC has an accuracy of 1 × 10−4°C. This unit has the ability to sweep the output current by a triangular input voltage, generated by a function synthesizer, HP 8904A by Hewlett-Packard. The amplitude of this voltage is directly translated to the laser sweep amplitude which determines the scanned frequency span.
4.1.2 Polarizers, gas cell and vacuum pumps
an angle of ∼5∘results in a ray separation of ∼15∘and, according to measurements, with no decrease in extinction ratio (that is the ratio of light that passes the angle that optimally should be blocked, over the total amount of light). Still, a distance for separation of the beams must be allowed before the light may enter the gas cell. In this setup, a distance of 15 cm is used and two beams are separated by approximately 3 cm.
The linearly polarized beam is entering a 25 cm long glass cavity filled with nitric oxide (NO). The windows of the cavity are made of CaF2which has a transmission of approximately 95% at the wavelength of use. To avoid excessive collision broadening, the pressure inside the cell should be significantly lower than the atmosphere, as will be de-termined in section 4.3.1. Two vacuum pumps are connected in series in order to generate as high vacuum as possible, one pre-vacuum pump, Leybold SCROLLVAC SC 5 D, and one turbo pump, Leybold TURBOVAC SL 80. The former has an ultimate pressure of 5 × 10−2mbar while the latter has an ultimate pressure of 2 × 10−10mbar. The pumps are connected to, what is referred to as, the vacuum buffer. This part of the system is separated from the gas cell by a valve which is closed during measurements so that pressure and gas concentration are preserved as effectively as possible.
As the beam exits the gas cell, it passes through the second polarizer which is of wire grid type. This polarizer is equipped with an integrated protractor with an accuracy of 0.1°. The protractor is used to determine the uncrossing angle.
4.1.3 Electromagnet and amplifier
In Faraday modulation the modulated property is, as already described, the energy of the states of the interacting substance, NO. This is modulated by the use of a magnetic field parallel to the propagation of light. To create this magnetic field we first generate a sine wave. This is done by the same lock-in amplifier that also is used to demodulate the signal after detection. This sine wave is amplified in an audio amplifier so that the current is strong enough to drive an electromagnet surrounding the gas cell.
The electromagnet consists of 765 turns of 1.0 mm thick copper wire. The electromag-net has been equipped with a capacitor bank tuning the natural resonance of the circuit to a frequency of 7.4 kHz such that a high AC current at this frequency may be produced. This frequency is suitable for FAMOS as it compromises the heat generation while significantly decreasing flicker noise.
The electromagnet is capable of producing an AC B-field of more than 2000 G; al-though the stronger magnetic field produced, the more heat is produced. This may become a problem if the magnet is run with a too large current for too long time. The electromag-net is equipped with a thermo-couple of standard type. It gives an estimate of the inner temperature of the coil. More on the lack of reliance of these temperature measurements and its consequences is presented in section 5.2.2.
4.1.4 Detector, trans-impedance amplifier and pre-amplifier
