gas detection by use of sagnac interferometer - qut...

Gas detection by use of Sagnac interferometer

Sean Robert McConnell

BappSc (Physics)

A thesis submitted at the Queensland University of

Technology, in the School of Physical and Chemical Sciences

in partial fulfillment of the requirements for the degree

of Master of Applied Science (Research).

July 2008

Keywords

Sagnac, Interferometer, Interferometry, Gas analysis, Gas analyser, Polarizability, Dy-

namic Polarizability, Lorentz-Lorenz equation, Optical dispersion.

iii

Abstract

Gas composition and analysis forms a large field of research whose requirements demand

that measurement equipment be as affordable, uncomplicated and convenient as possible.

The precise quantitative composition of an atmospheric, industrial or chemically synthe-

sised sample of gas is of utmost importance when inferring the properties and nature of

the environment from which the sample was taken, or for inferring how a prepared sample

will react in its application. The most popular and widely used technique to achieve this is

Gas Chromatography-Mass Spectrometry (GCMS) and, without a doubt, this technique

has set the standard for gas analysis.

Despite the accuracy of the GCMS technique, the equipment itself is bulky, expensive

and cannot be applied readily to field work. Instead, most field work is conducted using

a single gas detector, capable only of detecting one particular molecule or element at a

time. Presented here is an interferometric technique that theoretically, has the ability

to address all three issues of bulkiness, affordability and convenience, whilst not being

limited to one particular element or molecule in its analysis.

Identifying the unknown constituents of a gaseous mixture using the proposed method,

employs the optical refractive properties of the mixture to determine its composition. A

key aspect of this technique is that the refractive index of an arbitrary mixture of gases

will vary depending on pressure and wavelength1.

The Lorentz-Lorenz formula and the Sellmeier equations form the foundation of the

theoretical background. The optical refractive properties of air and other atmospheric

gases have been well established in the literature. The experimental investigations de-

scribed here have been conducted based on this, insofar as no analysis has been conducted

on gases that do not naturally occur in reasonable abundance in the atmosphere. However

this does not in any way preclude the results and procedure developed from applying to

a synthesised gas mixture.

1refractive index also varies with temperature, although the temperature will be held fixed

v

As mentioned, the platform of this technique relies on the pressure and wavelength

dependence of the refractivity of the gas. The pressure dependence of the system is easily

accounted for, in making this claim however it is still imperative the mixture be impervious

to contamination from the wider atmosphere. Wavelength dependence however is perhaps

slightly more difficult to accommodate. Multiple lasers, of differing wavelength form

the radiative sources which underpin the method developed. Laser sources were chosen

because of their coherence, making it easy to produce interference, when combined with

the inherent stability of the Sagnac interferometer, provides for a very user friendly system

that is able to quickly take results. The other key part of the experimental apparatus is

the gas handling system, the gas(es) of interest need to be contained within an optical

medium in the path of one of the beams of the interferometer. Precise manipulation of

the pressure of the gas is critical in determining concentration, this has been achieved

through the use of a gas syringe whose plunger is moved on a finely threaded screw, and

measured on a digital manometer. The optical setup has also been explored, specifically

in ruling out the use of such radiative sources as passing an incandescent source through

a monochromator or the use of LED’s to produce interference before settling on lasers to

produce the required interference.

Finally, a comprehensive theoretical background has been presented using classical

electromagnetic theory as well as confirmation from a quantum perspective. The theo-

retical background for this study relies upon the Lorentz-Lorenz formula. It is commonly

presented either from a classical or quantum perspective, in this work both classical and

quantum mechanical treatments are given whilst also showing how each confirms the

other. Furthermore, a thorough investigation into the dispersion functions of each of the

major components of the atmosphere has been compiled from the study of refractivity

on individual gases from other authors, in some cases, where no work has been done

previously, this has been derived.

The technique developed could be considered an ample addition to gas analysis tech-

niques in certain circumstances in terms of expense, convenience and accuracy. The

vi

system can predict relative quantities of constituents of the atmosphere to at least 3%.

The method described here would allow researchers more time to concentrate on actual

results and more resources to allocate to broadening intellectual horizons. This would

certainly justify further development.

vii

Contents

1 Introduction 1

1.1 Comparison to Other Gas Analysis Techniques . . . . . . . . . . . . . . . . 2

1.1.1 Optical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.2 Gas Chromatography Mass Spectrometry . . . . . . . . . . . . . . . 3

1.1.3 Interferometric methods and refractive index methods . . . . . . . . 4

1.2 Creating an interferometric gas analyzer, and interpreting the results . . . 6

2 Inferring quantities of constituent gases in a sample using interferome-

try 9

2.1 The interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Gas handling and measurement system . . . . . . . . . . . . . . . . . . . . 10

2.3 Operating theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Restrictions inherent to using interferometry for gas analysis . . . . . . . . 16

2.5 Optical properties of aerosols and their influence on results . . . . . . . . . 18

2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Theory of refraction of gases 23

3.1 The Lorentz-Lorenz equation . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 The classical determination of the dynamic polarizability . . . . . . . . . . 27

3.3 Quantum mechanical determination of the dynamic polarizability . . . . . 32

3.4 Quantum mechanical confirmation of the Lorentz-Lorenz equation . . . . . 35

3.5 An alternative quantum mechanical derivation of the Lorentz-Lorenz equation 40

3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4 System overview and operation 43

4.1 Range of detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2 Tables of dynamic polarizabilities and dispersion for atmospheric gases . . 44

4.2.1 Argon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

ix

4.2.2 Hydrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2.3 Helium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2.4 Nitrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2.5 Neon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2.6 Oxygen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2.7 Water Vapour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.2.8 Carbon Dioxide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2.9 Methane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.3 Minimum particle density required to detect a gas using interferometry . . 60

4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5 Experimental results and analysis 67

5.1 Properties of the linear system . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.2 Results: Variation of output with pressure . . . . . . . . . . . . . . . . . . 69

5.3 Refractive index vs. pressure for various wavelengths . . . . . . . . . . . . 72

5.4 Composition of the atmosphere using interferometry . . . . . . . . . . . . . 72

5.5 Uncertainty of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6 Conclusion 91

6.1 Experimental refinements . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

x

List of Figures

2.1 The Jamin interferometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 The Sagnac interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Variance of the refractive index with pressure . . . . . . . . . . . . . . . . 13

2.4 The photodetector output as a function of pressure. . . . . . . . . . . . . . 14

3.1 The distortion of the electron cloud of a particle [13]. . . . . . . . . . . . . 25

3.2 The polarizability near an absorption line. . . . . . . . . . . . . . . . . . . 30

3.3 The gradient of the potential of the Hydrogen atom. . . . . . . . . . . . . . 39

4.1 The Polarizability of Argon from various authors. . . . . . . . . . . . . . . 46

4.2 Comparison of Refractivity of Argon with data gathered from [32]. . . . . . 47

4.3 The polarizability of Hydrogen from various authors. . . . . . . . . . . . . 49

4.4 The polarizability of Helium from various authors. . . . . . . . . . . . . . . 51

4.5 The polarizability of Nitrogen from various authors. . . . . . . . . . . . . . 52

4.6 The Polarizability of Neon from various authors. . . . . . . . . . . . . . . . 53

4.7 The Polarizability of Oxygen from various authors. . . . . . . . . . . . . . 55

4.8 The Polarizability of Water Vapour from various authors. . . . . . . . . . . 56

4.9 The Polarizability of Carbon Dioxide from various authors. . . . . . . . . . 58

4.10 The Polarizability of Methane from various authors. . . . . . . . . . . . . . 59

4.11 Sample output of interferometer, run 1. . . . . . . . . . . . . . . . . . . . . 63

4.12 Sample output of interferometer, run 2. . . . . . . . . . . . . . . . . . . . . 63

5.1 Photodetector output, 200 mm cell, 532 nm . . . . . . . . . . . . . . . . . 70

5.2 Photodetector output, 200 mm cell, 543.5 nm . . . . . . . . . . . . . . . . 70



5.5 Photodetector output, 200 mm cell, 632.8 nm . . . . . . . . . . . . . . . . 71


5.7 Refractive index vs. pressure, various wavelengths, 0 atm. . . . . . . . . . 73

xi

5.8 Refractive index vs. pressure, various wavelengths, 1 atm. . . . . . . . . . 73

5.9 n vs. P for various wavelengths. . . . . . . . . . . . . . . . . . . . . . . . . 73

5.10 Variance of fringes as ξ changes . . . . . . . . . . . . . . . . . . . . . . . . 84

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List of Tables

4.1 Refractive index and Polarizability for Argon (Ar). . . . . . . . . . . . . . 45

4.2 Refractive index and polarizability for Hydrogen (H2). . . . . . . . . . . . . 48

4.3 Refractive index and polarizability for Helium (He). . . . . . . . . . . . . . 50

4.4 Refractive index and polarizability for Nitrogen (N2). . . . . . . . . . . . . 52

4.5 Refractive index and polarizability for Neon (Ne). . . . . . . . . . . . . . . 53

4.6 Refractive index and polarizability for Oxygen (O2). . . . . . . . . . . . . . 54

4.7 Refractive index and polarizability for Water (H2O). . . . . . . . . . . . . . 56

4.8 Refractive index and polarizability for Carbon Dioxide (CO2). . . . . . . . 57

4.9 Refractive index and polarizability for Methane (CH4). . . . . . . . . . . . 59

4.10 Minimum number density to elicit interferometric detection at λ = 555 nm

and L = 0.5 m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.1 Gradient of refractive index vs. pressure at various wavelengths . . . . . . 74

5.2 Composition of Earth’s atmosphere at 0% RH . . . . . . . . . . . . . . . . 77

5.3 Composition of Earth’s atmosphere at 72% RH, 22◦C . . . . . . . . . . . 78

5.4 Polarizability of Water Vapour, CO2 and air at experimental wavelengths. 79

5.5 Polarizability of N2, O2 and Ar at experimental wavelengths. . . . . . . . 80

5.6 Uncertainty in gradient of refractive index vs pressure . . . . . . . . . . . 85

xiii

Statement of original authorship

The work contained in this thesis has not been previously submitted for a degree or

diploma at any other higher educational institution. To the best of my knowledge and

belief, the thesis contains no material previously published or written by another person

except where due reference is made.

Sean Robert McConnell

xvi

Acknowledgements

Whenever I am asked, “what do you do for a living?” and I respond, “I’m a

Physicist,” the common reaction is usually, “wow, you must be smart!” To

which it is my custom to reply, “I’m stupid enough to study Physics!”

I would of course like to thank my supervisor, Esa Jaatinen. His very calm nature and

gentle approach has allowed me a lot of freedom in creating something which I feel is not

just an investigation into some very important physical properties, but also I feel it is an

expression of my own creativity, something which not a lot of first time reasearchers are

given license to use. I must also thank Ian Turner for his help on the linear algebra side

of this work, without which I would have surely have not made it this far. My colleagues,

Jye Smith, Scott Crowe, Richelle Gaw, Afkar Al-Farsi, Dan Sando, Dan Mason, Michael

Jones, Martin Kurth, Kristy Vernon and David Hopper have all made this experience

that much more bearable. Finally, I’d like to thank my high school physics teacher, Ian

Kelk. Although he probably would rather forget I existed at all, I’m sure he would be

very proud of me now.

xvii

List of Abbreviations

GCMS . . . Gas Chromatography Mass Spectrometry

RHS . . . Right hand side

LHS . . . Left hand side

IRED . . . Infra-red emitting diode

IR . . . Infra-red

UV . . . Ultra-Violet

STP . . . Standard Temperature and Pressure

LED . . . Light Emitting Diode

CRO . . . Cathode Ray Oscilliscope

OPD . . . Optical path difference

xix

1

1 Introduction

The accurate determination of constituent elements or molecules, and their concentration,

in a gaseous mixture is an essential part of many industrial applications, chemical and

physical research and atmospheric and meteorological studies. A key requirement of any

composition analysis method is that it be accurate and versatile; accurate because, for

example, even very subtle changes in atmospheric composition can have greatly varying

consequences on human health, or the health of a certain species in an environment which

is exposed to gases that may be dangerous in certain quantities; and versatility because

it can become expensive and impractical to employ a variety of methods to study gases

simply because the components of interest or conditions have changed.

The ideal method for analyzing any gas or mixture of gases would be one which could

quantitatively identify all constituents of a mixture, no matter what they are or in what

proportion they come. In terms of the spectrum of analyzers, some methods are able

to detect quantities of a single element or molecule, some are able to detect a number

of gases sharing similar properties and the most advanced methods come very close, if

not meet the characterstics required of an ideal gas analyzer. The leading candidates to

be considered an ideal gas analyzer are currently, gas chromatography mass spectrometry

(GCMS) or a combination of multiple spectroscopic techniques. The purpose of this paper

is to investigate the potential of using an interferometric technique in identifying gaseous

mixtures, and how it compares to existing approaches by determining how accurate and

useful it can be.

Although studied previously as a gas analyzer, and we will see in what forms shortly,

interferometry has not been investigated as a potential analyzer of more than simple

gaseous mixtures. Both [36] and [6] appear to pre-empt or call for further investigation into

this possibility in their work, with both inferring that the calculation of the refractivity

of a mixture having known their constituents could be applied inversely to calculate the

proportion of its constituents knowing the refractivity.

2 1 INTRODUCTION

The determination of the refractive index of gases, specifically air, as a combination of

multiple gases, and of course the individual constituents of air itself will be shown. The

refractive index of air, its variance with pressure, temperature and wavelength has been

well established in the literature [4; 28; 38; 19; 20; 21; 58].

To date, interferometry has not received wider consideration as a technique for gas

analysis, perhaps because most research has been built around other methods. The aim

of this work is to quantitatively provide a review of the effectiveness of interferometry as

a gas analyser, and to provide conclusions as to whether such a system can be commer-

cialised or replace other more expensive or restrictive methods. This will be achieved by

experimentally determining the quantities of a gaseous mixture, specifically air, using the

optical dispersion of the each species of the mixture. Furthermore, a study of the con-

straints shall be undertaken, to act as a guide as to what circumstances interferometric

techniques are most effective in determining quantities.

It is useful at this point to investigate how other techniques work, to examine if

an interferometric method could compete with traditional methods, and identify what

limitations, advantages and disadvantages there are in using an interforemeter to analyze

a gas sample.

1.1 Comparison to Other Gas Analysis Techniques

A well regarded extensive reference on gas analysis [74], devotes only a single page to

interferometric methods. The objective of this section is to compare briefly the versatility

of each method and specifically, how many different types of gases can be detected and

the accuracy of each method.

1.1.1 Optical methods

Although interferometry is itself an optical method it is not by any means the only opti-

cal method. Spectroscopic techniques exist for the identification of atoms and molecules

1.1 Comparison to Other Gas Analysis Techniques 3

according to transitions between quantised energy levels. Vibrational levels correspond

to far IR, electronic levels correspond to visible and near UV spectra, and vibro-rotation

transitions produce spectra in the near IR. Vibro-rotational transitions are principally

used in IR analysers [74].

The operating theory behind the following spectroscopic methods is reasonably straight-

forward. A broad spectrum source is split into a reference and analyzer beam. The

analyzing beam pases through a cell containing the gas of interest, and recombines with

the reference beam at the detector. The difference in output is due to the absorption prop-

erties of the gas in the cell, in this way, quantities and species of gases may be identified.

Infra-Red analysers are able to detect most gases with a few exceptions. O2, N2, H2, Cl2,

the inert gases and organic vapours do not have a unique spectrum in this region, and

so cannot be detected using IR methods, however anything else with a covalent bond is

detectable, ie. the heteroatomic gases (e.g., CO,CO2, SO2,NO,HC and NH3)[65] . Ac-

curacies are usually quoted for CO and CO2, with an accuracy down to 10ppm for CO2

[65].

Ultra Violet analysers have a similar function to IR techniques, but are able to exploit

different forms of molecular excitation and thus different compounds compared to IR

methods. Specifically, UV analysers are particularly proficient at detecting Cl2, SO2,F2

and O3 as well as aromatic Hydrocarbons and important pollutants like H2S, with an

accuracy similar to IR methods. Spectroscopic techniques could well be a powerful tool

in gas detection, especially so when an IR instrument is paired with a UV instrument,

greatly increasing the number of gases detectable [74]. Some of the limitations though with

spectroscopic methods are of course its inability to detect certain species, and expense,

expecially if as proposed, one were to combine multiple instruments.

1.1.2 Gas Chromatography Mass Spectrometry

GCMS is a seperating process, followed by an analyzing process. The gas chromatography

stage passes a sample into a flow stream of an inert gas, through a column containing

4 1 INTRODUCTION

certain substances (for example, charcoal is used to seperate O2 from H2[42]) designed to

impede the flow of the components in the mixture, such that individual components leave

the column at different times. A multiple column arrangement with many seperating sub-

stances is able to seperate and detect a greater range of gases. Many of the commercially

available designs can easily claim an accuracy of less than 1 ppm [74].

The gas chromatograph needs to be calibrated with a pure sample of each constituent

gas before it is able to measure a mixture. The time it takes for a particular pure sample

to travel the length of the column and arrive at the detector is known as the retention

time, and this is different for each different gas. Once a catalogue of retention times

for pure samples has been collated, the experimenter may then use this knowledge in

the analysis of a mixture. The mass spectrometry stage removes this requirement, for

once the constituent of the mixture has been seperated out by the chromatograph, upon

entering (the evacuated chamber of) the mass spectrometer, it is ionised in an electron

beam and passed through a constant magnetic field. The atomic mass of the substance will

determine the radius of the path it will take in the magnetic field, heavier substances will

trace a larger radius path and vice versa. In this way, it is not a difficult step to infer from

the knowledge of atomic mass, exactly what that particular substance is. Some difficulty

arises when two different gases have the same atomic mass number, however without

going into it here, there are methods around this problem. Accuracies are common in the

hundredths of parts per million[42], easily making a combination GCMS system the most

accurate and versatile system available. These systems however are bulky and therefore

not well suited to in situ measurements, furthermore, their expense can make acquiring

such a machine somewhat prohibitive.

1.1.3 Interferometric methods and refractive index methods

The focus of this section will be to review the manner in which other authors have used

interferometry or a sample’s refractive index as a gas analyzing technique.

An index of refraction method [55], aimed at natural gases and other oil field applica-

1.1 Comparison to Other Gas Analysis Techniques 5

tions, forces high pressure gases to flow through an optical cell, which is illuminated from

the side using a GaAs Infra-Red emitting diode (IRED). The light from the IRED passes

through sapphire, which is joined to the optical cell, with the intention of producing total

internal reflection inside the sapphire interface. The angle of reflection is dependant on

the refractive index of the gas in the cell, and the angle at which the light is reflected is

picked up on a detector array. This system, providing a light source that can be selective

of wavelength, could be modified into a detector capable of determining the concentration

of an arbitrarily high number of gases. The accuracy of such a system can be determined

from Snell’s law, the more accurately one can determine changes in the angle of reflection

(quoted incidentally by [55] at ∆Θ = 0.55◦), the better one can determine a change in

refractive index (and thus concentrations) in the cell.

Changes in the concentration of anaesthetic gases have also been interferometrically

measured using a Jamin interferometer [71]. In this technique, an interference pattern is

monitored for any changes in the concentration of the anaesthetic gases. Any variance

in the pattern impels a feedback system to alter the flow rates of one or another of the

anaesthetic gases until the interference pattern is returned to its “reference”. Although

it is a simple system consisting of no more than 2 gases (Oxygen and the required anaes-

thetic), it could readily be modified into a far more versatile gas analysis system similar

to the one described here.

Another more recently developed, and quite novel technique[50] describes the use of

holography to measure changes in the refractive index of a gas as a function of pressure.

The configuration described uses a Mach-Zender interferometer to generate a grating in

a photorefractive crystal, but simultaneously, one of the beams of the interferometer is

split, and guided through a Jamin interferometer2. The outgoing beam from the Jamin

Interferometer is then guided, parallel with the Mach-Zender beam3 toward the photore-

2The Jamin Interferometer contains a cuvette filled with a particular gas, the beam is guided throughthe gas, with the intention of varying the refractive index of the gas and changing the path length thebeam travels

3from which it was originally split

6 1 INTRODUCTION

fractive crystal, creating two Bragg gratings in the crystal. Using an IR readout beam,

changes in the phase of the grating can be detected. The changes in the phase are directly

related to the changes in the refractive index of the gas in the cell by the following formula

∆n =∆φ

2π

λ

l(1.1)

where φ is the phase, λ the wavelength of the beam used by the Mach-Zender Interferom-

eter, l the length of the cuvette in the Jamin Interferometer and n the refractive index of

the gas. If one can know how the refractive index will change, in this case with pressure,

then one can predict changes in phase of the grating4.

The error in detecting changes in refractive indices of this method are quoted to be

∼ 1.3×10−10. In terms of changes in mole fractions of a gas in the cuvette, this corresponds

to at best ∆N ∼ 0.00047. As a reference, the mole fraction of Nitrogen in the air is ∼ .78.

The advantage of this optical system is that it appears it is able to measure n more

precisely than other techniques.

The advantage of the technique described here is that it could easily be applied to all

of the interferomtric uses described in this section, as well as the quantitative detection

of gases. In some ways it would be useful to think of the technique described here as an

agglomeration of some of the key aspects of the methods developed elsewhere, making for

a more versatile, useful system.

1.2 Creating an interferometric gas analyzer, and interpreting

the results

The ensuing chapters of this work will center on how the Sagnac interferometer is set up to

serve as a tool for analyzing the quantities of the constituents of a gas. We will also expand

upon the essential tools required to effectively handle a gas, to allow it to be analyzed

interferometrically. Beyond this, the background theory of what essential properties of

4Changes in phase are measured as changes in reflected intensity of the IR readout beam

1.2 Creating an interferometric gas analyzer, and interpreting the results 7

matter that allow for optical dispersion, the key aspect upon which this technique relies,

to be realised, and what differentiates the dispersion properties of one atom or molecule

from the other shall be presented. Also provided is a comprehensive list of the dispersion5

of most of the major constitutuents of Earth’s atmosphere, a compilation that to date

has not been made. Finally, we will see the results of our technique in determining

the concentration of the constituents of a gas, how accurate it is, how it compares to

other techniques, and any potential improvements and insights a gas detector by Sagnac

interferometer can give.

An investigation of this type, i.e. one that asks whether an interferometer can function

effectively as a multiple gas analyzer has not to date been explored. The intention of this

work is to provide the reader with a basis for understanding how versatile an interferometer

can be as a gas analyzer, the results shown here certainly provide justification for it to be

used as such in a far greater role than it has enjoyed previously. With the modifications

suggested, the Sagnac interferometer could well form part of any chemist’s, environmental

scientist’s or meteorologist’s arsenal of tools in quantitatively differentiating mixtures of

gases.

5In a standardized equation form.

8 1 INTRODUCTION

9

2 Inferring quantities of constituent gases in a sample

using interferometry

2.1 The interferometer

The main focus of this excercise is to identify gas mixtures through the use of interfer-

ometry. As will be shown, it is a simple task to extract the refractive index of a sample

a gas when placed in the path of one of the beams of an interferometer.

The first question to ask is, “which interferometer is best suited to this task?” The

underlying principle is to employ equipment that is relatively inexpensive, but also main-

tain a high degree of accuracy and stability. Since gas concentration will be inferred from

the change in phase of the interferometer output, any changes due to other effects need to

be minimized. There are many choices available and the simplest and most obvious choice

would be a Michelson interferometer, however unless a purpose built Michelson is used,

it is very difficult to set up a Michelson interferometer on an optical bench that retains

a high level of stability without an active feedback system to reduce vibrational effects.

Two other interferometers considered were the Jamin type and Mach-Zender. The Mach-

Zender was discounted as the two beams travel different paths, and therefore, any change

in path length, due to a disturbance of either mirror, will affect one beam independently

of the other. This is undesirable, as the interference pattern produced has the potential

to change should one of the mirrors be disturbed. The Michelson interferometer exhibits

an identical problem.

The Jamin type interferometer would probably have been the ideal choice. The two

beams of the interferometer both strike the two optical surfaces (see figure 2.1) thus any

disturbance affecting one of the beams will equally affect the other, and more importantly,

because of the geometry, the disturbances will be registered at the detector at the same

time. This property greatly increases the stability of the Jamin interferometer. Another

interferometer with similar properties, and the instrument used for this work, is the

102 INFERRING QUANTITIES OF CONSTITUENT GASES IN A SAMPLE USING

INTERFEROMETRY

Sagnac interferometer. Like the Jamin, each beam strikes every optical surface once.

However, unlike the Jamin, if the mirrors A and C are disturbed, the time it takes for

these disturbances to reach the detector is not equal. From the diagram, we can see that if

a disturbance occurs at say mirror A, it would take t1 = d1

cseconds for the disturbance to

reach the detector travelling through beam one and t2 = d3+d2+d1

cseconds when travelling

through beam two. This decreases the stability of the Sagnac interferometer against

the Jamin, although the time scales for these fluctuations are far shorter than for any

changes produced due to fluctuations in the gas. The Sagnac is still admirable for its

stability and simplicity, thus it was decided to employ it as the interferometer for this

work. Furthermore, the Sagnac interferometer has a greater rotational stability compared

to the Jamin, as shown in [46].

Figure 2.1: The Jamin interferome-ter

Figure 2.2: The Sagnac interferom-eter.

2.2 Gas handling and measurement system

As has already been mentioned, a critical aspect of this excercise is to minimize changes

in the refractive index of the gas in the cell that are not due to controlled changes in

pressure. The experiment requires the pressure of the gas in the cell to change over a

2.2 Gas handling and measurement system 11

large range. In measuring this range, the experimenter must be confident that there are

no leaks. Furthermore, for studies of non-atmospheric gases, the gas handling system

must be impervious to contamination by the external atmosphere. These requirements

dictate a perfectly sealed system.

In the experiment performed, a series of 4 optical cells were used, placed in the path

of one of the beams of the interferometer. Each of the cells were connected with the

other via clear plastic tubing, which were sealed around the joints with an epoxy resin.

The tubing was connected to a hand vacuum pump and manometer, the vacuum pump

is necessary for two reasons, first to evacuate the cell of any atmospheric gases, should

a non-atmospheric mixture need to be analysed, and to provide a large pressure range

to ensure that a sufficient number of fringes pass the detector. A valve system is also

employed for the purposes of studying non-atmospheric gases, once the cells and tubing

have been evacuated, the valves may be opened to allow the gas of interest to flow into the

cell. Finally, a gas syringe (60 mL) is also used to provide fine control of the pressure. The

mixture of interest is firstly allowed to be injected into the fully open syringe, once full,

the syringe is placed in a mount, with the plunger attached to a finely threaded screw,

to allow the plunger to be closed at a very accurately determined rate. This is critical

for an accurate determination of the pressure. Whilst the manometer can measure to a

specified uncertainty, the threaded screw allows the experimenter to know precisely how

many turns of the screw will produce a given pressure change, to a much greater accuracy

than the manometer6.

As will be shown in Chapter 5, an accurate determination of the range of pressure

used, and the fringes that pass the detector over this range is critical.

The measurement system involves simply a photodiode and CRO. The recombined

beams at the interferometer output move in and out of phase, as the vacuum pump

increases the pressure7 in the cell. This produces a sinusoidal variation of output intensity

6approximately 5 times more accurately7Our vacuum pump was quite versatile in that it could also be used in reverse, as an ordinary pump.


INTERFEROMETRY

with pressure, as observed on the CRO. One simply counts the number of peaks or troughs

that appear on the screen for a given pressure range. We will expand on how one counts

exactly the number of fringes and fractional fringes later on in sections 4.3 and 5.5.

2.3 Operating theory

The principle behind the operation of this interferometer is that one monitors the output

of the recombined beam at the detector as a function of the varying pressure in the gas

cell. The Lorentz-Lorenz equation tells us that the refractive index of a gas depends upon

wavelength, temperature, pressure and composition [18].

n(ω) ≃ 1 +2πPNA

RT

∑

k

xkα(ω)k (2.1)

Where n(ω) is the refractive index at angular frequency ω, P,NA, R, T are the pressure,

Avogadro’s number, the gas constant and temperature respectively and xk and α(ω)

are the mole fraction of the kth constituent of the mixture and the polarizability at ω

respectively. For a fixed wavelength and ambient temperature, the only variable available

to the experimenter for adjustments is the pressure of the mixture in the cell. As the

number density of the gas changes, so does the refractive index in the cell. This change

in the refractive index shifts the path length of the beams of light. As the pressure in

the cell changes, the output at the detector will vary from maximum to minimum. In

the case where the output begins at some maximum, varies to a minimum and returns

to a maximum for a continuously increasing (or decreasing) pressure, the path length

difference over this pressure range would equal one integer wavelength.

The magnitude of this pressure difference ∆P required to change the path length by

one integer wavelength depends on the polarizability of the sample for the given wave-

length. If we were to plot the pressure dependence of the path length, a graph similar to

figure 2.3 is obtained. The polarizability for a particular gas for a particular wavelength

2.3 Operating theory 13

Pressure (P)

OPD(mλ = nl)

P1 P2

∆P

mλ

n1l

n2lgradient = mλ

P

Figure 2.3: Variance of the refractive index with pressure at approximately S.T.P. Al-though, according to the formal definition the refractive index is non-linear with tem-perature and pressure, we may treat it as such for the purposes of this experiment, seesection 3.1.

can be inferred by measuring the gradient of figure 2.3.

Dividing the gradient by the length of the cell will give the ratio between n and P . The

gradient of this line changes with every different wavelength laser that is used. How much

this gradient changes is indicative of the constituents in the gas. Therefore, by taking

measurements over multiple wavelengths, and recording the gradient for each, we can set

up a linear system of equations that can be used to reveal the quantities of the constituents

of the mixture. So, if there are say 5 unknown constituents in the mixture, then we

will need to run the experiment with at least 5 different wavelength lasers, otherwise

the linear system will be underdetermined, and there will be infinitely many solutions,

or combinations of relative quantities of each component that could explain the results

taken.

Let’s now take a look at the development of the idealised linear system. First, the


INTERFEROMETRY

Figure 2.4: The photodetector output as a function of pressure

If the output from the interferometer were measured by a photodiode as the pres-sure in the cell was varied, its response would look something like that shown here forthe same pressure range as in figure 2.3.

refractive index can be written in terms of its constituent components, expanding the sum

in equation 2.1

n(ωj) ≃ 1 + β[P (x1α(ωj)1 + x2α(ωj)2 + x3α(ωj)3 + ...xkα(ω)j)] (2.2)

Where P, α and xi are the pressure, dynamic polarizability and mole fraction respec-

tively for a fixed frequency ωj where β = 2πNA

RT. From figure 2.3 we know that the gradient

of the line divided by the length of the cell gives us the ratio of the refractive index to

pressure. Using equation 2.1, this may be expressed as

β(x1α(ωj)1 + x2α(ωj)2 + x3α(ωj)3 + ...xkα(ω)j) =grad

L=( n

P

)

ωj

(2.3)

Here, x1 . . . xk take the same meaning as in equation 2.1, ωj is the polarizability

measured at the jth wavelength and L the length of the optical cell. In matrix form

2.3 Operating theory 15

β

α(ω1)1 α(ω1)2 .. .. α(ω1)k

α(ω2)1 α(ω2)2 .. .. α(ω2)k

: : :: :: :

: : :: :: :

α(ωj)1 α(ωj)2 .. .. α(ωj)k

x1

x2

:

:

xk

=

(

nP

)

ω1

(

nP

)

ω2

:

:(

nP

)

ωj

(2.4)

Or

βAX = B (2.5)

Note that j ≥ k.

In section 4.1 the minimum quantity of a certain gas detectable by this apparatus

will be established8. The range that the elements of Matrix B can fall within to provide

realistic values for X is quite small. The constraints the linear system places on this range

will also be discussed in chapter 5. The linear system is in fact quite unstable, having

a very high condition number9, and if the experimentally determined values for B stray

too far from the exact solution, then these errors will be translated to X, but magnified

significantly. The gradient of the graph n vs. P for k different increasing wavelengths

for a known mixture decreases as wavelength increases. We would not expect then, that

the gradient, determined experimentally for wavelength k = 1 would be less than that of

k = 2 and so on up to the kth wavelength. Any instability in the interferometer will affect

the determination of the number of fringes that have passed the detector. The degree of

uncertainty in the output of the interferometer will significantly affect the calculation of

the fractions of fringes that pass the detector. Because the differences in the gradient of

8Section 4.1 assumes in practise that the output of the interferometer is perfectly stable, and thatthere are no errors introduced in measurements, however this is not the case.

9See section 5.5 for a discussion on condition number


INTERFEROMETRY

figure 2.3 are quite small, determining fractions of a fringe shift with good precision is

critical to ensuring that the components of B exhibit their expected behaivour.

This is mentioned as a note to any future experiments, that after each measurement,

a quick check of the constant of proportionality between n and P is made and compared

to the previous shorter wavelength, if the gradient has not decreased, then the results

should be re-taken. The potential sources of these problems in our apparatus are� The source: Many sources, especially lasers, require a certain amount of time for

their output intensity and frequency to stabilize. Often waiting a couple of minutes

before measuring sends this variability to a sufficiently low level.� The optical bench: An interferometer is an extremely sensitive piece of equipment,

for this experiment to perform reliably, much care must be taken as to ensure that

the optical bench is stabilized.� Changes in medium over path: The refractive index of air is not the same at all

points along the path of the beams, and changes due to wind currents. Enclosing

the apparatus in a box should aid in reducing this problem.

Given that the difference in the gradient of the graph of n relative to P for 400 nm

and 700 nm for air at 20◦ is only a matter of 5.5 × 10−11 (or 2% of the actual), it is

vital that the above steps are taken to ensure that the value for ∆P is determined to be

as close to the actual value as possible. As mentioned, these sources of uncertainty do

have the potential to move measurements outside the range of feasible results through a

miscalculation of the number of fringes passing the detector.

2.4 Restrictions inherent to using interferometry for gas analy-

sis

At this point, the potential pitfalls of using interferometry as a gas analysis device are

discussed. One potential scenario which could hamper the accuracy of results is a lack

2.4 Restrictions inherent to using interferometry for gas analysis 17

of knowledge of the constituent gases in the sample. Unless data on polarizability is

gathered on every gas that could possibly exist at commonly encountered temperatures

and pressures and an equivalent number of sources of light are available to ensure that

our linear system of equations is not underdetermined, then there is no way to be 100%

certain that the quantitative results are accurate; or even that it is truly known which

gases are in the sample. The only way to be certain of the results is if the dynamic

polarizability of every gas in existence is already known and there are at least as many

wavelengths of light with which to run the experiment.

Of course, in typical experimental testing, some knowledge is assumed about what is

being investigated. Furthermore, the scope of this work is narrowed to common atmo-

spheric gases, of which there is a great abundance of data regarding polarizability. It is

only in extreme situations in which this system, theoretically at least, could produce a

result in which the experimenter could not take confidence.

Another way in which the experimenter can sidestep the problem of being corralled

into using multiple wavelengths of light, is by grouping the dynamic polarizability of a

number of gases already in the system, effectively treating a group of gases, each with their

own dynamic polarizability as a single gas. This method is particularly useful when it is

not necessary to know the quantities, relative to each other, of the gases being “grouped.”

Circumstances may be such that the experimenter wishes only to know the absolute

percentage of a gas of particular interest. One such example would be in measuring

humidity. In this case, the common components of the atmosphere, such as Nitrogen and

Oxygen, are “grouped.” The gas of interest, Water Vapour, is treated as the second gas

in the system. This dramatically reduces the time required in gathering results given that

it now requires only 2 wavelengths of light to run the experiment.

Other problems which may arise in more complicated situations, such as a sample

of a high number of gases, is the range of wavelengths of light used. Using a number

of wavelengths that cover a short spectral range would not produce the same degree of

accuracy as using the same number of wavelengths spread over a much larger spectral


INTERFEROMETRY

range. Failure to take a large range of wavelengths will result in the condition number

for matrix A increasing, and consequently the accuracy of matrix X decreasing10.

2.5 Optical properties of aerosols and their influence on results

Aerosols are thought to play some role in the accuracy of the results of gas analysis

[45], so a brief description of the optical properties of aerosols is provided here. Mie

theory describes the optical interactions of small spheres with an electromagnetic field.

It is not within the scope of this work to provide a rigorous analysis of the Mie solution

to Maxwell’s equations, provided however are the most salient details. Starting from

Maxwell’s equations for a conductor

∇× H − ǫ

c

∂E

∂t=

4πσ

cE (2.6)

∇× E +µ

c

∂H

∂t= 0 (2.7)

Here the symbols used take their usual interpretation. Taking the curl of equation 2.7,

and using the vector identity ∇×∇× A = ∇ (∇.A) −∇2A gives

−∇2E +µ

c

∂

∂t∇×H = 0 (2.8)

One may wonder why the term representing the divergence of the Electric field has

dissappeared. The third of Maxwell’s equations, ∇.E = 4πǫρ, when differentiated with

respect to time and the term in ∇.∂E∂t

solved simultaneously upon taking the divergence

of equation 2.611 will realise a linear first order differential equation in ρ whose solution

shows that ρ attenuates rapidly with time for an appreciable σ, much faster than the

10equation 2.511Using the fourth of Maxwell’s equations ∇.H = 0.

2.5 Optical properties of aerosols and their influence on results 19

time it takes for one oscillation of (most) electromagnetic waves. This is mentioned, as it

justifies the approximation ∇.E = 0. Substituting for ∇× H from equation 2.6 gives

∇2E − 4πµσ

c2∂E

∂t− µǫ

c2∂2E

∂t2= 0 (2.9)

We can see, that due to the non-zero conductivity, the term in ∂E∂t

implies a damp-

ening of E as it propagates through the material, unlike a similar analysis of non-

conducting12 media, ie. gases. Materials with any appreciable conductivity, aerosol’s

included, will exhibit this absorption property. Continuing for a plane monochromatic

beam E = E0e−iωtek.r and a similar term for H, equations 2.6 and 2.7 become

∇×H + iωǫ

cE =

4πσ

cE (2.10)

∇×E − iµω

cH = 0 (2.11)

Taking the curl of equation 2.11 upon re-arranging we see that

ic

ωµ∇2E = ∇× H (2.12)

Substituting into equation 2.10 upon re-arranging we find that13

∇2E +µω2

c2

(

ǫ+ i4πσ

ω

)

E = 0 (2.13)

Which is the well known Helmholtz equation14. As will be shown in Chapter 3, we

12or poorly conducting13Noting for these manipulations that for a monochromatic wave dE

dt= −iωE and d2

E

dt2= ω2

E, similarlyfor H.

14This is the ansatz for Mie’s solution for a spherical conductor


INTERFEROMETRY

have simply replaced the dielectric constant with a complex valued dielectric constant.

Also implying a complex valued refractive index. If the definition of the speed of light is

c =√µ0ǫ0, then the definition of the refractive index is n =

√µrǫr. From the above we

see that n is given by

n2 = µ

(

ǫ+ i4πσ

ω

)

(2.14)

We replace n by x (1 + iy) where y is known as the extinction co-efficient15 giving

x2 (1 + iy)2 = µ

(

ǫ+ i4πσ

ω

)

(2.15)

Equating real and imaginary parts

x2(

1 − y2)

= µǫ, x2y =2πσµ

ωx2(2.16)

substituting from the RHS of equation 2.16 y2 = 4π2σ2µ2

ω2x4 into the LHS of equation 2.16

yields

x4 − µǫx2 − µ2σ24π2

ω2= 0 (2.17)

We must take the positive root given that both x and y must be both real and positive,

thus

15related to the absorption coffecient χ = 4πλ

y

2.5 Optical properties of aerosols and their influence on results 21

x2 =µǫ

2+

√

µ2ǫ2

4+

4σ2µ2π2

ω2(2.18)

y2 = 1 − µǫ

µǫ

2+√

µ2ǫ2

4+ 4σ2µ2π2

ω2

(2.19)

We now follow essentially the same treatment as will be adopted later in section 3.2

except in this case the equation of motion16 we are dealing with is not concerned with

bound charges, but free charges. The reason we take this approach may be intuitively

realised by contemplating what we see when we look at aerosols and gases. All gases are

completely transparent in visible wavelengths, aerosols on the other hand, depending on

what aerosol we are considering are not. Clouds absorb and block sunlight,17 soot and

particulate emissions do an even better job of blocking and absorbing visible light. The

reason that this is possible is that there are free electrons in the particulate available

to absorb and scatter light of any wavelength above a critical wavelength (usually in

the U.V.). Thus we have for our equation of motion the following ordinary differential

equation, minus any term in r that would otherwise approximate the attraction an electron

would have to a nucleus as a function of its distance (r) from the nucleus.

m∂2r

∂t2+ b

∂r

∂t= qE0e

−iωt−k.r (2.20)

Where m is the mass of the electron, b the damping co-efficient and q the charge of

the electron has the particular solution

r (t) = − qE0e−iω t−k.r

ω (iβ + ω)m(2.21)

Compare the form of equation 2.21 with equation 3.21 and one would notice there is

16See section 3.2 for justification of this ordinary differential equation of motion.17If only there were a silver lining to the labourious task of writing a thesis!


INTERFEROMETRY

no term representing a resonant frequency. For the commonly encountered aerosols, one

must remember that particle concentrations are many orders of magnitude lower than

for the gas in which they are suspended, so even if transmission does occur, the higher

refractive index of the particle only contributes to a change in path length given by the

size of the particle and the number of particles in a volume. Since the number of particles

in the volumes encountered in this work is extremely low, the effect on changing the path

length is completely negligible[75]. One final observation [12] of equation 2.21 shows that

when the frequency is sufficiently low, y > 1 the material is highly reflective. At higher

frequencies, y becomes very much smaller, as does x. The critical frequency at which this

occurs is, for the most part in the UV. Thus most optical activity of aerosols is absorbtive

and scattering effects, considering this work is only being done in the visible wavelengths,

their impact on results is neglible, thus it is safe to proceed in the knowledge that aerosols

do not have an appreciable impact on the results.

2.6 Conclusions

In this section the reasons for choosing the Sagnac interferometer as the instrument of

choice for calculating quantities of a gas in a mixture interferometrically have been es-

tablished. Other types of interferometers were discussed as well as the benefits and de-

tractions inherent to each system. The operating theory of the Sagnac interferometer in

this application, and the type of measurements that are made were discussed, including

an insight into what external effects produce errors in the results. Finally, some questions

have been asked of potential fundamental problems that may affect results, such as the

presence of aerosols. In the following chapter, a full theoretical treatment of the properties

governing dispersion, which is the foundation upon which an interferometric gas analyser

relies upon, is presented.

23

3 Theory of refraction of gases

3.1 The Lorentz-Lorenz equation

Before further discussion about the experiment itself, we must delve into the theoretical

foundations that underpin this work. The primary question is, how is the refractive index

of a gas determined? The fundamental equation governing all work henceforth is the

Lorentz-Lorenz equation [6].

α(ω) =3

4πd

n(ω)2 − 1

n(ω)2 + 2(3.1)

Where α(ω) denotes the dynamic polarizability, d the density and n(ω) being the

frequency dependent refractive index. Re-arranging equation 3.1 in terms of n gives.

n(ω) =

√

√

√

√

8πdα(ω)3

+ 1

1 − 4πdNAα(ω)3

(3.2)

At this point the density will be written in terms of temperature and pressure. There

are two manipulations that must also be made to simplify further analysis. Firstly, in

situations where n ≃ 1 equation 3.2 may be written in its approximate form and secondly,

for a mixture of gases, the contribution of a constituent gases polarizability is scaled

according to its mole fraction and added to the scaled polarizabilities of other constituents

of the mixture [72]. Thus, the form of the Lorentz-Lorenz equation that applies to this

work is

n(ω) ≃ 1 +2πPNA

RT

∑

k

xkα(ω)k (3.3)

Where P is the pressure of the gas in the box, T the temperature, and xk the mole

fraction of the kth constituent of the mixture. In the strictest of definitions, there would

24 3 THEORY OF REFRACTION OF GASES

be a correcting term Z, the compressibility appearing in the denominator of the RHS of

equation 3.3, however under the conditions of the experiment, this value is so close to

one as to be of negligible importance to this work considering the gases and quantities

studied. It is of extreme importance not to forget, that when substituting d in terms of

P and T not to ignore the factor of 103 that is introduced when eliminating n(moles)

from the equation. When calculating the number of moles in a sample, the mass is always

written in grams, the d quoted in equation 3.1 has units of kg

m3 .

To this point, what may appear as a very simple equation to describe the variation

in refractive index of a gas sample has been provided. Before getting too far ahead of

ourselves, let us now look at the development of equation 3.1, and from here take an

in-depth look into the dynamic polarizability. It is the polarizability that allows the

experimenter to distinguish between components of a gas sample, so the polarizability

will be a main focus of the theory to follow.

But first, the derivation of the Lorentz-Lorenz equation. The classical origins of the

Lorentz-Lorenz equation from electrodynamics is well established in the literature [12; 10],

whilst the more rigorous quantum mechanical theory of dispersion is usually found within

the discipline of quantum chemistry [27], though it will be dealt with briefly here. A

similar approach to the latter is presented here.

We begin by envisioning our target particle, in a volume of a great number of particles,

to be surrounded by a sphere. Inside the sphere, which contains the single molecule is

a region in which the polarization is non uniform. Outside the sphere, we consider the

polarization to be constant, considering the vast ensemble of particles surrounding the

molecule. The quantity of interest is the effective field, which is denoted as Ee, this is the

field that an individual molecule or particle experiences when subjected to a local field E.

Let us imagine an arbitrary molecular or atomic particle subjected to an electric field E.

The electron cloud and nucleus are displaced in such a way that a majority of negative

charge resides at one end, whilst a majority of positive charges are at the other, creating

a classical dipole, see figure 3.1.

3.1 The Lorentz-Lorenz equation 25

The contribution to the total field is determined by calculating the gradient of the

potential produced by the applied field on the molecule. A molecule in a gas can be

viewed as residing in a medium of homogeneous polarization. In an imaginary sphere

surrounding the molecule, the potential produced by the molecule using the Gauss law is

zero, as there is no net charge inside the sphere.

Conversely, the region inside the sphere can be viewed as being of uniform polarization,

and outside the sphere as being charge free. For a uniformly charged sphere, in the absence

of any polarizing field, Poisson’s equation gives

∇2V0 =−4πρ

ǫ0=∂2V0

∂x2+∂2V0

∂y2+∂2V0

∂z2(3.4)

And subsequently, assuming ρ = 1, each partial derivative within the Laplace operator

equals 4π3

. The potential produced by both an applied field and this charge distribution

is simply the dot product of both18 .

V = −P.∇V0 (3.5)

Where P in this case is the polarization. The point of this excercise is to determine the

strength of the field at the centre of the sphere in the direction of the applied field which

18Noting that the field produced by any potential is given by E0 = −∇V0

E

r

dq

Figure 3.1: The distortion of the electron cloud of a particle [13].The distortion produces an uneven charge distribution on the surface of the sphere.


can be assumed to be in the x direction (P = Pxi). Thus, the field produced by the above

potential is found by taking the gradient with respect to x.

−∂V∂x

= P∂2V0

∂x2(3.6)

The remaining terms ∂2

∂x∂y= ∂2

∂x∂z= 0 at the centre of the sphere, as are the components

of P in the y and z directions. As mentioned, ∂2V0

∂x2 = 4π3

so the field induced by an applied

field is

E′ =4π

3P (3.7)

When this internal field is added to the induced field, the magnitude of the field experi-

enced by the particle is,

Ee = E +4π

3P (3.8)

and introducing the well established equation for the dipole moment of a molecule

exposed to a sufficiently low incident field

p = αEe (3.9)

we can now begin building the more familiar aspects of the Lorentz-Lorenz equation.

The total electric moment per unit volume is given by

P =NaP

RTαEe (3.10)

Where Na is Avogadro’s number, P the pressure, R the gas constant, and T the

temperature. When substituting Ee for equation 3.8 and introducing

P = ηE (3.11)

3.2 The classical determination of the dynamic polarizability 27

3.10 becomes

η =NaPRT

α

1 − 4πNaP3RT

α(3.12)

Where η is the dielectric susceptibility. From Maxwell’s equations it is known that in

matter, the Electric displacement is given by

D = ǫE (3.13)

but it may also be written in terms of the polarization

D = E +4π

3P (3.14)

Substitution of equation 3.13 for D and equation 3.11 for P gives upon elimination of

E

ǫ = 1 + 4πη (3.15)

substituting this into equation 3.10 with n2 = ǫ gives us

α =3RT

4πNaP

n2 − 1

n2 + 2(3.16)

Which is essentially equation 3.1 with d defined as d = NA

RT.

Now that the Lorentz-Lorenz equation has been derived, we can investigate its key

component, the dynamic polarizability α(ω).

3.2 The classical determination of the dynamic polarizability

The dynamic polarizability of a substance, through equation 3.16, gives the degree of dis-

persion of light through the substance. The polarizability is the cornerstone in identifying


the components of a mixture, as the polarizability for each molecule or atom is unique.

Exact theoretical calculations of the dynamic polarizability for diatomic and poly-

atomic molecules can be time consuming, given that one must solve the Schrødinger

equation for such complicated systems. Presented here is the classical approach for deter-

mining the dynamic polarizability. The dynamic polarizability for the atmospheric gases

will be presented from experimental and theoretical calculations from external sources.

Although modern calculations of polarizability are made with the assistance of quan-

tum mechanics, it is worthwhile noting that classical physics can produce a model of

electronic behaviour, in an external field, that is confirmed by the more rigorous quantum

mechanical development. An insight into this quantum mechanical treatment is provided

later in this chapter. The Lorentz force imparted on each electron of a non polar molecule

in an electric field E is given by

Fl = e(Ee +v

c× Be) (3.17)

Note here that c is a dimensionless quantity ∼ 3× 108. One can confidently eliminate

the effective magnetic field, given that the velocity of the electrons of the molecule are

significantly less than the speed of light. Thus, equation 3.17 takes the form

Fl = eEe (3.18)

In the sufficiently weak field approximation confirmed by a more rigorous quantum

mechanical treatment, it is assumed that the electron and nucleus interact similarly to

masses on springs, governed by Hooke’s law in the form

F = −qr (3.19)

As will be shown later, the mean position about which the electron oscillates can be


approximated by the Bohr radius19 [1] and the motion for the electron is described by the

damped, driven harmonic oscillator equation [9; 35]

md2r

dt2+ k

dr

dt+ qr = eEe0e

−iωt (3.20)

We are interested only in the particular solution, which is

rp =e

m

Ee

ω20 − ω2 − ikω

m

(3.21)

Note here that the subscript p indicates the particular solution, as opposed to the

homogeneous solution to equation 3.20. Furthermore ω0 =√

q

mis the absorption or

resonant frequency and Ee = Ee0e−iωt. Here, the assumption is made that each molecule

has only one effective electron, from this we know that the dipole moment of an atom or

molecule is given by

p = er (3.22)

and the total polarization of an ensemble of identical molecules is

P = Np = Ner (3.23)

Where N is the number of molecules per unit volume as per 3.10. Substituting r from

equation 3.21 gives

P =e2

m

NEe

ω20 − ω2 − ikω

m

(3.24)

19For the simple example of the Hydrogen Atom


Equating 3.24 with 3.10, we see that

NaP

RTαEe =

e2

m

NEe

ω20 − ω2 − ikω

m

α(ω) =e2

m(ω20 − ω2) − ikω

(3.25)

It is straightforward to show that if N = NaPRT

then α(ω) is given by the equation on

the right. Shown in figure 3.2 is the real component of the polarizability compared to the

undamped polarizability.

0

Omega

infinity

Alpha

Figure 3.2: The polarizability near an absorption line.Damped(dashed line) and undamped. In the damped curve [2], the maxima and minima

occur at ω =√

ω20 ± ω0k

m. Interestingly [33] has shown that these absorption lines are

present over the entire dispersion spectrum of air. The magnitude of these resonancesthough affect the refractive index by less than one part in 1013, which is too small to be

significant in this work.


For systems that have more than one resonance, the right hand term in 3.25 becomes

α(ω) =e2

m

∑

l

fl

ω2l − ω2 − iklω

(3.26)

Where fl are the oscillator strengths, given by solutions to the Schrødinger equation

in the following form.

fl =2meωl

3~e2|〈0|r|l〉|2 (3.27)

This would represent the transition from the ground state |0〉 to some excited state

|l〉 and r is the position operator, the average electronic distance from the center of

the nucleus as given by solving the Schrødinger equation for the given state |l〉 [10].

It must be noted that these oscillator strengths are dependent on temperature, as it

relies on the Boltzmann populations of the excited and ground states and the partition

function [37; 38]. Evidently then, the polarizability is an agglomeration of real and

complex components. The real component determines the observed dispersion and the

complex component, the absorption. The solution to equation 3.20 is interesting in that

the Kramers-Kronig relations tell us that if the real part of a linear passive system is

known then the imaginary part can be calculated and vice versa. In real terms, this

means that if one knows the absorption spectrum, it is straightforward to calculate the

dispersion spectrum. If the polarizability is of the form α(ω) = α′(ω)+ iα′′(ω), then α(ω)

can be expressed as [44]

α(ω) =Cp

πi

∫ ∞

−∞

α(z)

z − ωdz (3.28)

Where Cp is known as the Cauchy principle value[69] of the integral∫∞

−∞α(z)z−ω

dz. The

principle value is


Cp = πi∑

residues (3.29)

From equation 3.28 it is evident that there is a pole at z = ω. Thus the sum of the

residues is given by multiplying the kernel of the integral by z − ω and taking the limit

as z approaches ω, specifically

∑

residues = limz→ω(z − ω)α(z)

z − ω(3.30)

Equating the real and imaginary parts of equation 3.28 we subsequently find that.

α′ =P

π

∫ ∞

−∞

α′′

z − ωdz, α′′ = −P

π

∫ ∞

−∞

α′

z − ωdz (3.31)

Using the property that α′′ is an odd function and α′ is an even function

α′ =2P

π

∫ ∞

0

zα′′

z2 − ω2dz, α′′ = −2Pω

π

∫ ∞

0

α′

z2 − ω2dz (3.32)

Clearly, if we have knowledge of the dispersion of a gas (α′(ω)), we also have knowledge

of its absorption (α′′(ω)).

3.3 Quantum mechanical determination of the dynamic polar-

izability

To build a more complete picture of the polarizability from this point, we now investigate

some of the quantum mechanical interpretations of polarizability. It is impossible to build

up an understanding of the polarizability of every different molecule or atom without

employing quantum mechanics. We will investigate the development of the dynamic

3.3 Quantum mechanical determination of the dynamic polarizability 33

polarizability from quantum mechanical principles. The intention being to provide some

basis for a purely theoretical determination of polarizability20 for other molecules or atoms

as neccessary.

It has been shown[11] that the polarizability is given as a sum over n excited states as

α(ω) =1

~

∑

n

〈Ψ|p|Ψ0〉(ω2

n − ω) − iknω(3.33)

Where p is the dipole moment operator, and Ψ is the wavefunction that describes the

atom. Equation 3.33 gives the polarizability for one direction only, to expand the analysis

to three dimensions, α(ω) is modified to [3]

α(ω) =1

3(αxx + αyy + αzz) (3.34)

Given that this study focuses on atmospheric gases, which are mostly dipolar molecules,

the following form of the polarizability can be used [41]

α(ω) = α||(ω) +2α⊥(ω)

3(3.35)

Where α|| is the polarizability parallel to the internuclear axis (αzz) and α⊥ the po-

larizability perpendicular to the internuclear axis.

The solution to the time dependent Schrødinger equation, given here from perturbation

theory

(

H0 + λH ′)

Ψn = EnΨn, λ << 1 (3.36)

20[11] is an excellent resource for an in depth discussion of polarizabilities for a variety of configurations.


for anything other than atomic Hydrogen-like atoms21 is reasonably complicated, al-

though a number of methods exist for solving higher order problems. H ′ in the above

equation is the time dependent perturbation caused by interaction with plane polarized

light, which may be simply written as H ′ = −∑

ie

mjcAj .pj [30], where the j’s represent

the jth electron in the system, and the magnetic and electric fields are given by H = ∇×A

and E = −1c

∂A∂t

.

Furthermore, the Ψn’s in equation 3.36 represent Eigenfunction expansions of the

unperturbed solution to the Schrødinger equation, specifically

Ψn(x, t) =∑

n

an(t)Ψ(0)n (x, t) (3.37)

Which can be substituted back into equation 3.36. This equation is not solveable

analytically, however an approximation may be found by setting Ψn =∑

k Ψ(k)n and

En =∑

k E(k)n [15].

An example of solving the time independent Ψ for the Helium atom has been performed[30]

with the Hamiltonian for such a system given by

H = − h2

8π2m

(

∇21 + ∇2

2

)

− e2(

2

r1+

2

r2− 1

r12

)

(3.38)

Where ∇21 and ∇2

2 are the laplacian of the electrons 1 and 2, m is the reduced mass and

r1,r2 and r12 are the midpoint internucleus-electron22 distances for each electron and the

inter-electronic distances respectively. The wavefunction to the Helium atom described by

this Hamiltonian can be found, using the variational method, by choosing an appropriate

trial eigenfunction, ψ = A[

eZ(R1+R2) (1 + cR12)]

[30] where Z is a number, between 1 and 2

that minimizes dEdZ

, A is the normalizing factor, R1 = r1

a0, R2 = r2

a0and R12 = r12

a0, a0 being

the Bohr radius. Z is chosen between 1 and 2 because the outer electron in the Helium

21Helium is frequently used as a comparison of theoretical calculations of polarizability to experimentalresults for the simplicity of the solution to the Schrødinger equation

22Working in elliptical co-ordinates.

3.4 Quantum mechanical confirmation of the Lorentz-Lorenz equation 35

atom does not experience a 1 to 1 attraction to the nucleus while the inner electron does,

closely approximating the 1s state of the Hydrogen atom. This treatment assumes the

lowest energy state, which is the most probable at laboratory temperatures. However, the

polarizability for higher excited states can be calculated [51]. Solving the time dependent

Schrødinger equation is not relevant to this work, although can be accomplished if required

[5; 14].

The solution to these equations and the subsequent calculation of the theoretical

polarizability is beyond the scope of this paper, but it does provide an avenue for a)

confirmation of experimental results and b) a method for determining polarizability for

molecules for which there is not an extensive body of research [43; 7; 8; 76; 34; 49; 3; 31;

68; 47; 70; 56; 41; 73; 66; 67].

3.4 Quantum mechanical confirmation of the Lorentz-Lorenz

equation

Given that the Lorentz-Lorenz equation is a derivation completely from classical mechan-

ics, it is important to ensure quantum mechanics yields the same result, otherwise doubt

may be cast on the validity of the equation. Presented here are two methods of a quan-

tum confirmation of the classical equation 3.20. First consider a Hydrogen like atom in

an electromagnetic field. The perturbed Hamiltonion is

H = H0 + W (t) (3.39)

Where H0 = P 2

2m+ V (r) is the time independent Hamiltonian with V (r) = ~

2µr2 − e2

4πǫ0r

[29] the familiar total electrical potential energy of the electron a distance r from the

nucleus23 and P the quantum mechanical momentum operator.

Although W (t) can contain a magnetic field component, for light matter interactions,

23taking the derivative of V (r) and equating to zero yields the bohr radius a0 as the solution for r


we need only concern ourselves with the electric dipole Hamiltonian, which is

W (t) = − e

mP .A (3.40)

Where A is H = ∇×A and E = −1c

∂A∂t

as before. Expressing A more completely

A(r, t) = A0kei(ky−ωt) + A∗

0ke−i(ky−ωt) (3.41)

Thus the electric dipole Hamiltonian becomes

W (t) = − e

mP .(

A0kei(ky−ωt) + A∗

0ke−i(ky−ωt)

)

(3.42)

The series expansion for e±iky is

e±iky = 1 ± iky ±O(y2)... (3.43)

Since k, being the wavevector and y is of the order of the atomic radius ky << 1 for

wavelengths near the visible part of the electromagnetic spectrum, the approximation can

be made that e±iky ∼ 1. Therefore W (t) becomes, with the substitution of A0 = −iE0

2ω

W (t) = − e

mP−iE0

2ω

(

eiωt − e−iωt)

(3.44)

Which simplifies to

W (t) =e

m

PzE0

ωsin(ωt) (3.45)

Using this in conjunction with Ehrenfest’s theorem [22] gives the rate of change of the


expectation values

d〈R〉dt

=1

i~〈[

R, H]

〉 =1

i~〈[

R, H0 + W (t)]

〉 (3.46)

d〈P 〉dt

=1

i~〈[

P , H0 + W (t)]

〉 (3.47)

Where R and P are the position and momentum operators respectively. Given that

H0 = P 2

2m+ V (r), we note that R commutes with the energy V (r) and

[

R, H0

]

is given,

in one dimension by

〈r|[

X, Px

2]

|Ψ〉 = 〈r|XPx

2 − PX

2X|Ψ〉 (3.48)

〈r|[

X, Px

2]

|Ψ〉 = −x~2 d

2

dx2〈r|Ψ〉 + ~

2 d2

dx2(x〈r|Ψ〉) (3.49)

Thus the commutation of the operators X and P 2 can be written

[

X, Px

2]

= 2~2 d

dx(3.50)

Substituting the full momentum operator P = ~

iddx

gives[

X, Px

2]

= 2i~Px. Thus we

have that

d〈R〉dt

=〈P 〉m

+e2E0

mωksin(ωt) (3.51)

Note that in d〈P 〉dt

= 1i~〈[

P , H0 + W (t)]

〉, P commutes with the first term in H0 as well

as W (t) whilst commutation with the second term in H0 gives −∇V , so that


d〈P 〉dt

= −〈∇V 〉 (3.52)

Taking the derivative with respect to time of d〈R〉dt

gives

md2〈R〉dt2

= 〈∇V 〉 + eE0kcos(ωt) (3.53)

In a spherically symettric atom, ∇V = dVdr

. Note from [12] that equation 3.20 specifies

that r is a distance only from an equilibirum position, not as might be thought, the

distance between the electron and nucleus. In the ground state, the equilibrium position

is the Bohr radius a0. Changing in variable for V, by substituting r = ρ + a0, ∇V then

takes the form

∇V = − ~

µ r3+ 1/4

e2

π ǫ0r2(3.54)

With the change or variable, this becomes

∇V = − ~

µ (ρ+ a0)3 + 1/4

e2

π ǫ0 (ρ+ a0)2 (3.55)

Recalling the original axiom, that the nucleus and electron act like masses on a spring.

This dictates that the restoring force must be linear. A linear approximation must be

made for equation 3.55. The simplest approximation is a Taylor series. However as can

be seen in figure 3.3, a Taylor series is not a good approximation over a range around

the Bohr radius. The best linear approximation to the function in this region is a Gram

Schmidt function approximation over an appropriately chosen range24. In this case, the

range chosen was from the Bohr radius, to the mean expectation value for the ground

24See the straight line in figure 3.3


VVVVTVVGSFA

r m4#10 - 11 6#10 - 11 8#10 - 11 1#10 - 10 1.2#10 - 10

dd r

V JmK1

0

1#10 - 8

Figure 3.3: The gradient of the potential of the Hydrogen atom.

state of Hydrogen (1.5a0). It is reasonable to assume that the electron will spend most of

its time oscillating in this region. The equation of this straight line is

∇V ∼ −1.7516 × 10−8 + 402.8148r (3.56)

The term proportional to r of the right side of equation 3.56 is of interest. Substituting

this into equation 3.53, with the change of variable gives.

md2〈ρ〉dt2

= 402.8〈ρ〉 + eE0kcos(ωt) (3.57)

Comparing this differential equation with the equation 3.20 and we can see that we

have arrived at a very similar result. The above ordinary differential equation being the

case for the Hydrogen atom, we can see immediately that the constant in Hooke’s law is

402.8, therefore, we have an ansatz for calculating the first absorption frequency of the

Hydrogen atom using the classical harmonic oscillator. Using µ = 9.1045× 10−31 kg, the


reduced mass, we find that

λ0 =2πc√

q

me

=2πc

√

402.89.1045×10−31

= 89.5767 nm (3.58)

It is well known[12] that the first of the resonant wavelengths for Hydrogen falls in the UV

range, exactly as predicted above. Even more interesting is the fact that this wavelength

is very close to the resonant wavelength for the decay of an electron from the n = ∞ to

n = 1 level of the Hydrogen atom, lending weight to the use of the classical harmonic

oscillator equation as a valid starting point for describing optical dispersion. Thus the

foundation of the Lorentz-Lorenz equation has been quantum mechanically confirmed in

the sufficiently weak field approximation.

3.5 An alternative quantum mechanical derivation of the Lorentz-

Lorenz equation

An alternative quantum mechanical treatment can be performed when taking the per-

turbed wavefunction describing the atom in an electromagnetic field [30]. Using equation

3.37

Ψn = Ψ0 +∑

n

an(t)Ψn (3.59)

The solution to the Schrødinger equation in this case involves an Eigenfunction ex-

pansion of the co-efficients an. Although not derived here, the solution of such an Eigen-

function expansion for the case being considered can be written as [30]

3.5 An alternative quantum mechanical derivation of the Lorentz-Lorenz equation 41

an(t) =iW21

2c~A0.〈2|e.r|1〉

(

eiW21+ǫ

~t

W21 + ǫ+ei

W21−ǫ

~t

W21 − ǫ

)

(3.60)

Where W21 = hν21 is the energy difference between states 2 and 1 and ǫ = hν. The dipole

moment is

〈p〉 = Re〈Ψ∗n|e.r|Ψn〉 = 〈1|p|1〉+

(

∑

n

iW21

c~〈1|p|2〉〈2|p|1〉.A0

(

1

W21 − ǫ− 1

W21 + ǫ

)

sin(ǫ

~t)

)

(3.61)

It is important to note that 〈1|p|1〉 is the permanent dipole of the atom, and as

such gives the static polarizability, in the absence of any perturbing field. The dynamic

polarizability is the term on the right of equation 3.61. Recall that E0 = 1c

ddtA0, from the

previous quantum mechanical treatment, the perturbing field is given by E = eE0cos(ǫ~t),

ǫ~

= ω. Substituting the above equation for A0sin( ǫ~t) with the corresponding term in

E0, specifically E0 = ǫc~A0sin( ǫ

~t). Making this substitution and noting that

1

W21 − ǫ− 1

W21 + ǫ=

2ǫ

W 221 − ǫ2

(3.62)

We realise

〈p〉 = 〈1|p|1〉 +2

h

∑

n

ν21E0

ν221 − ν2

〈1|p|2〉〈2|p|1〉 (3.63)

Using equation 3.34 and noting that 〈1|p|2〉〈2|p|1〉 is the square of the mean expecta-

tion value of the diploe moment for the hybridisation of the wavefunction of ground state

|1〉 and excited state |2〉 we may rewrite the right of the above equation in terms of the

dynamic polarizability as


α(ν) =2

3h

∑

n

ν21|〈p12〉|2ν2

21 − ν2(3.64)

The numerator within the sum is exactly the same as the numerator given in equation

3.26, therefore this alternative quantum mechanical treatment yields the same result. In

both of these quantum mechanical treatments, it is important to realise that we have as-

sumed that there is no damping co-efficient, such that would produce regions of anomolous

dispersion. Recall that the main aim of this work is focused in the optical part of the

spectrum, a region within which ω is a very long way away from any absorption regions.

Note that in equation 3.64 there is no damping term in the denominator, this is of no

concern as one can come to the same result if the damping term k in equation 3.20 is set

to 0.

3.6 Conclusions

Two methods have been shown by which quantum mechanics confirms the classical

Lorentz-Lorenz equation, firstly using Ehrenfest’s theorem, and secondly from a rigor-

ous investigation of time dependent perturbation to the Schrødinger equation. All the

theoretical foundations behind classical optical dispersion has been quantum mechanically

derived. A full calculation of the dynamic polarizability of an example molecule, might

have been given, however this in itself is the focus of another branch of study, and has

been extensively covered in other literature[5; 64; 54; 30; 51].

43

4 System overview and operation

4.1 Range of detection

It is useful to answer at this point the most critical question of all: How sensitive, and

to what minimum concentration of a constituent gas, can the apparatus detect? The

challenge is essentially one of being able to determine a refractive index. To do this, first

consider a box of volume V that is completely empty of any matter. We know that for

this enclosure, the refractive index n = 1 for a vacuum. To this box, particles are added,

one by one. The sensitivity is then given by the concentration when the interferometric

apparatus determines that n > 1. To quantify this the theory of the refractive index for

gases, as outlined in Chapter 3, and the constraints our apparatus places on our ability

to determine a change in refractive index need to be considered.

The minimum detectable concentration is not the same for all of the spectrum of possi-

ble constituents of a mixture. Say for example that one of the constituents of our mixture

has values of polarizability vastly greater than any other constituent. In determining how

many particles would be required to induce a detectable change of n above the vacuum

level, a constituent with a very high value of polarizability would require a much lower

concentration to induce a detectable change in refractive index above n = 1.

In the following the dynamic polarizability of each major constituent of the atmosphere

and the corresponding minimum detectable change in concentration for each will be given.

Following from this, it will also be shown how combinations of gases affect the range of

detection.

44 4 SYSTEM OVERVIEW AND OPERATION

4.2 Tables of dynamic polarizabilities and dispersion for atmo-

spheric gases

Provided in this section is a list of tables25 and graphs, compiled from theoretical and

experimental data, which detail the polarizability and refractive index of the main con-

stituents of our atmosphere. This data is required because the polarizability at a par-

ticular wavelength of a given gas needs to be known to fill the elements of the matrix

in equation 2.4 and allow the system to operate as a gas analyser. The data has been

provided for the visible part of the spectrum with measurements made at STP excepting

water vapour, which was measured at 293.16 K and 1333 Pa [19]. Where possible, data is

provided in equation form gathered from previous experiments, however some equations

are the result of a least squares fit to raw data from references such as [32]. All the

data in the tables have been standardized to r the refractivity, and λ, the wavelength in

metres. Furthermore, the figures provided are the graphical representation of the data in

the corresponding table. Where possible, the most recently gathered equation or data is

used as the reference for the work carried out in this paper. It should be noted that the

values of polarizability quoted from one author to the next vary by such a small amount

that their uncertainty is negligble when carrying out the final uncertainty analysis 26.

For example, [40] has shown that uncertainties from curves fitted to refractivity, and the

actual refractivity of various species usually varies by no more than a few parts per 108.

Given that there are other sources of uncertainty from the experiment that are orders of

magnitude greater than this, uncertainty due to function regression is negligible.

25Given in the tables is the term r(λ), which is the refractivity, and is given simply by r = n − 1.26In chapter 5

4.2 Tables of dynamic polarizabilities and dispersion for atmospheric gases 45

4.2.1 Argon

Table 4.1: Refractive index and Polarizability for Argon (Ar).λ is given in metres. Note: Cuthbertson also made measurements of the dispersion ofArgon, however, no ambient temperature and pressure was recorded, thus it has not

been included.equation r(λ) α(λ)(m−3) Reference

i 0.0002792 + 1.6×10−18

λ2 1.6527 × 10−30 +9.4736×10−45

λ2 + 7.5760×10−63

λ4

[16]

ii 0.0002778 + 1.558×10−18

λ2 1.6446 × 10−30 +9.2240×10−45

λ2 + 7.1835×10−63

λ4

[63]

iii 2.5281×109

8.7882×1013− 1

λ2

+

2.5281×109

9.1×1013− 1

λ2

+ 5.9593×1010

2.6964×1014− 1

λ2

2.9594 ×10−27[( 2.5281×109

8.7882×1013− 1

λ2

+

2.5281×109

9.1×1013− 1

λ2

+

5.9593×1010

2.6964×1014− 1

λ2

+ 1)2 − 1]

[48]

iv 6.7867×10−5+ 3.01829×1010

1.44×1014− 1

λ2

4.01702 × 10−31 +1.7866×10−16

1.44×1014− 1

λ2

+ 2.6960×10−6

(1.44×1014− 1

λ2 )2

[59]


Figure 4.1: Polarizability of Argon from various authors.Note the very close similarity between Peck and Larsen’s results.


Figure 4.2: Refractivity of Argon from various authors.From this graph, it has been decided to employ Peck’s equation for polarizability as thereference for Argon.


4.2.2 Hydrogen

Table 4.2: Refractive index and polarizability for Hydrogen (H2).λ is given in metres.

equation r(λ) α(λ)(m−3) Reference

i 0.0001388 + 0.8λ2×1018 +

1.36λ4×1032

8.2126 × 10−31 +4.7352×10−45

λ2 + 8.0500×10−59

λ4

[40]

ii 0.0001362 + 1.0367×10−18

λ2 8.0629 × 10−31 +6.1362×10−45

λ2 + 3.1803×10−63

λ4

[23]

iii 1.4896×1010

1.8070×1014− 1

λ2

+

4.9037×109

0.9200×1014− 1

λ2

( 6.5656×10−7

1.8070×1014− 1

λ2

+8.8155×10−17)

1.8070×1014− 1

λ2

+

( 7.1155×10−8

0.92×1014− 1

λ2

+2.9021×10−17)

0.92×1014− 1

λ2

+

4.3229×10−7

(1.8070×1014− 1

λ2 )(0.92×1014− 1

λ2 )

[60]

iv − 1.03252×10−16

1.2813×1014− 1

λ2

[39]


Figure 4.3: The polarizability of Hydrogen from various authors.

Given its status as the most recent of the data, Peck’s curve is the one used asthe standard for Hydrogen.


4.2.3 Helium

Table 4.3: Refractive index and polarizability for Helium (He).λ is given in metres.


i 1470.0910×107

423.98×1012− 1

λ2

6.3957×10−7

(4.2398×1014− 1

λ2 )2+

8.7011×10−17

4.2398×1014− 1

λ2

[53]

ii 2.7212×1010

7.8086×1014− 1

λ2

2.1914×10−6

(7.8086×1014− 1

λ2 )2+

1.6106×10−16

7.8086×1014− 1

λ2

Curve fitto [32]

iii 1.5908×1010

4.5741×1014− 1

λ2

7.4891×10−7

(4.5741×1014− 1

λ2 )2+

9.4155×10−17

4.5741×1014− 1

λ2

Curvefit to

Selected[32] Data


Figure 4.4: The polarizability of Helium from various authors.

As can be seen, the level of quality and quantity of the data for Helium is not asgreat as the two gases preceding. The two fitted curves to the [32] data is accurate,but imprecise, having an RMS of only 0.7080, the selected points on the other hand areinaccurate but precise, having an RMS of 0.9953. Again, Peck’s curve appears to be asatisfactory compromise between all the data, Ghosh included. The curves fitted to [32]data were done using a non-linear curve fitting program.


4.2.4 Nitrogen

Table 4.4: Refractive index and polarizability for Nitrogen (N2).λ is given in metres.


i 6.8552×10−5+ 3.2432×1010

1.44×1014− 1

λ2

4.05756 × 10−31 +1.9197×10−16

1.44×1014− 1

λ2

+ 3.1126×10−6

(1.44×1014− 1

λ2 )2

[61]

Figure 4.5: The polarizability of Nitrogen from various authors.

It is of note to point out that many theoretical calculations have been performedon the Polarizability of Nitrogen also, presented here is one of the works of [70]. Many ofthese theoretical works reference [32] for comparison against their results, which as canbe seen are an excellent fit for Peck’s curve once again.


4.2.5 Neon

Table 4.5: Refractive index and polarizability for Neon (Ne).λ is given in metres.


i 2.8854×1010

4.33×1014− 1

λ2

2.4638×10−6

(4.33×1014− 1

λ2 )2+ 1.7078×10−16

4.33×1014− 1

λ2

[25]

Figure 4.6: The Polarizability of Neon from various authors.

Compared to other atmospheric gases, there is little to go by in terms of data forNeon. It appears as though the work of Cuthbertson is the only experimental work doneon the dispersion of Neon.


4.2.6 Oxygen

Table 4.6: Refractive index and polarizability for Oxygen (O2).λ is given in metres.


i 3.8212×1010

1.4431×1014− 1

λ2

4.3211×10−6

(1.4431×1014− 1

λ2 )2+

2.2617×10−16

1.4431×1014− 1

λ2

[52]

ii − 2.09579×10−16

1.3373×1014− 1

λ2

[39]

iii 0.0002674 + 1.2×10−18

λ2 +1.06×10−31

λ4

1.5829 × 10−30 +7.1044×10−45

λ2 + 6.2756×10−58

λ4

[40]

iv 0.0002651 + 1.9431×10−18

λ2 1.5692 × 10−30 +1.1504×10−44

λ2 + 1.1174×10−62

λ4

[23]


Figure 4.7: The Polarizability of Oxygen from various authors.

From the graph, and given it is the most recent of the data (plus it’s simplicity)the data of [39] is chosen as the reference for Oxygen.


4.2.7 Water Vapour

Table 4.7: Refractive index and polarizability for Water (H2O).λ is given in metres.


i 3.01733 × 10−6 +2.7003×10−20

λ2 − 3.3092×10−34

λ4

1.4580 × 10−30 +1.3049×10−44

λ2 − 1.5992×10−58

λ4

[19]

ii − 1.7259×10−16

1.1873×1014− 1

λ2

Curve Fitto [32]

Figure 4.8: The polarizability of Water Vapour from various authors.

Given that the Data quoted in [32] comes from work performed more than 60years ago, it has been decided to accept the work of [19] as the standard for thepolarizability of Water Vapour.


4.2.8 Carbon Dioxide

Table 4.8: Refractive index and polarizability for Carbon Dioxide (CO2).λ is given in metres.


i 1.5449×106

5.8474×1010− 1

λ2

+

8.3092×1010

2.1092×1014− 1

λ2

+

2.8764×109

6.0123×1013− 1

λ2

(5.9187 × 10−26(2.8918 ×1024λ2+2.5751×1073λ10−2.6127×1038λ4+7.1378×1051λ6−8.5687×1062λ8−8.5970× 109))/((5.8474×1010λ2 − 1)2(2.1092 ×1014λ2 − 1)2(6.0123 ×1013λ62 − 1)2)

[57]

ii 0.0004375 + 2.58×10−18

λ2 +2.3×10−32

λ4

2.59×10−30+ 1.5277×10−44

λ2 +1.3621×10−58

λ4

[40]


Figure 4.9: The polarizability of Carbon Dioxide from various authors.

Once again, given its more recent status, the work of Peck will be used as thestandard for Carbon Dioxide.


4.2.9 Methane

Table 4.9: Refractive index and polarizability for Methane (CH4).λ is given in metres.


i 5.5940×1010

1.3006×1014− 1

λ2

9.2607×10−6

(1.3006×1014− 1

λ2 )2+

3.3109×10−16

1.3006×1014− 1

λ2

[24]

ii − 3.4180×10−16

1.3430×1014− 1

λ2

Curve fitto [32]

Figure 4.10: The Polarizability of Methane from various authors.

Due to a lack of any more recent and reliable data, the work of Cuthbertson willbe used as the reference for Methane.


4.3 Minimum particle density required to detect a gas using

interferometry

Using the data provided in the previous section for the 9 main constituents of the atmo-

sphere, we now seek to quantify the smallest concentration of each substance that can be

detected using interferometric techniques, this value will be used to give a lower bound

on the range of uncertainties for xi (the constituent concentrations) and will determine

to within how many parts per million of a sample, this apparatus can identify a gas27. As

alluded to at the beginning of the chapter, the minimum detectable concentration will be

a function of polarizability and therefore wavelength.

Providing one knows the amplitude of the oscillation of the output of the interferometer

as a function of pressure28, it is easy then to calculate the change in phase that occurs

for a given change in output. Determining the amplitude of the oscillation requires a few

calibration runs before proceeding with the experiment itself, the amplitude of oscillation

will be different for each different wavelength, given that each laser will have different

output powers and the different detector response to each wavelength. Furthermore,

alignment will not be exactly the same when one laser is removed and replaced with

another.

As established in section 2.3 we seek the linear dependence of n with P . We assign a

generic sinusoidal function, say O = Asin(B×P +C)+D, where A,B,C,D are arbitrary

constants, O the output of the CRO, and P the pressure, to the points in figures 4.11 and

4.12 then ∆P = 2πB

for one complete oscillator cycle.

Equation 4.1 shows that for a change in output voltage ξ (due to a change of pressure

δP ) one can infer the change in phase or fringe shift, denoted as δm, the fractional fringe.

δm =1

2πarccos

(

Vpp − 2ξ

Vpp

)

(4.1)

27The upper bound on uncertainties will be examined in chapter 528Vpp, as shown in figures 4.11 and 4.12

4.3 Minimum particle density required to detect a gas using interferometry 61

Where as will shortly be established, Vpp is the peak to peak voltage of the CRO. The

phase shift δm has values between 0 and 12, when counting fringes when at least one half

fringe has passed the detector. If we substitute ξ = Vpp this would evidently correspond

to a change in path length of λ2, and the output of the interferometer would have shifted

from a peak to a trough. Similarly for δP

δP =P

2πarccos

(

Vpp − 2ξ

Vpp

)

(4.2)

It is clear that ∆P corresponds to one whole integer fringe shift, and that δP corresponds

to a change of δm of a fringe29. Algebraically this can be written as

∆P = 2δP

δm(4.3)

the constant of proportionality between n and P (denoted as g) is then g = δmδPλ = λ

∆P.

Finally, the uncertainty in this gradient is

∆g

g=

∆(δm)

δm+

∆P

P

∆g

g= ∆ξ

2π2

Vpp

√

1 −(

(Vpp−2ξ)2

V 2pp

)

1

δm+

∆P

P(4.4)

∆PP

can be derived from the digital manometer, and over the full range for the device

used here, is 0.05%, and ∆ξ is determined by the smallest change in voltage the CRO can

detect, which is 0.001V 30. From equation 4.4 it is evident that the most accurate results

will be obtained when Vpp is large enough to make the first term in equation 4.4 negligible

when compared to ∆PP

. As will be shown in section 5.5, the accuracy can be increased

further still by expanding the range of measurement to include multiple fringes (where

δm ∼ 35 − 50) over a very large pressure range (where P ∼ 12

atm.), meaning that even

29Of course, if one has changed the pressure in the cell by an amount such that the output has variedfrom a peak to a trough and has come to rest somewhere between the next trough and peak, then δm

corresponds to a fringe shift of 1

2+ δm.

30See section 5.5


if Vpp is small, the large number of phase/fringe shifts will reduce the relative error in g.

Further discussion is provided in section 5.5.

As mentioned at the beginning of this section, it is useful to know exactly how small a

concentration of each atmospheric gas is detectable. To gain an estimate of the minimum

detectable concentration of a gas, the minimum number density, denoted as Nmin, that

can be detected in the situation where there is only one type of gas in the cell is used

to provide a lower bound on the uncertainty in xk from equation 2.1. It is calculated as

follows, starting from the smallest change in refractive index that can be determined.

(nmin − 1)L = δmλ, nmin =δmλ

L+ 1 (4.5)

Where L is the length of the optical cell. Relating this to the minimum concentration N

(number density) of a gas required to raise the refractive index of a gas in a cell initially

at vacuum, to nmin is given by substituting equation 4.5 into the Lorentz-Lorenz equation

thus giving

Nmin(α, λ, L) =3

4πα

(δm)2

L2 λ2 + 2(δm)L

λ(δm)2

L2 λ2 + 2(δm)L

λ+ 3(4.6)

Table 4.10 gives typical values of Nmin for each different atmospheric gas at 555 nm,

using for δm those presented in table 5.631 for a cell length of 0.5 m. Furthermore, table

4.10 assumes that there is only a single gas within the cell, not a mixture, the values in the

table represent the minimum number density required to produce a detectable fringe shift,

as opposed to determining relative quantities, whose error is associated with the linear

system. The values in table 4.10 assume that the apparatus can detect fringe shifts to

31In table 5.6, the percentage error in the total number of fringes is used, to obtain δm the value inthe table is multiplied by the number of fringes


Figure 4.11: Sample output of interferometer, run 1.

The fitted equation corresponding to this figure is O = 3.088 sin (0.001244P − 111.8) +4.558. The period of oscillationis 2π

0.001244= 5050.8 Pa. Both this figure and figure 4.12

are an example of what an experimenter would view on a CRO as the pressure changes.

Figure 4.12: Sample output of interferometer, run 2.

The fitted equation corresponding to this figure is O = 3.094 sin (0.001280P + 10.01) +4.57.The period of oscillation is 2π

0.001280= 4908.7 Pa. Note that although there appears

to be quite a good fit to the data points, the quality of this fit is not good enough toproduce a realistic calculation of relative quantities of gases in a mixture. See chapter 5for further discussion.


Table 4.10: Minimum number density to elicit interferometric detection at λ = 555 nmand L = 0.5 m

*note 2.6867 × 1025 is the number density of standard dry air.Gas N, (m−3) N

2.6867×1025

Ar 1.0221 × 1018 3.8 × 10−8

H2 2.1444 × 1018 8 × 10−8

He 8.5422 × 1018 3.2 × 10−7

N2 9.9822 × 1017 3.7 × 10−8

Ne 4.4454 × 1018 1.6 × 10−7

O2 1.0999 × 1018 4.1 × 10−8

H2O 1.1788 × 1018 4.4 × 10−8

CO2 6.6286 × 1017 2.5 × 10−8

CH4 6.7650 × 1017 2.5 × 10−8

within 1/100th of a fringe, an achievable degree of accuracy with modern fringe analysing

equipment [17].

At first appearances, the minimum number density for each of the substances listed

in the table appears quite good, in fact compared with the STP number density for air,

2.6867×1025(m−3), we can see that the orders of magnitude indicate detection below one

part per million. However, it should be noted that determining the concentration of a

single gas is significantly more accurate than determining the concentration of multiple

gases, this should not however be seen as a barrier to employing an interferometer as a

multiple gas detector, as will be discussed in section 5.

Furthermore, one must bear in mind that a longer cell length will reduce the minimum

number of particles that can be detected. If we have a cell of length L, initially under

perfect vacuum, we need to add a certain quantity of gas to the cell before the path length

changes by a detectable amount. If we now have a much longer cell, say 10L, we need

only add a tenth the amount of gas to produce the same change in path length.

This leads to the questions: what restrictions are there on cell length? and why not

use as large a cell as required? We cannot arbitrarily continue increasing the length of

our cell, a cell of length greater than 500 mm is probably the largest one that an ordinary

lab could accomodate, this is only going to yield an increase in accuracy of ∼ 1 order


of magnitude32 given that there exist problems due to beam divergence and mechanical

stability that impede the use of large cells and correspondingly, large interferometers.

There is the option of forcing the beam passing through the cell, to bounce back

and forth multiple times through the cell with the use of retroreflectors. This seems

like a workable solution, however it has two drawbacks. The first drawback is that the

beam will diverge much further than it would otherwise, leading to problems of spatial

containment 33. The second problem is that now we have effectively placed a Fabry-Perot

interferometer within our Sagnac Interferometer. One of the main reasons for employing

the Sagnac interferometer is the increased stability. Adding another interferometer into

the system will reduce stability to the level of a Fabry-Perot interferometer at least34,Xiao

et. al. [77] employ just such a system to some good effect, although with a Fabry-Perot

interferometer, active stabilization of the cavity is required.

This then leads to the question of how one would be able to detect trace gases, such

as Methane or Neon (see table 5.2) accurately in a mixture of many gases be overcome?

Recalling equation 4.6, it may be assumed, for cell lengths of the order of centimeters,

and optical wavelengths that

Nmin ∼ δmλ

2παL(4.7)

Quite clearly then, the only way one can reduce the minimum detectable concentration is

either through decreasing the wavelength or by increasing the cell length. One must be

wary of destroying the practicality of this system. Noted at the outset is the portability

of an interferometric gas detector, and as such cell length of greater than 500 mm might

become somewhat unwieldy and restrict the device to the confines of a laboratory.

Thus we can see that for single gases, the minimum concentration to induce fringe

shifts requires of the order of 1018 molecules or atoms of gas per m3, this corresponds to

32From cell length used in the actual experiment33Any method, such as this, where the cell length is increased will face this problem of beam divergence,

it is not specifically restricted to the use of retroflectors on a short path length cell34The combined system would in fact be less stable than either the Fabry-Perot or Sagnac individually


∼ 0.01 Pa. Obviously, not too bad for some applications, but as will be seen for mixtures

of gases, an interferometric detector is unable to compete with a GCMS device unless

extremely long path lengths are used.

4.4 Conclusions

Presented in this section are the polarizability graphs that are necessary for determining

the effect that each atmospheric gas has on the dispersion of air. These graphs and their

equations form part of the linear system required to determine relative concentrations of

gases in a mixture. Also shown is the method for calculating the minimum detectable

concentration of a single gas, highlighting some of the key aspects affecting the uncertainty,

as well as establishing a lower bound on the uncertainty for a mixture35. These effects

include the dynamic polarizability, cell lengths, Output amplitude, and the accuracy of

the manometer. In the following section, the values of polarizability for the principle

constituents of the atmosphere will be substituted into equation 2.4, as well as the values

for the gradient of the graph of n vs. P for each different wavelength. Using the postulates

of linear algebra, this system will be manipulated to determine the relative quantities of

each component of the atmosphere, and we will see how accurate an interforemeter is at

determining these quantities, or at least, how accurate the system allows it to be.

35The implication being that it is impossible to detect one particular gas more accurately when it is ina mixture than when it is the only component

67

5 Experimental results and analysis

We turn our attention now to confirmation of the theory, which will be done by an analysis

of the composition of air. The work of other authors [6; 36], shows that using a knowledge

of the composition of a gas, we can calculate how the refractive index of that gas will

change as a function of pressure for a constant wavelength. The converse must also be

true and this is what will be discussed in this section. The outcomes from this section

include showing how the linear system is manipulated to elicit the relative composition of

each constituent, as well as determining how accurate the methodology is. The previous

section dealt with limitations to the accuracy resulting from measurement uncertainty,

such as counting fringes, and measuring pressure. In this section it will be shown that

the uncertainties of the xk’s, the mole fraction of the kth gas in a mixture, of equation

2.1 are not simply a product of measurement errors, but also of error introduced by the

linear system.

An indicator of the uncertainty introduced by the linear system is the condition num-

ber, a quantity that infers how stable the linear system is, so for example, if small changes

in matrix B36 of equation 2.4 result in large changes in x then the system would be said

to be unstable, and correspondingly have a high condition number37. As mentioned ear-

lier on, we do need a knowledge of what it is we are seeking, but how much of each

component can be determined by the experiment.

5.1 Properties of the linear system

When dealing with any linear system (equation 2.4), it is important to understand how

accurate the experimental data must be in order to determine the quantity of interest, in

this case matrix X. The condition number of matrix A determines how variations in B,

the measured values, will affect the results of X, which yields the concentration of each

constituent of the mixture. The elements of row 1 of matrix A are, α11 to α1n being the

36Matrix B is the vector on the right of the equals sign37Condition numbers approaching 105 are considered high

68 5 EXPERIMENTAL RESULTS AND ANALYSIS

polarizability of gas 1 to gas n at wavelength 1. The elements of row 2 are, α21 to α2n

being the polarizability of gas 1 to gas n at wavelength 2, and so on until we have reached

the final wavelength.

In the experimental analysis, A is a 6×5 matrix, using six different wavelengths to de-

termine the 5 most abundant constituents of the atmosphere. The 6 wavelengths used are

532, 543.5, 594, 612, 633 and 780 nm respectively. The constituents being determined are

Argon, Nitrogen, Oxygen, Carbon Dioxide and Water Vapour, in that order. Ordinarily

the condition number for a non square matrix cannot be found. But the solution to this

system is a least squares approach, so it is required that the condition number of ATA be

found38, as is required by the least squares method. Thus using the equations given in

Chapter 4, matrix A takes the form

10−31

1.674 1.773 1.610 2.670 1.502

1.672 1.771 1.608 2.668 1.500

1.667 1.766 1.601 2.658 1.494

1.666 1.764 1.599 2.655 1.492

1.664 1.763 1.597 2.652 1.490

1.657 1.754 1.587 2.636 1.479

(5.1)

And Matrix ATA

10−31

16.669 17.652 16.004 26.566 14.928

17.652 18.694 16.948 28.133 15.809

16.004 16.948 15.365 25.506 14.332

26.566 28.133 25.506 42.339 23.792

14.928 15.809 14.332 23.792 13.369

(5.2)

38AT is the transpose of Matrix A

5.2 Results: Variation of output with pressure 69

The condition number of which is 650000. Such a high condition number implies that

a change of 1 unit to the values of B will result in a change of ≃ 6.5×106 to the values of

X. Clearly this implies that the values of B must fall within a very precise range of the

theoretical values, lest the results of X become utterly implausible. Furthermore, since

equation 2.4 is an exact system, small variations in B will not only produce results that

vary significantly from the actual values, but given its exactness, will produce non-sensical

results, such as negative relative concentrations or values greater than 1 for individual

elements of X, as well as the sum of all the elements of X being greater or less than 1.

This may be better understood by realising that it is impossible for any one component

of a mixture to comprise more than 100% of the mixture, or for the sum of all the mole

fractions not to add to 100%.

Not only must the solution of the system be of a least squares form, but it must also

come coupled with some linear constraints and maximum and minimum values for each

element, ie. no element can be less than 0 or greater than 1 and the sum of all elements

must equal 1. This analysis will be discussed in greater detail in the subsequent sections.

5.2 Results: Variation of output with pressure

A point of clarification must be made before contemplating the following data. In the

previous theoretical foundation discussed in chapters 2 and 3, all the work has been

performed based on fitting a curve to a set of data points for a reasonably small pressure

range (within about 3-5 kPa of ambient atmospheric pressure), a range that encompasses

only a few periods of the interferometer output, for example in figures 4.11 and 4.12. The

data that follows is essentially the same application over many more output periods. The

reasons for taking a greater number of output periods is to reduce the relative uncertainty

in determining the number of fringes counted by the detector. The technique used is to

reduce the pressure in the cell by a specific amount, in this case approximately one half

of an atmosphere, and slowly allow the external atmosphere to re-enter the cuvette whilst


Figure 5.1: Photodetector output, 200 mm cell, 532 nmThe pressure change required to shift the path length by one integer wavelength is thus59700

59.9466= 995.8863 Pa

Figure 5.2: Photodetector output, 200 mm cell, 543.5 nmThe pressure change required to shift the path length by one integer wavelength is thus5180050.88

= 1018.0818 Pa

monitoring the interferometer output on a CRO.

This method is employed because it can produce far greater accuracy in determining

the pressure change required to induce a path length difference of 1 wavelength. To

illustrate this, imagine that we vary the pressure in the cell only enough to see the output

rise and fall through the shift of a single fringe. Thus for a system that has a fringe

uncertainty of 1/100th of a fringe, the relative uncertainty is 1100

. However, if the pressure

range is increased so that 100 fringes are counted by the detector, the relative uncertainty

in the measurement will be 110000

. Although it is essentially exactly the same method as

described previously, it provides more accurate results, this is shown in figures 5.1 to 5.6.

5.2 Results: Variation of output with pressure 71

Figure 5.3: Photodetector output, 200 mm cell, 594 nmThe pressure change required to shift the path length by one integer wavelength is thus

5030045.04265

= 1116.7194 Pa


43.4425= 1150.9467 Pa

Figure 5.5: Photodetector output, 200 mm cell, 632.8 nmThe pressure change required to shift the path length by one integer wavelength is thus50300

42.1244= 1194.0830 Pa



36.6573= 1475.8316 Pa

5.3 Refractive index vs. pressure for various wavelengths

From the data in the previous section the elements of matrix B may now be calculated,

these are given in the second column of table 5.1. Provided also is a further level of

approximation in figure 5.9. The reason for providing this line of best fit, is to provide

a platform from which to extrapolate results to unmeasured wavelengths. The results of

both the actual experimental values and the values given by the line of best fit in figure 5.9

are compared. The reason for doing so is that in one instance, the raw values are entered

into the linear system, equation 2.4 and then the values at the same wavelengths from the

best fit curve are entered into the linear system, to determine whether an improvement

in the accuracy of the results has occured. Given that there are already well established

data with which to compare the composition of the atmosphere with (see table 5.2), any

improvements in the results using the best fit curve will easily be seen.

5.4 Composition of the atmosphere using interferometry

Figure 5.9 shows how the refractive index of air changes with pressure at various wave-

lengths. With this established, the composition of the atmosphere that the results predict

5.4 Composition of the atmosphere using interferometry 73

Pressure Pa0 500 1,000 1,500 2,000

n

1.000000

1.000001

1.000002

1.000003

1.000004

1.000005

Refractive index vs. Pressure

Figure 5.7: Refractive index vs.pressure, various wavelengths, 0atm. To show that in fact theselines are diverging, two graphs areincluded with the same range, withthe graph on the left starting atvacuum and the graph to the rightstarting at 1 bar.

532nm 543.5nm 594nm 612nm 632.8nm 780nm

Pressure Pa100,000 100,500 101,000 101,500 102,000

n

1.000265

1.000266

1.000267

1.000268

1.000269

1.000270

1.000271

1.000272

Refractive Index vs. Pressure

Figure 5.8: Refractive index vs.pressure, various wavelengths, 1atm.

Figure 5.9: n vs. P for various wavelengths with fitted curve. The equation of the line ofbest fit is n

P= 444.4948λ2 − 7.007 × 10−4λ+ 2.9186 × 10−9 is shown.


Table 5.1: Gradient of refractive index vs. pressure at various wavelengthsλ (nm) Gradient of n vs. P

(Pa−1)× 109, actual experi-mental values

figure 5.9 approximation(Pa−1) × 10−9

Method λL∆P

532 2.6710 2.6716 532×10−9

0.2×995.8863

543.5 2.6692 2.6691 543.5×10−9

0.2×1018.0818

594 2.6596 2.6592 594×10−9

0.2×1116.7194

612 2.6587 2.6563 612×10−9

0.2×1150.9467

632.8 2.6506 2.6532 632.8×10−9

0.2×1194.0830

780 2.6426 2.6425 780×10−9

0.2×1475.8316

can be determined, by substituting in the values from column 3 of table 5.1 as matrix B

in to equation 2.4

α(ω1)1 α(ω1)2 .. .. α(ω1)k

α(ω2)1 α(ω2)2 .. .. α(ω2)k

: : :: :: :

: : :: :: :

α(ωj)1 α(ωj)2 .. .. α(ωj)k

x1

x2

:

:

xk

=1

β

(

nP

)

ω1

(

nP

)

ω2

:

:(

nP

)

ωj

(5.3)

Both sides of equation 2.4 have been multiplied by 10−30 and 1β, β = 1.5419× 1021 has

been multiplied into B39.

39rounding has also been applied


1.6737 1.7726 1.6097 2.6702 1.5021

1.6724 1.7711 1.6079 2.6676 1.5003

1.6674 1.7658 1.6011 2.6579 1.4937

1.6659 1.7642 1.5991 2.6550 1.4917

1.6644 1.7625 1.5970 2.6518 1.4895

1.6568 1.7544 1.5867 2.6359 1.4790

xAr

xN2

xO2

xCO2

xH2O

=

1.7323

1.7312

1.7249

1.7243

1.7191

1.7139

(5.4)

The evaluation of these results was performed with Maple, using the “LSSolve” op-

eration contained within the “with(Optimization)” package. Maple is a mathematical

computer program, similar to other programs such as Mathematica and Matlab. The

“LSSolve” operation is a least squares solving operation designed for use on linear sys-

tems whose solution is not exact. Part of the LSSolve operation is that the user may define

certain limitations on the results. The first limitation are the linear constraints, which

is simply entered as “a:=Matrix([[1,1,1,1,1]], datatype = float); b:=Vector([1], datatype

= float); lc := [a, b];” this confines the 5 values of x to collectively sum to 1, as would

be expected. The second is that none of the values of x can be negative, the line of

code which one would enter to obtain the results in equation 5.5 is LSSolve([A, B],lc,

assume=nonnegative); after having previously defined “lc”. The results of this analysis

are shown in equation 5.5 with the corresponding 1-R2 value.


4.1801 × 10−6,

0.03936

0.7365

0.2242

0.0

0.0

=

xAr

xN2

xO2

xCO2

xH2O

(5.5)

Before entertaining any analysis, the results from the line of best fit in figure 5.9 will

be considered and is given in equations 5.5 and 5.6. The values in the columns of equation

5.6 as well as subsequent similar equations are the values of the second column of equation

5.4.

1.1615 × 10−6,

0.03299

0.7390

0.2280

0.0

0.0

(5.6)

Compare this to the actual composition of the atmosphere at the time of the experiment.

The data was taken with a relative humidity (RH) of 72% at, as mentioned in the graphs

an ambient temperature of 22 degrees celsius. Using the Goff-Gratch equation (equation

5.7, an equation that determines the saturation vapour pressure, gives a mole fraction of

1.85% Water Vapour.


Table 5.2: Composition of Earth’s atmosphere at 0% RHMolecule/Atom Mole Fraction

N2 0.7808O2 0.2094Ar 0.009325CO2 ∼ 0.0003Ne ∼ 0.00002He ∼ 5 × 10−6

CH4 ∼ 2 × 10−6

H2 ∼ 5 × 10−7

Kr ∼ 1 × 10−7

log10 Ps = 3.0057 − 7.9030

(

373.16

T− 1

)

+ 5.0281 log10

(

373.16

T

)

+

− 1.3816 × 10−7(

1011.314(1− T373.16 ) − 1

)

+ 8.1328 × 10−3(

10−3.4915( 373.16T

−1) − 1)

(5.7)

Where Ps is the Saturation vapour pressure of Water, and T the absolute temperature.

For an ambient T of 22◦ C the Saturation vapour pressure is 2658 Pa. Given that the

relative humidity is 72% the actual vapour pressure of water is 1914 Pa. The mole fraction

of water in the atmosphere at the time of the experiment is given by Dalton’s Law

Pw = xw × P (5.8)

And in this case Pw

P= 1914

103446= 0.0185. And subsequently the mole fraction of Dry air

is xd = 1− 0.0185 = 0.9815.The composition of dry air is given by [62] in table 5.2 whilst

the composition of the atmosphere for the conditions encountered in the experiment are

given in table 5.3.


Table 5.3: Composition of Earth’s atmosphere at 72% RH, 22◦CMolecule/Atom Mole Fraction

N2 0.7661O2 0.2055H2O 0.01877Ar 0.009150CO2 ∼ 0.0003Ne ∼ 0.00002He ∼ 5 × 10−6

CH4 ∼ 2 × 10−6

H2 ∼ 5 × 10−7

Kr ∼ 1 × 10−7

Of course, the most obvious omission is that of any trace of Carbon Dioxide or Water

Vapour in both analyses. But clearly the result for each is within ∼2% of the actual

values at the time of the experiment. It stands as a testament, in the face of such a high

condition number, that the interferometer can deliver results that come within 2% of the

actual values.

Further analysis can be performed based on the earlier discussed method of “grouping”

gases and treating them as a single gas. In this case, all the gases that comprise dry, CO2

free air are grouped together, which leaves the remainder of the ungrouped gases as CO2

and Water Vapour. The results aim to show how well the detector can perform in a

situation when it is desirable to know how much of a certain type of non-atmospheric gas

is present in the air. The refractivity of dry CO2 free air has been determined [19] and is

denoted as rd0%, it is given at 1 atm and 15◦ as

rd0% =

(

5.7921 × 1010

2.3802 × 1014 − 1λ2

+1.6792 × 109

5.7362 × 1013− 1

λ2

)

× 0.9997597 (5.9)

Using the Taylor expansion of equation 3.1640centred on n = 1, the polarizability of

40α = 3RT4πNaP

n2−1

n2+2= RT

2πNaP

(

(n − 1) − 1

12(n − 1)2 + O

(

(n − 1)3))

, taking only the first term in the

expansion for n2−1

n2+2.


Table 5.4: Polarizability of Water Vapour, CO2 and air at experimental wavelengths.Wavelength(nm)

α × 10−30 Dry,CO2 free air(m3)

α × 10−30WaterVapour(m3)

α × 10−30

CO2(m3)

532 1.7381 1.5021 2.6702543.5 1.7366 1.5003 2.6676594 1.7310 1.4937 2.6579612 1.7293 1.4917 2.6550632.8 1.7276 1.4895 2.6518780 1.7191 1.4790 2.6359

this mixture is.

α(λ) = 6.2490×10−27×(

5.7921 × 1010

2.3802 × 1014 − 1λ2

+1.6792 × 109

5.7362 × 1013− 1

λ2

)

×0.9997597 (5.10)

Using this, and the previously established formulas for the polarizability of CO2 and

Water Vapour, the polarizability for dry, CO2 free air, CO2 and Water vapour, at the 6

wavelengths used in the experiment are given in table 5.4

The values in table 5.4 form an alternative matrix A from the above calculations, and

subsequently, matrix X now only contains three elements, xDryAir, xH2O and xCO2. The

elements of matrix B remain unchanged. The results are given in equation 5.11

1.4994 × 10−6,

0.9667

0.03200

1.3370 × 10−3

(5.11)

Clearly what these results demonstrate is that the configuration is not well set up for

detecting small quantities of gases, that is, below ∼ 1-2%. As a final level of analysis, it

is assumed that the atmosphere is comprised entirely of Nitrogen, Oxygen and Argon41

41which it essentially is


Table 5.5: Polarizability of N2, O2 and Ar at experimental wavelengths.Wavelength(nm)

α × 10−30 N2

(m3)α × 10−30 O2

(m3)α × 10−30 Ar(m3)

532 1.7726 1.6097 1.6737543.5 1.7711 1.6079 1.6724594 1.7658 1.6011 1.6674612 1.7642 1.5991 1.6659632.8 1.7625 1.5970 1.6644780 1.7544 1.5867 1.6568

as well as demanding the results give the quantity of Water vapour in the air is as mea-

sured (1.877%)42. Using the formulas for polarizability from chapter 4 we have α at the

experimental wavelengths given in table 5.5.

The results of this analysis are

1.5107 × 10−6,

0.7588

0.2092

0.01877

0.01323

(5.12)

All of which fall within 1% of table 5.3. This demonstrates that for gases in quantities

above ∼1%, an interferometric detector can return reasonably accurate results. Further

this demonstrates why an interferometric gas analyser is reasonably well set up to detect

simple mixtures of gases, as the number of gases in the mixture decreases, the accuracy

increases. This has also been shown elsewhere [71] to be the case.

42Performed using a wet/dry thermometer

5.5 Uncertainty of results 81

5.5 Uncertainty of results

Returning to the linear system

a

(n×m)

x

(m× 1)

=

1

β

B

(n× 1)

, n > m, aX =

1

βB (5.13)

If the system is overdetermined, as in the case presented, then the exact solution to

the system is

X =1

βa†B (5.14)

Where a† is the pseudo-inverse of a, we require the psuedo-inverse as opposed to the

standard inverse a−1 considering that a is non-square. The pseudo-inverse is defined as

a† =(

aTa)−1

aT (5.15)

The uncertainty of X, written in terms of vector norms, is thus given by the following

bound [26]

||δX||||X|| ≤ σmax

σmin

( ||δb||||b||

)

(5.16)

Where σmax and σmin are the maximum and minimum Eigenvalues of aTa, the ratio

of σmax and σmin describe the stability of the linear system, or as has already been es-

tablished, the condition number. However, this formula is derived from the perturbation

theory for the least squares problem, whereas what this analysis deals with is a perturba-

tion to the constrained, bounded least squares problem, and a rigorous theoretical model


for a perturbation to such a system is well beyond the scope of this work43

We therefore confine our attention to a more practical method for estimating uncer-

tainty. To do this, the measured values of B used to generate the solution in equation 5.6

are subtracted from the exact values of B generated by table 5.3. The relative difference

between the exact and experimental values of B, are given by the following

∆∗B =||A.xexact −B||

||A.xexact||(5.17)

It is expected that the value of ∆∗B to be very close to the value of δBk from equation

5.19, which we will determine shortly. If this is the case, or even if the values of δBk are

greater than ∆∗B, then it is assumed that the absolute error in X is approximately the

average difference between the exact values for X from table 5.3 and the values given in

equation 5.6. This is essentially a comparison of the errors determined from measurements,

such as counting fringes and measuring pressure, to the errors determined from the linear

system.

The values for the Bk, the constant of proportionality between n and P for each

wavelength, are given by

Bk =mkλk

0.2Pk

(5.18)

Where mk is the number of fringes detected when the pressure is varied over the given

range Pk, λ the kth wavelength and 0.2 m being the length of the cell. Subsequently the

43The closest the author has seen to such a theoretical formulation for this sytem is merely the solu-

tion to a linearly constrained (but not bound) least squares system. No such reference could be founddiscussing perturbations.


uncertainty in Bk is

δBk =

∣

∣

∣

∣

∂Bk

∂mk

∣

∣

∣

∣

δmk +

∣

∣

∣

∣

∂Bk

∂Pk

∣

∣

∣

∣

δ(Pk) (5.19)

Which may also be written

δBk = Bk

(

δmk

mk

+δ(Pk)

Pk

)

(5.20)

The values for δmk

mk, δPk

Pkand δBk for each wavelength are provided in table 5.6, although,

the manufacturer specifications claim that the uncertainty in all pressure measurements

is .05% .

The calculation of δmk

mkis somewhat subjective however. In determining fractions of

a fringe, the method employed was to note the detector reading at the time when the

pressure in the cell stops changing, or in other words, reaches 1 atm. The value of this

point compared to the value of the whole previous fringe is used to infer what fraction of

a fringe this corresponds to. So for example, if the reading when the manometer reads

zero (once the pressure in the cell has reached 1 atm) is at half the height of the previous

whole fringe, this would be said to correspond to one quarter of a fringe. Mentioned above

is that this method is used to infer what fraction of a fringe this corresponds to, but it is

not a simple linear calculation.

Figure 2.4 to illustrates this point. Assuming arbitrary units of pressure on the hori-

zontal axis, note that the graph is drawn such that the output voltage stops just after a

trough. In calculating what fraction of a fringe has passed after this trough, one would

measure first the peak to peak voltage Vpp and secondly the difference in Voltage between

the trough and the final position of the output, which is denoted as ξ. It can be shown

that this fractional fringe δm is given by


Figure 5.10: Variance of fringes as ξ changes.

One can see that as ξ approaches either 0 or 1 whole fraction (of Vpp), the frac-

tion of a fringe begins to change much quicker than around Vpp

2.

δm =1

2πarccos

(

Vpp − 2ξ

Vpp

)

(5.21)

This equation has already been established (equation 4.1). As discussed in equation

4.1, this always results in a number between 0 and 12, the experimenter should be able to

determine by eye, fringe shifts of half integer steps, equation 5.21 calculates accurately the

amount of fringe shifts between half integers. As you would expect, for shifts in fringes

approaching one quarter of a fringe, this equation becomes approximately linear, however

if the beginning or the end of the output of the interferometer is very near a peak or

trough, careful use of this equation is required, as the rate of change in m is much greater

as ξ approaches 0 or Vpp, as shown in 5.10.


Table 5.6: Uncertainty in gradient of refractive index vs pressure

Wavelength(nm)

δBB

(%) δmm

(%) δ(P )P

(%)

532 0.0573 0.00073 0.05543.5 0.068 0.0018 0.05594 0.0527 0.00027 0.05612 0.0531 0.00031 0.05632.8 0.0533 0.00033 0.05780 0.068 0.0018 0.05

The uncertainty in this reading is given by the uncertainty in the voltage reading by

the CRO and the photodetector device44, which is 0.001 Volts, hence the uncertainty in

the number of fringes is the uncertainty in the voltage, divided by the potential difference

between the maximum and minimum voltages of the previous fringe divided by two, given

that we are counting half integer fringes.

We seek a comparison of δB to ∆∗B. For the sake of brevity, 1β

has been multiplied

into the answers and rounding has been performed.

A.xexact = 10−30

1.7334

1.7318

1.7262

1.7245

1.7227

1.7141

(5.22)

44The CRO can measure to the nearest µ V, however the detector CRO system is only as accurate asthe most inaccurate device, that being the photodiode at ∼ 1 mV


The experimental values,B, for the fitted curve to figure 5.9 are again.

B = 10−30

1.7327

1.7310

1.7246

1.7227

1.7207

1.7138

(5.23)

r = |A.x− b| =

6.5560 × 10−4

7.7834 × 10−4

15.36 × 10−3

1.7833 × 10−3

2.0176 × 10−3

3.3764 × 10−4

(5.24)

The average of each element of r divided by each corresponding element of A.xexact is

6.8660 × 10−4, thus the average percentage error is 0.07%. It has been shown that when

using the values derived from experiment, the results are perturbed by about 0.07%,

which is roughly equal to the values calculated in table 5.6. Given this confirmation, it is

reasonable to then say that the absolute error in this experiment is, as established earlier,

||xexact − xexp||, which is


∆X =

0.02382

0.02715

0.02246

0.0003759

0.0187

(5.25)

Assuming the worst case scenario, and taking the highest value of ∆X, which is the

uncertainty in N2 = 2.7%. It can be said of the system that it is accurate to approximately

3% of the actual mole fraction of each constituent.

One lingering question remains, is an error calculation performed in this manner re-

liable? Previously, the condition number of the calculations was established to be of the

order of 105, the uncertainty in B is definitely not less than 1 part in 105. Recalling

that a condition number of the order of 105 will magnify errors by about 105, given that

the errors in B are of the order of 10−5 this would correspond to an absolute error of

1, or 100% in the elements of X. Such a question however, applies only to a linear sys-

tem, and the results found here have been made with the assistance of a program that

allows us to place linear constraints and bounds on the solution. In the absence of any

theoretical calculation for the uncertainty in a linearly constrained and bounded system,

we are left with almost no alternative but to compare the results to that of previously

established theory, and as the above work shows, we are indeed safe in doing so, given

that two independent methods for calculating the error in B, first by residuals and second

by traditional experimental error, both give very similar values.

In the absence of a rigorous theoretical check of this estimate of error, we are left

with only the option of actually perturbing the system by the very values that have been

claimed to produce the results to within 3% of the actual values. To do this, we will make

use of the constrained and bounded least squares system that produced equation 5.12,


but apply random perturbations of ∼ 0.05% to the theoretical values of B and repeat, in

this case it was chosen to perform 50 repetitions. The average values from this operation

of Nitrogen, Argon and Oxygen were 76.3%, 3.86% and 17.7% respectively. Compare this

with the exact quantities and we see that Nitrogen deviates by only 0.3%, Argon by 3%

and Oxygen by 3%. Assuming, as before, the worst result, we see again that the error is

approximately 3%.

The results of this calculation comes with a word of warning, individual calculations

can still produce reasonably implausible results, the maximum and minimum values for

Nitrogen among all 50 calculations was 94.7% and 64.3% respectively, the maximum and

minimum values for Argon among all 50 calculations was 30.9% and 0% respectively and

for Oxygen, the maximum and minimum values was 21.6% and 1.5%. Although this may

be attributed to the fact that this is a random calculation, as with all experiments, care

should be taken to make multiple measurements, to average out statistical fluctuations.

Finally, why can’t equation 5.16 provide an adequate value for the uncertainty for

this experiment? It is important to realise that equation 5.16 does not in fact provide a

best estimate of the uncertainty, it yields only an upper bound of the uncertainty. Upper

bounds are useful in some situations, but in this case it is not appropriate, the reason

being that the ratio of the maximum and minimum singular values of B is of the order of

(105)45 when we consider that the error calculated for b is of the order of 10−4. According

to equation 5.16, the most the uncertainty could possibly be is 1000%. This is in no way

wrong, it simply means that there is no way of knowing how much less than that value the

uncertainty actually is. Furthermore, equation 5.16 quotes the uncertainty in the length

of the vector X, not the absolute uncertainty in that vector. Mentioned earlier was the

fact that during the repetitive analysis of the uncertainty in X the magnitude of Argon

reached as much as 30.9%, which interestingly is in fact of the order of 1000% greater

than its actual concentration. However, as shown, the average value of the concentration

45This value also depends on the number of decimal places one takes matrix A to, the rule of thumbis that matrix A should be quoted to no less than 4 decimal places, any less results in the uncertainty ofB varying by more than the calculated uncertainty of B

5.6 Conclusions 89

of Argon is indeed much lower.

5.6 Conclusions

It has been shown in this chapter how the results are calculated using the linear, con-

strained and bounded least squares system. The results show that an interferometer

is capable of estimating concentrations to within less than 3%, and with some of the

improvements suggested in the following section, may be even better.

Also shown is the inherent stability of the linear system, and how this impacts on

the overall uncertainty and how it compares to the standard measurement uncertainties,

such as fringe counting and pressure measurements. In the remaining section, attention

will be given to what the results mean for the applicability of an interferometer as a gas

detector, potential expansion of its uses, and how the interferometer could be modified to

counteract the error magnifying effects of the linear system due to the condition number46.

46And how with an appropriate choice of wavelength, the condition number might be reduced

91

6 Conclusion

6.1 Experimental refinements

Evidently, the only hurdle lying in the path of approaching ever more accurate results47

is being able to measure pressure and fringes to a higher resolution. But how accurately

can pressure be measured? To what fraction can we measure the change in optical path

length of a beam of light? The answers to these questions, effectively result in changing to

the RHS of equation 2.4 and will be discussed shortly, but what can be done with Matrix

A to improve our results?

With such a high condition number, in its present state, greater range in the values

in Matrix A would certainly assist in improving the accuracy of the system. In chapter

3 and borne out in the graphs provided at section 4.2, it was found that the gradient of

the polarizability increases with decreasing wavelength, meaning that as the wavelength

decreases, the difference between polarizability, for corresponding wavelengths increases.

Increasing the differential polarizability between constituent gases would assist in reducing

the condition number of A. But we do not necessarily need to look to shorter wavelengths

for a way to reduce the condition number of A.

The underlying reasoning for searching through higher frequencies can equally be

applied to selecting wavelengths that occur in regions of anomolous dispersion, many of

which lie both in the infra-red and ultra-violet for atmospheric gases. Strictly speaking,

one must be careful however not to become too dependent on seeking regions of anomolous

dispersion for reducing the condition number of A, as this transforms the problem from

an interferometric/refractometric domain into a spectroscopic domain. However if the

experimenter is interested in knowing only if a particular gas is present in the sample,

then using a radiative source close to a resonant frequency of the gas of interest would make

the presence of the gas immediately recognisable in the interference pattern produced48.

47To the limits established in table 4.1048Close to a resonant frequency, one would notice an extremely high number of fringe shifts over a

given pressure range when compared to fringe shifts over the same pressure range at another wavelength.

92 6 CONCLUSION

Other basic challenges also exist in dealing with IR light, such as the ease of alignment

provided with visible wavelengths and that ordinary glass cuvettes are highly reflective of

IR light.

The only real restriction on how accurate an interferometric gas detector can be made

lies only in determining shifts in fringes, which, as mentioned in chapter 4, can be reduced

to significantly small levels, providing accurate determination of the height of the previous

fringe shift. Measurements of pressure too can be made to very accurate levels, and

although not used in this work, a system was developed that would allow the experimenter

to determine changes in the pressure of the cell to less than one pascal.

It simply involved the use of a gas syringe, whose plunger was attached to a threaded

screw. The pressure in the cell changes with changes to the volume in the cell by Boyle’s

law. The percentage uncertainty in the pressure is proportional to the percentage uncer-

tainty in the volume which is proportional to the percentage uncertainty in the distance

that the plunger of the syringe is from the stop. Because the plunger is attached to a

finely threaded screw, small changes in the distance from the stop correspond to large

changes in the angle swept out by turning the screw. This angle can be measured very

accurately and subsequently the pressure in the cell can be very accurately measured also.

Based on this, it is the author’s belief that the accuracy of this multiple gas analyzer

could with a small amount of additional work, easily reach an accuracy of around 1000

ppm, with some simple improvements in pressure measurement at the wavelengths chosen.

Although at this stage, with its current level of accuracy, it still retains use as a bulk gas

analyzer, more than useful for simple mixtures where quantities need only fall within ∼1%

accuracy, a level of accuracy which is more than useful in certain applications [71; 50].

Finally, during the course of any experimentation and research, various other unex-

pected properties of the system being investigated became apparent. One such property

is the ability of an interferometer to accurately determine changes in pressure, to a much

higher degree of precision than could be obtained with a manometer or barometer. Given

the availability of fringe counting hardware and software, that is capable of distinguishing

6.1 Experimental refinements 93

changes down to about 1/100th of a fringe, it would seem like this could be a useful ap-

plication of an interferometer in circumstances were very precise pressure measurements

need to be taken. This might be an area of research worth pursuing in the future.

94 6 CONCLUSION

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