minimum refractometrically detectable concentrations of single atmospheric gases and simple mixtures...

$: Minimum refractometrically detectable concentrations of single atmospheric gases and simple mixtures with a Sagnac interferometer$
Minimum refractometrically detectable concentrationsof single atmospheric gases and simple mixtures

with a Sagnac interferometer

Sean McConnell* and Esa JaatinenApplied Optics Program, Queensland University of Technology 2 George Street, Brisbane,

Queensland, 4000, Australia

*Corresponding author: [email protected]

Received 14 January 2009; revised 8 May 2009; accepted 12 May 2009;posted 3 June 2009 (Doc. ID 106363); published 12 June 2009

Theminimum quantities of the ninemost abundant, isolated, atmospheric gases that are detectable witha refractometer are calculated. An examination of the applicability of refractometric techniques for de-tecting and analyzing gaseous mixtures is discussed and a comparison made against other establishedtechniques. Traditionally, most gas analysis performed with an interferometer is in determining the dis-persion or refractivity of a known sample, presented here is the inverse approach, where refractivities aremeasured to determine the concentrations of particular species within a gas. The method, and experi-mental results for determining the minimum quantities of a particular species detectable in a mixturehas been explored, as well as the complications, such as the indistinguishability of dynamic polarizabil-ities of different gases and the subsequent demands for accurate pressure and fringe measurements ofusing interferometric techniques. It is shown that the concentration of a single (isolated) gas, in units ofnumber density, can be determined to within approximately 1–10 × 1018 m−3, and a mixture of the threemost abundant gases, N2, O2 and Ar, to within 3:4 × 104 parts in 106 ðppmÞ when a minimum detectablefringe shift of λ=100 is assumed. © 2009 Optical Society of America

OCIS codes: 120.3180, 010.1290.

1. Introduction

Gas analysis and identification forms a large field ofresearch and is also crucially important in a varietyof commercial, industrial, and healthcare applica-tions. The desired outcome is to develop measure-ment equipment that is able to determine thecomponents of mixtures and their concentrationsas well as being as affordable, uncomplicated, andconvenient as possible. Many techniques exist for thedetection and analysis of gases, the most widely usedof which being gas chromatography/mass spectrome-try (GCMS), however GCMS systems are usually ex-pensive, bulky, and are not readily applied to fieldwork without a resultant loss in sensitivity [1] Sincethe sample has to be taken to a laboratory-based

GCMS, it is unable to provide a real-time monitoringsolution as can be required in quickly changing en-vironments. As a result, other techniques such as in-terferometric/refractometric approaches may bemore suitable for in-situ analysis.

Although unable to match the sensitivity of aGCMS, the interferometric technique presented hereis still applicable to a variety of gas analysis situa-tions, especially in circumstances where trace quan-tities are not of interest. Some work has been done inexamining the use of interferometers as gas detec-tion devices [2–4] and, theoretically, an interferom-eter should be able to function as a gas analysisdevice that determines the quantities of the knownconstituents of a gas mixture. Unlike a GCMS how-ever, it cannot be used to determine the constituentgases. A comprehensive outline of noninterfero-metric gas analysis methods can be found by consult-ing [5,6] or [7].

0003-6935/09/183481-09$15.00/0© 2009 Optical Society of America

20 June 2009 / Vol. 48, No. 18 / APPLIED OPTICS 3481

Previously, a single-wavelength Jamin interferom-eter has been used to detect changes in a gas mixture[4]. In that work, a servo loop used the interferometersignal to adjust the gas mixture so as to keep it at afixed ratio.In the work presented here, we expand on this con-

cept by usingmultiple wavelengths, so as to allow thedetermination of the quantities of the individual con-stituents themselves. For this method to work, themolar refractivities of each of the constituents mustbe known in order for their quantities to be calcu-lated from the refractive index change measuredby the interferometer. This investigation is appliedto gases of the atmosphere, and in Section 5, wheremultiple gases are analyzed, N2, O2, and Ar are usedas the sample mixture, with the intent of providingan indication as to how effective the Sagnac interfe-rometer is at distinguishing multiple gases.In determining these quantities, this method,

using a Sagnac type interferometer, varies the pres-sure of the gaseous sample, at six different wave-lengths, permitting a greater number of constituentsof a mixture to be determined. In terms of sensitivity,an accuracy of �1% [5] is considered an acceptablelimit for gas analysis instrumentation.

2. Interferometer

The Sagnac interferometer (shown in Fig. 1) was cho-sen for this investigation as it exhibits a superiorpassive stability [8] over other equivalent interfe-rometers. High stability is required to measure smallfringe shifts at the interferometer output, particu-larly in environments with high levels of vibrationalnoise.As is discussed in Section 4, accurate determina-

tion of fringe shift is the limiting factor in evaluatingthe quantities of the constituent components of themixture. The Sagnac’s stability is achieved throughthe split beam striking every optical surface, hence,any instability over one beam path will be trans-mitted equally to the other. As both beams contain

the same phase jitter, when the beams recombineat the beam splitter, most of the phase noise cancelresulting in a much lower noise level, allowing betterdetermination of fringe shift to bemade. The only dif-ference in the optical path length for both beams isdue to the cell containing the gas to be analyzed.

The procedure involves measuring the total fringeshift recorded at the detector as the pressure of thegas mixture in the cell is varied over a specific range.The total fringe shift measured at the detector, for agiven pressure range, depends on the mixture in thecell, and the wavelength of light used. Therefore, byvarying the pressure and by measuring the totalnumber of fringes (whole and fractions) that passthe detector, the variation of the refractive indexacross this pressure range can be determined. If thisis repeated over several wavelengths, the results canbe combined to yield the quantities of the gases in themixture. At the very minimum, one unique wave-length is required to determine the quantity of eachconstituent in the mixture.

3. Detectability of a Single Gas

Here we assume that the minimum possible frac-tional fringe shift that the interferometer can mea-sure is 1=τ of a whole fringe. For simplicity, weassume τ to be the same for all wavelengths used.By inspection of Fig. 1, it is evident that when start-ing with a perfect vacuum both inside and outsidethe cell, the path length difference of the beams(neglecting the thickness of the optical windows ofthe cell) is 0, and the normalized output registeredby the detector is 1 (arb. units). As increasing quan-tities of gas are gradually added to the cell, the out-put of the detector will decrease to a minimum value,which corresponds to a path length difference of λ=2from the initial state.

The smallest detectable change in refractive index,nmin, from the initial vacuum, is related to the smal-lest detectable change in fringe position by Eq. (1):

nmin ¼ λτLþ 1; ð1Þ

where L is the length of the cell in Fig. 1, and λ thewavelength. In practice, τ is of the order of 100 [9].

Furthermore, N, the number density correspond-ing to the amount of particles per unit volume produ-cing a shift in fringe of 1=τ, is related to the refractiveindex by the Lorentz–Lorenz equation [10,11]:

n2− 1

n2 þ 2¼ 4π

3NαðλÞ; ð2Þ

where αðλÞ is the dynamic polarizability of the singlegas in question. Thus, the detectable concentration ofa gas can be expressed as

Nðλ;LÞ ¼ λ2παðλÞτL : ð3Þ

Fig. 1. Sagnac interferometer.

3482 APPLIED OPTICS / Vol. 48, No. 18 / 20 June 2009

It can be seen that the minimum detectable con-centration of a gas is a function of the dynamic polar-izability, α, the wavelength of light, λ, and the lengthof the optical cell for a fixed τ.

4. Minimum Detectable Concentrations of SingleGases

As Eq. (3) shows, the detectability of a single gas de-pends on four variables, the cell length, fractionalfringe τ, the wavelength, and the dynamic polariz-ability, only the latter of which is independent ofthe apparatus. Table 1 lists the dynamic polarizabil-ities for commonly encountered atmospheric gases,valid for the visible and near infrared.Using the values of αðλÞ at the specified wave-

lengths, it is now possible to utilize Eq. (3) to deter-mine the detectability limits of each of the gases inTable 2; these results are displayed in Table 2 withτ ¼ 100 and L ¼ 0:5m. As shown in Eq. (3), the mini-mum detectable quantity varies inversely with both

L and τ. Therefore, theoretically the minimum de-tectable quantity can approach 0 as either L or τ ap-proach ∞. In practice, with modern fringe countingequipment it is difficult to measure fringe fractionsgreater than τ ∼ 100 [9]. Similarly, due to manufac-turing difficulties, space considerations, and beamdivergence, cell lengths much in excess of 0:5mare impractical. For these reasons, τ ∼ 100 and L ¼0:5m are assumed.

As can be seen, interferometric techniques for de-tecting gases can be quite accurate for inferringquantities of a single gas to levels around one partper 10 million. The situation is complicated thoughwhen dealing with mixtures.

Table 2 shows that for isolated gases, very low con-centrations can be detected. Comparing the values inTable 2 to the number density of standard air (as in-dicated in column 3), an interferometer can detectchanges in pressure within the optical cell to approxi-mately a few parts in 108 of an atmosphere. Such anaccuracy could be very useful for any experiments in-volving the study of critical temperatures andpressures.

5. Detectability of Multiple Gases

Defining detectability for multiple gases requiresquantifying a particular species not by its numberdensity, but by its mole fraction. The number densityis usually reserved for describing the concentrationof a species within the mixture as a whole, whereasthe mole fraction, in units of parts per million, is themethod most commonly used to describe the quanti-ties of the constituents of the atmosphere, relative to

Table 1. Dynamic Polarizability of Selected Atmospheric Gasesa

Gas Dynamic Polarizability (m3) Reference

Ar 4:01070 × 10−31 þ 1:7866×10−161:44×1014− 1

λ2þ 2:6960×10−6�

1:44×1014− 1λ2

�2

[13]

H2

�6:5656×10−7

1:8070×1014− 1λ2þ8:8155×10−17

�1:8070×1014− 1

λ2þ

�7:1155×10−8

0:92×1014− 1λ2þ2:9021×10−17

�0:92×1014− 1

λ2þ 4:3229×10−7�

1:8070×1014− 1λ2

��0:92×1014− 1

λ2

� [14]

He 8:7011×10−174:2398×1014− 1

λ2þ 6:3957×10−7�

4:2398×1014− 1λ2

�2

[15]

N2 4:0576 × 10−31 þ 1:9197×10−61:44×1014− 1

λ2þ 3:1126×10−6�

1:44×1014− 1λ2

�2 [16]

Ne 1:7078×10−164:33×1014− 1

λ2þ 2:4638×10−6�

4:33×1014− 1λ2

�2

[17]

O22:0958×10−161:3373×1014− 1

λ2[18]

H2O 1:4580 × 10−30 þ 1:3049×10−44λ2 −

1:5992×10−58λ4 [19]

CO2 ð5:9187 × 10−26ð2:8918 × 1024λ2 − 2:6127 × 1038λ4 þ 7:1378 × 1051λ6−8:5687 × 1062λ8 þ 2:5751 × 1073λ10 − 8:5970 × 109ÞÞðð5:8474 × 1010λ2 − 1Þ2

ð2:1092 × 1014λ2 − 1Þ2ð6:0123 × 1013λ2 − 1Þ2Þ

[20]

CH43:3109×10−161:3006×1014− 1

λ2þ 9:2607×10−6�

1:3006×1014− 1λ2

�2 [21]

a Not all of these equations are given explicitly in the references, some have required a least squares fit to data points.

Table 2. Minimum Number Density to Elicit InterferometricDetection at λ ¼ 555 nm, τ ¼ 100, and L ¼ 0:5m

Gas N × 1018, (m−3) NNSTP

× 10−8

Ar 1.0221 3.8H2 2.1444 8.0He 8.5422 32N2 0.99822 3.7Ne 4.4454 16O2 1.0999 4.1H2O 1.1788 4.4CO2 0.66286 2.5CH4 0.67650 2.5


every other component, this convention is ob-served here.The technique used in this study for determining

the smallest detectable quantity of a particular spe-cies in a mixture is carried out as follows. Equa-tion (2) permits a linear system of equations to bewritten that describe multiple species in a mixture,which for laboratory temperatures is expressible as[12]

nðλÞ≃ 1þ 2πPNA

RT

Xj

xjαðλÞjZj

; ð4Þ

where P is the pressure, NA is Avogadro’s number, Ris the gas constant, T is the temperature, and xj, Zj,and αj are the mole fraction, compressibility, and dy-namic polarizability of the jth constituent, respec-tively. Note that the multicomponent form of theLorentz–Lorenz equation [Eq. (4)] has the depen-dence of the number density on pressure and tem-perature explicitly given. Using Eq. (4) at a fixedtemperature, one can see that when varying the pres-sure in the cell (of the gas in Fig. 1) and measuringthe total fringe shift over this pressure range, for avariety of wavelengths, a linear system of equationsis created with the mole fractions, xi, as the un-knowns. Therefore, to obtain a unique solution to thislinear system, the number of different wavelengthsused in the measurement process must be equal orgreater than the number of different gases withinthe gas mixture.Once this linear system is established, the values

found for xi will be compared to known values, andthemean squared difference between the establishedvalues and the experimentally-measured values willprovide an indication of the minimum amount ofeach species that is required to be present in the mix-ture for it to be accurately determined. In otherwords, this would be the smallest quantity detectableby a refractometric gas detection system for a mix-ture of gases.In this study, the gases considered are in concen-

trations of ∼1% or greater. Trace gases are too smallin quantity to elicit detection by this apparatus,which is discussed in Section 6.

6. Minimum Detectable Concentrations of MultipleGases

Expressing Eq. (4) in matrix form yields

2πNA

RTA:x ¼ B; ð5Þ

where the column vector x represents the xi inEq. (4), and the matrix A is the dynamic polarizabil-ity, whereAij is the polarizability of the jth species atthe ith wavelength. The column vector B representsthe ratio of n to P [again from Eq. (4)] for the ithwavelength. When solving for x, Eq. (5) is particu-larly vulnerable to small perturbations of B, a mea-sure of the degree of this sensitivity is the well-known condition number, Cn [22,24], given by

Cn ≡σmax

σmin; ð6Þ

where σmax and σmin represent the largest and smal-lest eigenvalues of AT:A, respectively. For Eq. (5), Cnis of the order of 105. This implies that the relativeuncertainties in B magnify the uncertainty in x bya factor of ∼105. If constraints are not placed onthe permissible values of xi, unrealistic solutions tothe system will be obtained. The constraints thatcan logically be applied to x are the following:

1. The sum of the elements of x must equal1. That is, the total gas quantity must equal 100%.

2. No individual element of xmay be greater than1 or less than 0. That is, the quantity of a single gasin the mixture cannot exceed 100% or be lessthan 0%.

Finally, the ratio of n to P from experimental dataand the individual elements of A are required to com-plete the linear system. The elements of A can befound from Table 1 at the required wavelengths,while the process for obtaining ratios of n to P areoutlined in Subsection 6.A.

A. Calculations of nP

The performance of the method will be evaluated bydetermining the composition of standard air with aknown temperature, pressure, and humidity. Sincethe actual composition is known, the accuracy ofthe results can be determined.

Initially in the experimental arrangement, air ispartially evacuated (usually to ∼0:5 atm) from theoptical cell (the cell length used here was 200mm)shown in Fig. 1, and a valve attached to the cell isopened slightly, allowing air to slowly refill the cell.As the air enters the cell, the output of the inter-ferometer cycles through the peaks and troughs ofconstructive and destructive interference as the op-tical path length within the cell changes.

The interferometer output is monitored while it cy-cles through the peaks and troughs until the pres-sure has reached equilibrium and the total numberof fringe shifts,m, is measured. The change in refrac-tive index over this pressure range is given by

n ¼ mλL

: ð7Þ

Using Eq. (7), it is relatively straightforward to cal-culate n=P for this wavelength. As mentioned, thehigh condition number of Eq. (5) demands high pre-cision in both the measurement of P and the mea-surement of m. The magnitude of m is unlikely toconveniently be a whole or half integer of fringes; be-cause of this, the fractional fringe shift, the fractionof a fringe between half integers, must also be de-termined.

If we adopt the standard definition that the phasecorresponding to a shift of output from peak to trough


Fig. 2. Output of the interferometer at wavelengths (a) 532, (b) 543.5, and (c) 594nm. The horizontal axis represents time.


Fig. 3. Output of the interferometer at wavelengths (d) 612, (e) 632.8, and (f) 780nm. The horizontal axis represents time.


corresponds to a fringe shift of 1=2, then the followingformula can be used to determine shifts of fringe be-tween a peak and trough (or vice versa):

δm ¼ 12π arccos

�Vpp − 2ξ

Vpp

�; ð8Þ

where Vpp is the peak to peak magnitude of the pre-vious fringe (the symbol V was chosen as our outputis measured as a voltage) and ξ is the voltage differ-ence between the peak (or trough) and the finalfringe position. Clearly ξ can take values between0 and Vpp. Figures 2 and 3 show typical data gath-ered by this process at the six wavelengths used.Bear in mind, the figures are images captured fromthe output of an oscilloscope, hence the horizontalaxis represents time, though this is irrelevant andunhelpful as this would represent only the time ta-ken for the pressure to change from its initial valueup to zero. The pressure range is the only variable ofimportance. Variance in fringe frequency among thefigures is a result of flow rate change.In each of the figures, the pressure range over

which the data was gathered is given, as is the num-ber of fringes detected at that wavelength. Further-more, we define ΔP as the pressure required toinduce one integer shift in path length. P, the pres-sure range, can be found from Figs. 2 and 3 at theirrespective wavelengths. Using the expression

�nP

�i¼ λ

LΔP; ð9Þ

one can find the ratio of n to P for the ith wavelength.These values are plotted in Fig. 4. The results are gi-ven in Table 3 below. A least-squares fit to the datapresented in Fig. 4 is given by the function

nPðλÞ∼ 2:9405 − 7:6371 × 10−4λþ 4:8819 × 10−7λ2:

ð10Þ

This least squares approximation to the data are va-lid only for the wavelength range shown in Fig. 4.

Multiplying the values of columns two and three ofTable 3 by ðRTÞ=ð2πNAÞ, substituting the results forB in Eq. (5), and employing a constrained, boundedoptimization approach results in the quantities foreach of the three most abundant atmospheric gasesas shown in Table 4.

The results in Table 4 are only from a single mea-surement, however it is possible to obtain a picture ofwhat multiple measurements would produce by al-lowing the values of n=P to randomly vary throughtheir uncertainty, running the optimization proce-dure multiple times, and averaging the results. Therelative uncertainty in n=P is ∼0:05%, running theoptimization procedure 1000 times, produces the re-sults given in Table 5.

Comparing the results in Tables 4 and 5 to theseactual values shows that, for Table 4, the values arewithin approximately 1% of each constituents corre-sponding actual value. Comparison with Table 5shows that the constituents are within 2% of theircorresponding actual values. These values of uncer-tainty are obtained simply by the average of the sumof the difference between each constituent and itscorresponding actual value.

However, the uncertainty for the system as a wholeis determined by the fact that this is the solution of aleast-squares linear system. In keeping with the con-vention of the determination of uncertainties forsuch a system, the average of the sum of the rootmean square difference between the values incolumn four of Table 4, and columns one and twoof Table 4 and column two of Table 5 determinethe uncertainty of the system as a whole.

Fig. 4. Ratio of n to P as a function of wavelength with quadratictrendline.

Table 3. Ratio of Refractive Index to Pressure at Various Wavelengths

λ(nm)

nP ðPa−1Þ × 109,

experimental valuesLeast Squares Approximation to

nP ðPa−1Þ × 109

532 2.6711 2.6724543.5 2.6703 2.6696594 2.6600 2.6591612 2.6591 2.6560632.8 2.6490 2.6527780 2.6421 2.6418

Table 4. Quantities of Ar, O2, and N2 Detected via Refractometrya

Gas

Quantities UsingExperimentalValues of n

P

Quantities UsingEq. (10) for n

P

ActualConcentration(Adjusted for

H2O)

Ar 0 0 0.092N2 7.770 7.769 7.660O2 2.230 2.231 2.060

a All units are in parts per million ×105.


The calculation of uncertainty for Table 4 is as fol-lows. The difference between columns four and two ofTable 4 is

ΔAr ¼ 0:092; ΔN2 ¼ −0:11; ΔO2 ¼ −0:17:

ð11ÞAnd the average of the root mean square sum of thissystem is

13

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΔ2

Ar þΔN22 þΔO2

2

q¼ 2:22%∼ 2%: ð12Þ

Application of this method to Table 5 produces avalue of ∼3%.

7. Conclusion

Under the experimental limitations assumed here ofL ¼ 0:5m and τ ¼ 100, it was shown that the ap-proach could detect changes in single gases to a sen-sitivity of 10ppb. Consequently this interferometricgas analysis system could potentially be used as anextremely accurate manometer. The accuracy of∼3%for multiple gas mixtures can also be useful in someapplications, such as an anaesthetic gas detectionsystem [4] and a natural gas detection system foroil exploration [3].The difference in uncertainty between a single

measurement (Table 4) and series of simulated mea-surements (Table 5) has produced a situation wherethe experimentally-derived measurement is moreaccurate than the series of simulated multiple mea-surements. This is most likely due to the measure-ment of ΔP, the pressure range, in Eq. (9). Thoughthe uncertainty in pressure measurements is fixed,it is probable that in this single measurement thevalues measured were closer to the actual valuesthan usual.Improvements in the resolution of the interferom-

eter as a multiple gas analyzer can be made by over-coming impediments like the high condition numberof matrix A in Eq. (5). The high condition number is aresult of the slow rate of a change of the dynamic po-larizabilities as a function of wavelength over therange of wavelengths used. Therefore, if wavelengthsthat resulted in a greater difference in polarizabilitywere used, a much lower condition number wouldresult.It would be relatively straightforward to expand

the wavelength range, into the short infrared andlong ultraviolet. If such a procedure is able to reducethe condition number by even one order of magni-tude, the interferometer as a multiple gas analyzer

could enjoy a vast expansion of its applicability,due to its greater precision. To illustrate this, consid-er the following equation [23] describing the relativeuncertainty in x from Eq. (5) [24]:

∥δx∥∥x∥

≤ Cn

�∥δB∥∥B∥

�: ð13Þ

This bound on uncertainty shows that if the condi-tion number is reduced, then the upper bound on un-certainty in x is proportionately reduced. To reducethe condition number of matrix A [Eq. (5)], wave-lengths should be chosen that are close to com-monly-known regions of anomalous dispersion forthe gases of interest. In these regions, the refractiveindex rapidly decreases as wavelength decreases, of-fering a large variance in refractive index over a veryshort spectral range. This does imply, however, that aspectroscopic element has been introduced into theprocess to improve the accuracy of the approach asa gas analyzer.

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Table 5. Average Quantities of Ar, O2, and N2 over 1000Repetitionsa

Gas Average Quantity Standard deviation

Ar 0.205 0.401N2 7.863 0.322O2 1.932 0.302

aAll units are in parts per million ×105.


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