optimizing modulation transfer spectroscopy signals for frequency locking in the presence of...

Optimizing modulation transfer spectroscopy signalsfor frequency locking in the presence

of depleted saturating fields

David J. Hopper and Esa Jaatinen*Applied Optics Program, School of Physical and Chemical Sciences, Queensland University of Technology,

PO Box 2434, Brisbane, Queensland 4001, Australia

*Corresponding author: [email protected]

Received 28 January 2008; revised 10 April 2008; accepted 11 April 2008;posted 11 April 2008 (Doc. ID 92190); published 2 May 2008

A theoretical model of modulation transfer spectroscopy (MTS) that includes pump beam depletion is pre-sented and experimentally verified with data covering visible iodine transitions at 532, 543, and 612nm.This model is used to determine the values for pressure, interaction length, and saturation intensity thatyieldmaximumMTS signals for frequency locking to iodine transitions. The approach is demonstrated foriodine transitions at 532, 633, and 778nm, with the results showing that theoretically the frequency in-stability scales inversely to the absorption coefficient. © 2008 Optical Society of America

OCIS codes: 140.0140, 140.3425, 120.0120, 120.3930, 120.3940, 300.6190.

1. Introduction

Optical frequency references are laser systems thatemit lightat accurately knownwavelengths in the visi-ble part of the electromagnetic spectrum. These refer-ences areessential tools in fundamental investigationssuch as the creation of Bose–Einstein Condensates [1]and the high resolution spectroscopic probing of suba-tomic and atomic interactions [2]. They also haveimportant practical uses in the modern industrialworld. Most nations employ these standards to realizethecountry’smostaccuratemeasurementof lengthandtime [3] with optical frequency references routinelyused for accurate wavelength and dimensional mea-surements [3–9] and in optical clocks [10,11].The saturated absorption resonances of molecular

iodine (127I2) are the frequency references of choicesince there are more than 20,000 hyperfine transi-tions covering wavelengths from 500 to 780nm.Many of these transitions have been thoroughly in-vestigated, allowing their frequencies to be accu-rately determined with uncertainties at the kHz

level. Therefore, with a properly designed experimen-tal arrangement using a technique such asmodulationtransfer spectroscopy (MTS) [12–19], this level of un-certainty can be transferred to an optical field by lock-ing (stabilizing) the frequency of a laser to thesereferences. However, the absorption strength of iodinevaries quite markedly over the visible range of thespectrumwithsome532nmtransitionshavingabsorp-tion coefficients on the order of 0:3m−1 Pa−1 [17] whileat 778nm the absorption drops to 0:0000045m−1 Pa−1[20]. Therefore, it is unlikely that an optical frequencyreference designed at onewavelengthwill operatewellat another without adjustment of the many systemparameters.

MTS is a saturated absorption technique well sui-ted for frequency locking of optical references since itprovides large signals that are free of baseline offsets[12,17,21]. The frequency stability of lasers lockedusing MTS or other saturated absorption spectro-scopy techniques depends on the linestrength andon the linewidth of the hyperfine transition. In gen-eral, frequency stability is improved when the signalis increased and the width of the line reduced [21].Therefore, for frequency locking it is advantageousto optimize the magnitude of the MTS signal.

0003-6935/08/142574-09$15.00/0© 2008 Optical Society of America

2574 APPLIED OPTICS / Vol. 47, No. 14 / 10 May 2008

Theoretical modeling of the MTS process has re-vealed that the signal magnitude depends on severalinterrelated controllable system parameters, such asthe pressure of the gaseous medium, interactionlength, modulation settings, and saturation inten-sity. Other fixed parameters also influence the signaland these include the absorption coefficient, the sa-turation parameter, and the transition linewidth.Consequently, the exact blend of system parametersrequired to optimize the signal in a given situation isdifficult to estimate and often requires much experi-mental trial and error. While theoretical and experi-mental research has been performed to identify themodulation parameters required to optimize theMTS signal [16,21,22], no systematic approach existsfor determining optimal values for the remainingsystem parameters.We present a new theoretical model that accu-

rately determines the influence that the systemparameters have on the magnitude of the MTS sig-nal. This is the first MTS model, to our knowledge,that incorporates the non-negligible effects of satur-ating beam energy depletion, allowing for a more ac-curate determination of the conditions necessary forobtaining maximum signal magnitudes. Themodel isverified with experimental data covering a number ofdifferent iodine transitions. A methodology is alsopresented to guide the choice of system parameterssuch as iodine pressure, interaction length, andsaturation intensity to enable the design of anMTS arrangement that delivers optimum signalsfor frequency locking to iodine.

2. Low Pressure Regime—No Depletion of theSaturating Field

Saturation of the absorption of a medium is a well-known phenomenon that arises when the numberof photons in an interacting light field greatly ex-ceeds the number of absorbers. A simple rate equa-tion approach shows that the absorption coefficientexperienced by a field with an incident intensity ofI in an inhomogenously broadened medium is givenby Eq. (1a) [23]. The equivalent equation describinghomogenously broadened media is given in Eq. (1b):

αINHOMðIÞ ¼αp�

1þ IIsat

�1=2

; ð1aÞ

αHOMðIÞ ¼αp

1þ IIsat

: ð1bÞ

Here α is the absorption coefficient at low intensity, pis the pressure of the gaseous medium, and Isat is thesaturation parameter that is determined by the prop-erties of the medium [24]. In a typical saturatedabsorption spectroscopy arrangement, such as mod-ulation transfer spectroscopy (MTS) [16–19], the ab-sorption of the medium is saturated by an intensesaturating field (I ¼ Ipump), with the resulting

change observed by measuring the transmission ofa much less intense counter-propagating probe field.In the event that the probe and saturating fields areresonant with a hyperfine component of the medium,the resulting MTS signal can be written as

Sig ¼ KIprobeðexp½−αINHOMd� − exp½−αpd�Þ: ð2Þ

Here d is the length of the interaction and K is a con-stant dependent on the properties of the detector. Ty-pically, it is assumed that the level of saturation isweak (Ipump ≪ Isat) and that the overall absorptionis low (αpd ≪ 1). Under these two assumptions theMTS signal obtained is given by the low pressure–weak saturation approximation that is commonlyused [12,13,25]:

Sig ≈12KIprobe

Ipump

Isatαpd; ð3Þ

where Isat ¼ Is0ð1þ cpÞ2 is the saturation parameterof the transition [16,17,24,26] with c, the linear pres-sure broadening coefficient [24].

Thus the signal should increase when the level ofsaturation, the interaction length, or the pressure in-creases. However, this is not always observed in prac-tice. For example, the Rð56Þ iodine transition near532nm has an absorption coefficient of 0:28m−1

Pa−1 [17]. In many realizations of optical frequencyreferences at this wavelength that employ MTS, in-teraction lengths in the range of 100 to 1000mm aretypically used [4–7,17]. Equation (3) suggests thatchoosing the largest interaction length and the high-est possible saturation intensity should provide thelargest signal. However, this will not occur if the de-pletion of the probe and saturating fields by the ab-sorbing medium itself is so great that there is verylittle overlap of their energies within the volume,as depicted in Fig. 1. Therefore, a balance is requiredbetween the intensities of the two optical fields, theabsorption of the medium, and the length of the in-teraction to produce the largest possible signal.

3. Volume Interaction Model

To find the optimum pressure, beam intensities, andinteraction length, the theoretical description of theMTS process must include the effect of pump deple-tion. In MTS the signal accumulates as the probefield propagates through the interaction volume. Si-multaneously, this volume is saturated by the pumpfield with an intensity that is dependent on the dis-tance it has travelled within the medium and the ab-sorption of the medium. In this event the change inthe probe beam intensity as a function of propagationdistance, z, for inhomogenously and homogenouslybroadened media can be written as Eqs. (4a) and(4b), respectively:

10 May 2008 / Vol. 47, No. 14 / APPLIED OPTICS 2575

dIprobedz

¼ −αðf Þp�1þ IpumpðzÞ

IsatLðf Þ

�1=2

Iprobe; ð4aÞ

dIprobedz

¼ −αðf Þp1þ IpumpðzÞ

IsatLðf Þ

Iprobe; ð4bÞ

where IpumpðzÞ is the intensity of the saturating fieldas a function of z. Since the pump and probe counterpropagate, the pump intensity is given by

Ipumpðz; f Þ ¼ I0 exp½αðf Þpðz − dÞ�: ð5Þ

Theabsorptioncoefficientα foraniodinetransition isfrequencydependentandresults fromthe combinationofN Doppler broadened hyperfine transitions that areassumed to have equal strength. Using the usual con-vention α can be expressed as [23]

αðf Þ ¼ α0XNi¼1

exp�−ðf − f iÞ2

σ2�: ð6Þ

Here f isthelaserfrequency, f i isthecenterfrequencyofthe ithhyperfinecomponent,andσ istheDopplerwidthof thecomponent.Saturation isonlyexperiencedbytheprobe when the frequency of the laser is resonant withone of the hyperfine components. The resonance condi-tion is included in Eqs. (4a) and (4b) through the func-tion Lðf Þ, which is the sum of the LorentziandistributionfunctionsthatdescribestheNDopplerfreeresonancesof thetransition.This functionensuresthataway from resonance (Lðf Þ∼ 0) the probe encountersnegligible saturation. The function Lðf Þ can be ex-pressed as [23]

Lðf Þ ¼XNi¼1

γ2γ2 þ ðf − f iÞ2

: ð7Þ

Here γ is the half-width at half-maximum (HWHM) ofthe natural Doppler free hyperfine components.Equations (4a) and (4b) can be integrated, yielding

the transmission of the probe field in the presence ofa saturating field that is depleted due to absorption:

Trans ¼ IprobeðdÞIprobeð0Þ

¼ exp½−αðf Þpd�

×

0BBB@

1þ�1þ I0

IsatLðf Þ

�1=2

1þ�1þ I0 exp½−αðf Þpd�

IsatLðf Þ

�1=2

1CCCA

2

; ð8aÞ

TransHOM ¼ IprobeðdÞIprobeð0Þ

¼ exp½�αðf Þpd� ×1þ I0

IsatLðf Þ

1þ I0 exp½�αðf Þpd�Isat

Lðf Þ:

ð8bÞ

In most practical MTS realizations, the change inprobe beam absorption due to saturation is small.Consequently, it is usually necessary to modulatethe saturating field and use phase sensitive photo-detection to reveal the MTS signal. In the presenceof a modulated saturation field the probe beam ex-periences a periodically varying level of absorption.For amplitude modulated (AM) pump fields the sa-turation is periodically turned on or off, while for fre-quencymodulated (FM) pump fields the saturation isperiodically shifted away from resonance. After demo-dulationat themodulation frequencyω, theMTSsignalgeneratedhasamagnitudethatwilldependonthetypeof modulation used. The signal magnitude for bothmodulation forms for inhomogeneously broadenedmedia can be obtained from Eq. (8a):

SignalAM ¼ exp½−αðf Þpd� × ½Hðf Þ � 1�;SignalFM ¼ exp½−αðf − ωÞpd� ×Hðf − ωÞ

− exp½−αðf þ ωÞpd� ×Hðf þ ωÞ;

where

Fig. 1. Depletion of probe and pump fields as they counterpropagate through the absorbing medium.


Hðf Þ ¼

0BB@

1þ�1þ I0

IsatLðf Þ

�1=2

1þ�1þ I0 exp½−αðf Þpd�

IsatLðf Þ

�1=2

1CCA

2

: ð9Þ

It isrelativelystraight forwardtoshowthat intheweaksaturating field (I0=Isat ≪ 1) and low absorption re-gimes (αpd ≪ 1), Eq. (9) converges to Eq. (3).

4. Influence of Pressure

The pressure of the absorbing medium has a signifi-cant effect on determining the signal magnitude. Asthe pressure increases from zero the MTS signal in-creases until a local maximum is reached. Beyondthis optimum pressure, the signal starts to decreasedue to beam energy depletion and the increase in thepressure dependent saturation parameter Isat.A comparison between the signal magnitude pre-

dicted by the volume interaction model [Eq. (9)] andthe low pressure–weak saturation model [Eq. (3)] forarangeofdifferent iodine transitions is showninFig.2.Previously reported values for the system parameterswere used and are listed in Table 1, apart from thesaturating field intensity that was chosen to be1mW=mm2 for allwavelengths. At lowpressures, bothmodels predict the same linear relationship betweensignal magnitude and pressure. However, as the pres-sure increases, the level of beam depletion will also in-crease. This results in an overestimate of the signalstrength by the low pressure–weak saturation model.A comparison of Eq. (2) with Eq. (9) shows that,

while the volume interaction model is more rigorousand includes pump depletion, it is more complexmathematically than the low pressure–weak satura-tion model. In some situations, such as when a Four-ier decomposition of the MTS process is involved[12–15,25], it may become difficult to implementthe rigorous volume interaction model. A good ap-proximation to the volume interaction model at mod-erate pressures can be obtained by substituting theaverage saturating beam intensity within the med-ium, given by Eq. (10), into the low pressure–weaksaturation model [Eqs. (2) and (3)] instead of the in-cident saturating beam intensity:

Ipump ¼ I0 exp�−αpd2

�: ð10Þ

With this substitution the signal becomes

SigHybrid ¼ exp

26664

−αpd�1þ I0 expð−αpd=2Þ

Is0ð1þcpÞ2

�1=2

3775 − exp½−αpd�:

ð11ÞThis “hybrid” approach retains the simplicity of the

low pressure–weak saturation model but includespump depletion effects. The signal magnitudes pre-

dicted by this approximatemethodare shown inFig. 2,and it is evident that they agree well with the volumeinteraction model values at moderate pressures. Athigher pressures the two approaches diverge becausethe pump intensity can no longer be approximatedas a linear function of distance (i.e., exp½−αpd� ≠ 1−αpd).Figure3shows the ratioof thesignalmagnitudepredicted by the volume interaction model and the hy-brid approach for the 532nm transition for various sa-turating beam intensities. This figure shows theagreement between the two approaches diminishesat high pressures.

Fig. 2. MTS signal magnitude predicted by the various theoreti-cal models as a function of cell pressure for iodine transitions at532nm, 543nm, and 612nm. Note that the low pressure – weaksaturation data has been divided by 10 for the 532nm and 543nmgraphs and by two for the 612nm graph to allow it to be viewed onthis scale.

Table 1. Parameters Used in Volume Interaction Model Calculations

532nm [17] 543nm [26] 612nm [24, 29]

Hyperfine feature a[10] - R(56) b[10] - R(106) b[15] – P(48)α (m−1Pa−1) 0.28 0.33 0.11c (Pa−1) 0.15 0.1 0.214Is0 (mW=mm2) 0.1 0.174 0.425


As shown in Fig. 2, a feature of all the theoreticalmodels discussed is a pronounced signal maximumas a function of pressure. The pressure, pmax, atwhich this local maximum occurs depends on allother system parameters and can be determinedby differentiation of Eq. (9) with respect to pressure.This results in a complex algebraic function that can-not be rearranged to express pmax explicitly in termsof the other parameters. However, it is possible to ob-tain an approximate analytical expression with thehybrid approach [Eq. (11)] under some simplifyingapproximations. If the overall absorption is low(αpd < 1), Eq. (11) is approximately given by

SigHybrid ≈ 1 −αpd�

1þ I0 expð−αpd=2ÞIs0ð1þcpÞ2

�1=2

− 1þ αpd: ð12Þ

Taking the derivative of Eq. (12) with respect to pres-sure when the saturation is weak (I0 ≪ Isat) andequating to zero gives

pmax ¼2

�αd

�1þ I0

Is0

�� ðαdþ 2cÞ

�1þ I0

Is0

�1=2

�

ðαdÞ2 I0Is0

− 4αdc − 4c2:

ð13Þ

Since the agreement between the volume interac-tion model and the hybrid approach is good except athigh pressures, it is anticipated that Eq. (13) shouldyield a useful approximation to the pressure requiredto deliver a maximum signal for a given cell lengthand saturating beam intensity.Using a typicalMTS arrangement [19], signalmag-

nitudesweremeasuredasa functionofvaporpressurefor several iodine transitions. Here, the saturatingbeam was modulated using either, an acousto-optic modulator (AOM) for amplitude or frequencymodulation, or an electro-optic modulator (EOM) for

phase modulation (PM). The MTS signal magnitudewas obtained by measuring the probe transmissionwith a photo-detector and a phase sensitive lock-inamplifier. The vapor pressure of the solid iodine with-in the cell was controlled by either heating or coolingthe cell finger.

ThemeasuredMTS signal magnitude as a functionof pressure is shown in Fig. 4 for iodine transitions at532, 543, and 612nm. Also shown is the signal pre-dicted by the volume interaction model with the re-levant system parameters given in Table 1. Thecrosses in Fig. 4 show the pressures (pmax) predictedby Eq. (13) at which the signal is a maximum. Whilethe agreement is not perfect, it is clear that Eq. (13)approximates reasonably well the pressure requiredto obtain a maximum MTS signal.

5. Influence of Cell Length

The distance the saturating beam travels throughthe absorbing medium determines the level of energydepletion it experiences. Therefore, the pressure atwhich the maximum signal is obtained and the mag-nitude of the signal depends on the cell length. As anexample, the MTS signal magnitudes measured withtwo different cell lengths (0:1mand 0:2m) along withthe signals predicted by the volume interaction mod-el are shown in Fig. 5.

As shown in Fig. 5, with all other parameters keptconstant, the shorter cell requires a higher pressurethan the longer cell to produce a local signal maxi-mum that is smaller than that obtained with thelonger cell. Note that doubling the cell length doesnot double the signal size at pressures close to theoptimum value (of about 5Pa) or above. However,at lower pressures (p < 5Pa) the signal strength isapproximately proportional to cell length in accordwith the low pressure–weak saturation model[Eq. (3)]. In general, the signal strength is only pro-portional to cell length for pressures less than pmax—

the pressure required for the local maximum signal.As the cell length is increased, the pressure must

be lowered in order to reduce the pump depletion andobtain a maximum signal. This is depicted in Fig. 6,

Fig. 3. Ratio of theMTS signal predicted by the volume interactionmodel to theMTSsignalpredictedbythehybridmodelasa functionofcell pressure for three different saturating beam intensities.

Fig. 4. Measured and theoretically calculated MTS signal mag-nitude with AM and FM saturating fields as a function of pressurefor various iodine transitions.


which shows the signal predicted by the volume in-teraction model [Eq. (9)] for the 532nm transition asa function of pressure. Here iodine cell lengths of0.05, 0.10, 0.20, 0.50, and 1:00m have been used,as these lengths are similar to those previously usedin MTS systems [4–7,17,27,28]. Also included is thesignal produced when a 2:0m long cell is used. Fromthis figure it is apparent that the maximum signalplateaus as the cell length is increased. For example,doubling the cell length from 0:5m to 1:0m increasesthe maximum signal by less than 15%. Figure 7shows the maximum signal obtained as a functionof cell length when the cell is operated at pmax foreach length for three different iodine transitions.Here the maximum 532nm signal approaches a lim-iting value, and cell lengths that are 1m or longer arewithin 10% of that value. At the two longer wave-lengths the iodine absorption is much weaker and,as a result, the signal increases almost proportion-ally to cell length over the range considered here.Since large cells can be difficult to implement in prac-tice, in some situations it may be more convenient tochoose a shorter cell operated at a higher pressure andsufferthepenaltyof slightly lesssignal.However, coun-tering this argument is that lower pressures are gener-ally preferable as pressure broadening of the linewidthis reduced [29].

6. Methodology for Optimization of MTS Signals

For frequency locking applications the important fig-ure of merit is the frequency stability. An estimate ofthis is obtained by dividing the signal magnitude bythe hyperfine linewidth with the result acting as atheoretical signal to noise ratio (TSNR) [7]. It is wellknown that the linewidth becomes broader at highpressures and intensities [17,26] with the functionaldependencies described by Eq. (14):

~γ ¼ ðγ þ kppÞ�1þ Ipump

Is0ð1þ cpÞ2�

1=2: ð14Þ

Here γ is the natural linewidth; kp is the pressurebroadening coefficient of the transition [17,26–30];and the last bracketed term is the power broadeningfactor.

In Section 6, a methodology will be outlined toevaluate the system parameters that are necessaryto obtain the absolute maximum MTS signal magni-tude to achieve the lowest frequency instability pos-sible when the laser is locked. The operation of thismethodology will be demonstrated by determiningthe cell length, iodine pressure, and saturation inten-sity necessary for maximum MTS signal magnitudesfor iodine transitions at 532, 633, and 778nm. Thesethree wavelengths are chosen since 532nm and633nm are important optical frequency referencesfor metrology applications, while 778nm is impor-tant for the optical communications industry. Also,the applicability of the approach is suitably demon-strated since the iodine absorption is significantlydifferent at the three wavelengths.

From the prior discussion and from Eq. (9), it is ap-parent that the overall signal magnitude depends ona complicated blend of controllable and fixed para-meters. Once the particular iodine hyperfine compo-nent that is to be investigated has been selected, theunsaturated absorption coefficient α, the hyperfinelinewidth γ, and the saturation parameters Is0 andc are fixed known constants. The MTS signalstrength then depends on the operator adjustable sa-turation intensity Ipump, cell length d, iodine vapor

Fig. 5. Experimentally observed and theoretically predicted de-pendence of the 532nmMTS signal magnitude on pressure for celllengths of 0:10m and 0:20m.

Fig. 6. 532nm MTS signal magnitude dependence on cell pres-sure for various cell lengths.

Fig. 7. MTS signal magnitude dependence on cell length at532nm, 633nm, and 778nm, when optimum pressure, pmax, isused for each length. Note that the 778nm data has been magni-fied by a factor of 100 in order for it to be observed on this scale.


pressure p, and the modulation/demodulation para-meters. Optimization of modulation/demodulationparameters for the MTS process has been previouslyinvestigated [21,22] and those findings have beenused here. Thus, to deliver maximum MTS signalsfor the chosen transition only the saturation inten-sity, cell length, and pressure need to be determined.First, it is useful to consider the impact that the

saturation intensity and power broadening have onthe TSNR. From Eqs. (3) and (12) it is apparent thatwhen the saturation intensity is increased from lowvalues (Ipump ≪ Isat) that the MTS signal will in-crease with negligible increase in the linewidth[Eq. (14)]. Therefore, at low saturating intensitiesthe TSNR is approximately proportional to the sa-turation intensity. At high saturating field intensi-ties (Ipump ≫ Isat), increasing the saturatingintensity marginally improves the MTS signal mag-nitude but causes significant power broadening ofthe linewidth. As a result the TSNR decreases uponincreasing the saturating intensity at high values.An approximate expression for the TSNR can befound in the low (Ipump ≪ Isat) and high (Ipump ≫

Isat) saturating beam intensity regimes by dividingEq. (12) with Eq. (14):

TSNR∼Sig~γ ≈

αpdðγ þ kppÞ

�1þ Ipump

Isat

�1=2

� 1�1þ Ipump

Isat

� ;

TSNRðIpump ≪ IsatÞ ≈α:p:d

ðγ þ kppÞIpump

2Isat;

TSNRðIpump ≫ IsatÞ ≈αpd

ðγ þ kppÞ�

IsatIpump

�1=2

:

ð15Þ

Therefore, some value for Ipump will exist that max-imizes the TSNR. In the following, saturating beamintensities up to the value corresponding to theTSNR maximum will be considered.With the saturation intensity range established,

the optimum pressure, pmax, is evaluated usingEq. (13) for a range of cell lengths that are typicallyused. These pressure values and their correspondingsaturating beam intensities are then substituted intoEq. (9) to give the local maximum MTS signal that isthen divided by the linewidth [Eq. (14)] to yield theTSNR. These TSNR values are shown in Fig. 8 as afunction of saturating beam intensity for the threewavelengths.From the data for the 532nm transition shown in

Fig. 8, it is apparent that the TSNR is larger for thelonger cell lengths. However, doubling the cell lengthdoes not yield a proportional increase in TSNR. Forexample, only a 10% increase in the TSNR is ob-tained if the cell length is doubled from 1:0m to2:0m. The saturating beam intensity that corre-sponds to the peak TSNR decreases as the cell lengthincreases and, for the lengths considered here, is ap-proximately in the 2–10mW=mm2 range. The pres-

sure values that correspond to these lengths andsaturating intensities, as obtained from Eq. (13), fallin the range 1–30Pa with corresponding linewidthsof 1–3MHz. Therefore, for the 532nm a½10� compo-nent of the Rð56Þ iodine transition, the volume inter-action model suggests that using long cells willimprove the frequency stability. However, as onlymodest increases (i.e. 10%) are achieved in doublinglarge (1:0m) cell lengths, little is lost if a cell lengththat is easier to implement (∼0:5m) is used andoperated at its optimum pressure (∼5Pa) and satur-ating beam intensity (∼3mW=mm2). These operat-ing values compare well with those used for532nm optical frequency references that have beenreported in the literature [4–7,17].

Fig. 8. Dependence of theoretical signal to noise ratio on satur-ating beam intensity for iodine transitions at 532nm, 633nm, and778nm over a range of cell lengths.


At longer wavelengths the iodine absorption is low-er and, consequently, a higher vapor pressure andsaturating beam intensity is required to obtain max-imum TSNR. Figure 8 shows the TSNR calculatedusing the volume interaction model for the 633nma½1� component of the Pð33Þ iodine transition for arange of cell lengths. Similar trends to that observedat532nmarepresentat633nmwithboththeoptimumpressure and saturating beam intensity decreasing asthe cell length increases. For the lengths consideredhere, the peakTSNRs occur for saturatingbeam inten-sities in the range 20–750mW=mm2 with correspond-ing pressures of 30–240Pa leading to linewidths in therange3–20MHz.However,ascanbeseeninFig.8,dou-bling the cell length leads to a proportional increase inTSNR and, therefore, a proportional improvement infrequency stability. This is also indicated in Fig. 7,where the rate of change in the signal with respect tothe cell length is linear for lengths less than 5m. Im-practical cell lengths (∼100m) are required for the633nm signal to approach its absolute maximum va-lue. Thus for 633nmMTS optical frequency referenceslocked to iodine, the largest cell that is practically pos-sible shouldbeused tooptimize the frequency stability.At 778nm, the absorption coefficient is about 100

times smaller than that of the P33 transition at633nm and about 10,000 times smaller than thatof 532nm. As a result much higher pressures and sa-turating beam intensities are required to maximizethe TSNR as shown in Fig. 8. For the cell range usedhere, the a [8] component of the Pð36Þ transition re-quires saturating beam intensities in the range700−20; 000mW=mm2 and corresponding pressuresof 500–3000Pa leading to linewidths of 50–280MHz,to yield maximum TSNRs. As was found for the633nm transition, increasing cell length produces aproportional increase in TSNR at 778nm, so for opti-mum performance as large a cell as is practical shouldbeused. It is interesting tonote fromFig. 8 that the lar-gest rate of change in TSNR occurs in the first20mW=mm2 of the saturating beam intensity range.This is important as some 778nm laser sources usedmay not be able to produce the high saturating beamintensities required to maximize the TSNR for thisweak transition. In thatevent itworthnoting thathighTSNR (within 20% of maximum) can still be obtainedwith much lower saturating intensities.Comparing data for the different wavelengths in

Fig. 8, it is evident that the maximum TSNR for aparticular cell length scales according to absorptioncoefficient. However, the scaling is not strictly lineardue to the widely differing range of other systemparameters that determine the overall signal magni-tude. Table 2 summarizes the system parameters ne-cessary to obtain a maximum TSNR for a 0:5miodine cell. However, as a general rule of thumb, io-dine transitions with larger absorption coefficientswill lead to a larger TSNR resulting in improved fre-quency stability. Therefore, if the lasers are locked tothe iodine transition using an MTS arrangementwith the system parameters given in Table 2, the fre-

quency stability of the 532nm laser will be approxi-mately 20 times better than the 633nm laser and1000 times better than the 778nm laser when allsources of technical noise in the three systems areignored. While various measurements of the fre-quency instability of references based on MTS at532nm exist [4–7,17] and show that the instabilitycan be less than 10−14, less data is available that de-tails the measured stability of MTS references at633nm and 778nm. The measurements that havebeen reported on 633nm references locked to satu-rated iodine components using external cell techni-ques [9,28] do, however, show that the frequencyinstability is no better than 10−12 for nonoptimisedsystems which is consistent with the findings pre-sented here.

7. Conclusions

A theoretical model was presented that accuratelydescribes the influence of the system parameterson the MTS signal. This novel analysis includesthe effects of pump beam depletion allowing a moreaccurate determination of the signal magnitude thanpreviously possible. The calculated signals were foundtoagreewellwithexperimentalmeasurements ofMTSsignals at 532nm, 543nm and 612nmperformedwithboth AM and FM pump fields and two different celllengths. Using this analysis a simple algebraic expres-sionwasderived thatgives thepressure required to ob-tain a maximum MTS signal magnitude for a givenabsorptioncoefficient, cell lengthandsaturation inten-sity. Comparisons of this formula with experimentaldata showed good agreement.

Using thismodel, amethodologywasproposed toal-low the determination of the appropriate values forthe iodine pressure, cell length and saturating inten-sity that lead to the lowest frequency instabilitywhenthe laser is locked to a particular iodine transitionusing MTS. The calculations showed that the celllengths necessary for maximum signal magnitudesfor transitions with low levels of absorption such asat 633nm and 778nm, are too large to be practicallyimplemented.Therefore, in lowabsorption situations,the largest cell that is practical should be used sincethe improvement in frequency stability is propor-tional to cell length.Once the cell length has been cho-sen themodel can be used to determine the saturationintensity necessary to obtain the maximum TSNR.The saturation intensity necessary for maximumTSNR increases as the absorption coefficient de-

Table 2. Summary of System Parameters Necessary to ObtainOptimum TSNR with a 0:5m Iodine Cell

532nm 633nm 778nm

α (m−1Pa−1) 0.28 0.012 0.000045TSNR (arb.units)

7 × 10−2 4 × 10−3 8 × 10−5

p (Pa) 5 100 1000Isat (mW=mm2) 3 175 2500~γ (MHz) 1.4 8.6 100


creases. Thus for weak transitions intensities beyondthe range of the laser source used may sometimes benecessary. In this event it is worth noting that largesignals can still be obtained atmuch lower intensitiesdue to the rapid increase of the TSNR at low intensi-ties. For example, when a 0:5m cell is used forlocking to the 778nm transition, an intensity of2500mW=mm2 is required to optimize the TSNR.However, the TSNR is 80% of its maximum value ata 20mW=mm2 saturating intensity.After the cell length and saturating intensity have

been determined the optimum pressure can be eval-uated using the derived expression Eq. (13). Themethodology presented here for optimizing theMTS arrangement for frequency locking, is applic-able to any iodine transition for which the absorptioncoefficient, the hyperfine structure, and the satura-tion parameters are known.

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