ARTICLE IN PRESS

Optics & Laser Technology 42 (2010) 439–446

Contents lists available at ScienceDirect

Optics & Laser Technology

0030-39

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/optlastec

Spectral-line asymmetry in double-output-cavity lasers

Ahmed Aissani a,�, Belkacem Meziane b, Karim Battou a, Omar Ziane a

a Laboratoire d’�electronique quantique, Facult�e de physique USTHB, B.P. 32 El-Alia Bab-Ezzouar, Algiers, Algeriab Laboratoire de Physico-Chimie des Interfaces et Applications, Universit�e d’Artois, UFR Sciences, Lens, France

a r t i c l e i n f o

Article history:

Received 12 April 2009

Received in revised form

18 July 2009

Accepted 12 August 2009Available online 1 October 2009

Keywords:

Spectral line

Double-output laser

Frequency shift

92/$ - see front matter & 2009 Elsevier Ltd. A

016/j.optlastec.2009.08.010

esponding author.

ail address: aaissani@yahoo.fr (A. Aissani).

a b s t r a c t

The aim of this work is to show that the output from a double-output laser cavity exhibits distinctive

features, manifesting themselves through typical asymmetry in the corresponding spectral-line shapes.

Such asymmetry in an inhomogeneously broadened gas laser usually appears with a clear, but

unpredictable shift of the maximum intensity level either towards the high or low frequency sides with

respect to line centre. In the case of a double-output laser, the spectral profiles show opposite shifts.

This means that when the maximum intensity of one output moves towards the high frequency side of

the profile, the maximum intensity of the other moves towards the low frequency side. This gives rise to

a frequency shift with respect to the Lamb-dip for a classical laser. On the theoretical side, we apply the

standard disturbed Gaussian beam model to provide a quantification of the frequency shifts obtained at

both sides of the system.

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

It is well known that, for a low gain laser, the emission line isgenerally characterized by a dissymmetric Lamb-dip [1–3]. Manyreasons have been attributed to the origin of this asymmetry.Among these are: atomic collisions [4–6], the presence of an intracavity aperture for fundamental transverse mode selection [7], thegas pressure, and discharge current [8]. In addition to the previousreasons it is important to note that the feedback [9,10] andisotopic effects [11] can be the causes of the asymmetry. However,these can be eliminated by taking some experimental precautions.

These diverse causes contribute to the creation of populationand saturation inhomogeneities, which induce lens effects, thusmodifying both the wavefront curvature and the mode size.Furthermore, it was demonstrated that the frequency dependenceof these two parameters is at the origin of the line asymmetry[12]. In such a laser, population heterogeneity effects are oppositeto saturation ones [13–15]: the first effect gives rise to a highfrequency line asymmetry while the second gives rise to a lowfrequency one. Other works [16–25] have shown that thegeometrical parameters of the laser (cavity length, position anddiameter of the tube containing the amplifying medium, mirrorcurvatures, aperture diameter, and so on) do influence lineshaping. Somehow, these geometrical parameters modulate thecompeting effects between the two heterogeneities. Therefore,the appearing asymmetry is imposed by the predominantheterogeneity.

ll rights reserved.

In some cooperative cases, the cavity geometry favours mutualcompensation between population and saturation effects [26,27].These critical cases exhibit symmetric line shapes. However, thisfavorable situation is very sensitive to the system’s controlparameters. Slight changes in any parameter (for example, thedriving current or the tube position inside the cavity) rapidlyimpose line-symmetry breaking.

In metrology, line asymmetry is undesirable for frequencystandards design [28,29]. In order to make up a reliable standard,it is necessary that all the sources of the asymmetry be taken intoaccount in the theoretical model in order to minimize them andobtain a symmetric line shape. Otherwise, it is necessary to locatethe Lamb-dip frequency with regard to the central resonancefrequency of the laser. However, the Lamb-dip shows someflattened character with the consequent drawback of increasingmeasurements errors.

A straightforward method that circumvents such a drawbackconsists of inserting an absorption cell containing iodine ormethane vapor in the cavity [30]. The emission line is thencharacterized by a narrow peak, which constitutes a wavelengthreference. Such laser configurations, when properly servo-con-trolled on one of the absorption lines of iodine or methane, arewidely used in metrology [31–35]. These constitute the best lasersources for high precision length measurements [36].

In most lasers, the reflectivity of one of the two mirrors is equalto unity. Therefore, the comparison of the two outputs is notnecessary. However, this is necessary in a double-output laserwhere the two mirrors’ reflectivities are not equal to unity. Forinformation, one of the authors reported in a recent paper [37] auseful double-output configuration used for laser Q-switching

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A. Aissani et al. / Optics & Laser Technology 42 (2010) 439–446440

operation. We could think that the two outputs are equivalent andconsequently the two line shapes are strictly identical. In fact, it isnot the case.

The main objective of the present work consists in demon-strating, both theoretically and experimentally, that the max-imum of the intensity is not at the same frequency at the twooutputs of the laser. This phenomenon is called the ‘‘outputeffect’’.

It is important to note that this effect has already beenaccidentally observed [27,28]. It is also important to highlight thatto the best of our knowledge, no theoretical explanation has beengiven to such effect. In addition, it is the first time that a completeexperimental study has been devoted to this effect.

The model of the disturbed Gaussian beam [13] has alreadybeen successfully used to explain the variation of the Lamb-dipasymmetry within the discharge current. It is therefore interestingto use that model in order to confirm and quantify this ‘‘outputeffect’’. The theoretical approach consists in writing the disturbedGaussian field expressions at each position inside the laser and toiteratively link them through the usual propagation equations.Such a method straightforwardly leads to simply express the fieldat one output with respect to the field at the opposite side.

We carry out this operation for two symmetric frequencieswith respect to the amplifying-medium gain-contour in the caseof a classical Lamb-dip. We further proceed and calculateasymmetry factor whose final expression is evaluated in termsof simple ratio between the difference and the sum of theintensities. In so doing, we straightforwardly obtain the spectral-line asymmetry at one output with respect to the other.

The derived final expression takes into account the Gaussianbeam longitudinal perturbations in addition to the transverseones. These are included by simply making use of the evolutionequation that describes such perturbations all the way throughthe laser. Associated with adapted continuity conditions, theobtained expression is finally used to calculate the frequency shiftbetween the two outputs for the Lamb-dip.

2. Theoretical model

Let us consider a laser cavity consisting of: two concavemirrors M1 and M2 with curvatures R1 and R2 respectively,separated by a distance D; an amplifying tube of length Lg andan aperture with diameter 2b inserted near mirror M1. Thedistances between: M1 and the laser tube; the tube and M2 arerespectively L1, L2 (see Fig. 1).

In the following, we will derive a closed form expression of theoutput-signal asymmetry evaluated at M1 with respect to the

z1 z2z3 z4

Lg

Amplifying Medium

L1 L2

2b

Output 2 Output 1

3

D

21

M1 M2Aperture

Fig. 1. Schematic representation of the cavity.

output at M2, starting with known expressions of the forward andbackward resonant fields which write [4]

Ef ;bðr; zÞ ¼E0f ;b

172izexp 7i k

oo

cW2

o

� �z

h i� exp �

r2

172iz

� �

� exp �i ef ;b8r2

2Zf ;b

� �� �ð1Þ

where E0f,b are the amplitudes of the complex fields, ef,b thelongitudinal perturbation parameter (it includes phase andamplitude terms) and Zf,b the transverse one (it contains boththe size and the curvature of the mode). 2W0 is the beam-waistsize and o0 the central frequency. (r,z) are the reduced cylindricalcoordinates (r,z) expressed as

r ¼ r=w0; z ¼ cz=o0w20 ð2Þ

c is the vacuum light speed. k ¼ 2p/l is the wave number. Theupper sign in the used symbol ‘‘7’’ or ‘‘8’’ corresponds to theforward wave and the lower to the backward wave.

As indicated in Fig. 1, the cavity is made up of three alternateregions: n ¼ 1, 2, 3. In the empty regions 1 and 3, k is equal to o0/c, while in regions 2 it corresponds to kg.

Moreover, we suppose that the transmission coefficient at theboundary of two neighboring regions is equal to unity. Thus, forany region ‘‘n’’ we may write

nþ1Ef ;bðr; zjÞ ¼nEf ;bðr; zjÞ ð3Þ

where zj is the reduced coordinate at the boundary of twobordering regions. The index ‘‘j’’ can take the value 3 or 4.

Taking into account relations (1) and (3), we can express thebackward field at position z1 as a function of its value at positionz2, according to the following relation:

Ebðr; z1Þ ¼ Ebðr; z2Þ:1� 2iz2

1� 2iz1:expi

o0

cW0

� �2

ðl1 þ l2Þ þo0

cW2

0 ðkglgÞ

� �

�exp �r2 1

1� 2iz1�

1

1� 2iz2

� �� �exp �i eB þ

r2

2ZB

� �� �ð4Þ

where l1, l2 and lg are the reduced lengths correspondingrespectively to L1, L2 and Lg as deduced from relation (2) and thefollowing expressions:

eB ¼ ð1eb1� 1eb3

Þ þ ð2eb3� 2eb4

Þ þ ð3eb4� 3eb2

Þ

ZB ¼ ð1Zb1� 1Zb3

Þ þ ð2Zb3� 2Zb4

Þ þ ð3Zb4� 3Zb2

Þ ¼ Zb1� Zb2

ð5Þ

The continuity relations for the transverse parameter Zb lead tothe following equality: nZbj ¼

n+1Zbj. These assumptions are notvalid for the longitudinal parameter eb.

The intensity variable is simply deduced from the squaremodulus of the field amplitude appearing in Eq. (4). It writes

Iðr; z1Þ ¼ Ið0; z2Þ:1þ 4z2

2

1þ 4z21

exp �2o0

cW2

0 kig lg

h i

exp �2r2

1þ 4z21

þ 2eiB þ

1Zib1r2

" #ð6Þ

where I(0,z2) is the axial intensity of the backward field at positionz2, and the index i represents the imaginary part of the differentquantities.

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A. Aissani et al. / Optics & Laser Technology 42 (2010) 439–446 441

From Eq. (1), we readily extract the relation between the axialintensity and the transverse one as

Iðr; z2Þ ¼ Ið0; z2Þexp �2r2

1þ 4z22

þ Zib2r2

" #ð7Þ

It is worth mentioning that because of diffraction phenomena,the line asymmetry undergoes some sensible changes that closelydepend on the detector position [24]. These elements must beconsidered with careful attention in order to avoid any misleadinginterpretation.

In the following, we will take into account the transverse sizeof the field to evaluate the total output-beam power.

The output power at position z1 is given by

Pðz1Þ ¼

Z 2p

0

Z b

0Iðr; z1Þrdrdj ¼ 2pW2

0

Z b=W0

0Iðr; z1Þrdr ð8Þ

Indeed, the presence of the diaphragm in front of mirror M1,limits the integral operation to the aperture dimension 2b.

Taking into account Eqs. (6) and (7), we obtain:

Pðz1Þ ¼ Pðz2Þ �2� ð1þ 4z2

2 ÞZib2

2� ð1þ 4z21 ÞZi

b1

24

35

� 1� exp �b2

W20

�2� ð1þ 4z2

1 ÞZib1

ð1þ 4z21 Þ

!" #

� exp �2o0

cW2

0

� �� ki

g lgh i

� expð2eiB Þ ð9Þ

This last expression clearly shows that the total laser-outputpower P(z1) at M1 depends upon the total output power P(z2) atM2. We also note that the first term corresponds to the Gaussianshape factor, the second to the transmission factor through thediaphragm aperture, while the third term takes into account themedium gains and losses and the last one refers to saturationeffects.

Since the laser output at M2 contains no aperture, the overallpower P(z2), is obtained from an integral over an infinite area ofthe backward intensity Ib(r,z2). The power is then given by thefollowing expression:

Pðz2Þ ¼pW2

0 ð1þ 4z22 Þ

2� ð1þ 4z22 ÞZi

b2

" #Ið0; z2Þ ð10Þ

Let us now consider two symmetric frequencies o1 and o2

with respect to amplifying-medium central frequency o0

o1 �o0 ¼ o0 �o2ðwith o14o04o2Þ

At a given output i of the laser, (i ¼ 1 or 2), one expresses theline-asymmetry factor by the following ratio:

Di ¼Piðo1Þ � Piðo2Þ

Piðo1Þ þ Piðo2Þ¼

Fi � 1

Fi þ 1ð11Þ

where

Fi ¼Piðo1Þ

Piðo2Þð12Þ

represents the ratio between the output powers at o1 and o2,evaluated either at M1 or at M2. This ratio is equal to unity in thecase of a symmetric line shape.

Taking into account Eq. (9), we find the following relation:

D1 ¼F1 � 1

F1 þ 1¼

F2:f ðo1;o2Þ � 1

F2f ðo1;o2Þ þ 1ð13Þ

where f(o1,o2) is a function of the different parameters thatappear in Eq. (9).

D1 is asymmetry factor at output 1 (i.e. at z2), expressed interms of the asymmetry at output 2 (i.e. at z1). We easily see thatD1 is different from D2 except in the case where f(o1,o2) ¼ 1, asituation that rarely happens to be the case in a laser.

As mentioned in paragraph 1, we have demonstrated theore-tically the existence of the output effect already observedexperimentally. The combination of relations (13) and (9) showsclearly that the asymmetry of the line shapes is not the same atthe two outputs of the cavity. As a direct consequence, this resultsin a difference in the frequency of the Lamb-dip between the twooutputs of the laser. The following section is devoted to derivingsuch a frequency shift.

2.1. Application of the model to the Lamb-dip

In a gas laser with low gain, the emission line is characterizedby a minimum intensity value at the so-called Lamb-dipfrequency oLD. This later is generally different from the centralfrequency o0 of the laser line. In addition, it has been shownexperimentally [6,14] that the line shape closely depends on thegeometrical parameters of the cavity, and particularly on theamplifying-tube position inside the cavity. For example, it isinteresting to evaluate the variation of the Lamb-dip frequency atthe two outputs of the laser cavity, versus the position z3 of thetube. In so doing, we use the following steps:

(i)

First, we write the total laser-output power and obtain asecond order-limited development that gives its value in thevicinity of the centre-frequency o0. This is equivalent toassimilating the shape of the Lamb-dip to a simple parabolicform.

(ii)

Second, we take advantage of the fact that at the Lamb-dip,corresponding to minimum output intensity, the gain profilehas a zero value derivative, to give a simpler and clearerpicture of the obtained frequency shifts.

Before proceeding further, we note that the quantities Zbi,eb

i,kg

appearing in Eqs. (9) and (10) that give the power at each output,are frequency dependent. It is then useful to introduce a reducedfrequency x, expressed as

x ¼o�o0

Gð14Þ

where G is the Doppler half-width and o0 the central frequency asmentioned above. o0 is defined according to Zr(o0) ¼ 0, where Zr

is the real part of the dispersion function Z(o)[15].In the following, we will introduce Pi(x), representing the

output power at position zi(i ¼ 1 or 2).In the vicinity of the Lamb-dip i.e. the central frequency

corresponding to x ¼ 0, we may, as stated before, write a secondorder Taylor development of the output power Pi(x) of the form

PiðxÞ ¼ Pið0Þ þ xdPi

dx

� �x¼0

þx2

2!

d2Pi

dx2

� �x¼0

ð15Þ

ARTICLE IN PRESS

0.0

0.5

1.0

1.5

2.0

Output 2

Output 1

Out

put p

ower

(a.

u)

A. Aissani et al. / Optics & Laser Technology 42 (2010) 439–446442

The minimum of the Lamb-dip satisfies

dPiðxÞ

dx

� �x¼xLD

¼ 0 ð16Þ

where xLD is the reduced frequency at the Lamb-dip.Deriving again Eq. (15), and taking into account Eq. (16), we

obtain

xLD ¼ �dPi

dx

� �x¼0

d2Pi

dx2

� �x¼0

�ð17Þ

This last relation allows for a direct evaluation of the differencebetween the Lamb-dip frequency oLD and the central frequencyo0. In such a way that applying Eq. (17) to the two laser outputs,one obtains the difference (xLD2�xLD1), which in turn directlyquantifies the frequency difference (oLD2�oLD1) between the twolaser outputs.

-1.0 -0.5 0.0 0.5 1.0

Reduced frequency x

-1.0 -0.5 0.0 0.5 1.01.25

1.50

1.75

2.00

Output 2

Output 1

Out

put p

ower

(a.

u)

Reduced frequency x

Fig. 2. (a) Line shape (output power) versus the reduced frequency at the two

outputs. (b) Zoom of the line shape (a).

3. Numerical results

The numerical calculations are performed for a He–Ne laseroscillating on the 632.8 nm line. The cavity length (see Fig. 1) isD ¼ 420 mm. The output mirrors M1 and M2 have equal reflectiv-ity R1 ¼ R2 ¼ 0.64 and curvature radii equal, respectively, toRC1 ¼ 1200 mm and RC2 ¼ 600 mm. The amplifying tube has alength Lg ¼ 265 and a 3 mm inner diameter.

For a fundamental TEM00 transverse mode selection [7], anaperture is placed inside the cavity close to mirror M1. Thetruncate parameter defined as the ratio of the aperture to themode diameters is taken equal to 1.2. The choice of this ratio isderived from the works reported in references [26,38] in onehand, and from our own experience in the other hand.

Moreover, in the present experimental conditions, we havefound that an aperture diameter 2bo1.2W0 gives rise to a quasiGaussian mode, whereas for larger diameters, high ordertransverse modes oscillate.

We have plotted in Fig. 2a the line shapes obtained fromoutputs 1 and 2 of the laser corresponding to a positionz3 ¼ 55 mm of the amplifying tube. This latter has beenarbitrarily selected. Both lines show a high frequencyasymmetry. We may hence assume that the populationinhomogeneities play a predominant role with respect to thesaturation ones. These inhomogeneities are more accentuated atoutput 2 as demonstrated from zoomed representations in Fig. 2b.These findings were quite predictable since the beam intensity atoutput 2 is larger than at the opposite side, since more atomsparticipate to the laser mode in the corresponding direction.

Fig. 3 shows, for the same position of the tube (z3 ¼ 55 mm),the variations of the squared-mode-size difference between Wa

for an active and Wp for a passive cavity obtained at the twooutputs, with respect to the reduced frequency. Both plots have adispersion shape that exhibits a maximum value at the lowfrequency side: this gives rise to high losses [20]. In addition, thevariations are more important at output 2; thus confirming theobserved high asymmetry at this output.

Fig. 4 represents the deviation of the Lamb-dip frequency fromthe central one at the two outputs with respect to the amplifyingtube position. In the two cases, the difference is systematicallypositive. The asymmetry is of a high frequency kind, whatever thetube position inside the cavity. Note that the asymmetry is moresignificant at output 2 for reasons given hereafter.

When the amplifying tube moves closer to output 2, thefrequency difference decreases at this output and increases at theopposite one. This result is in adequate agreement with the mode-size behavior, since it is 1.5 larger at output 2 than at the output 1.

We easily deduce that the population inhomogeneities increase atoutput 2 while these decrease at output 1. An opposite behavior ispredicted for the saturation inhomogeneities.

In Fig. 5, we plot the frequency difference between the twoLamb-dips extracted from the two outputs, with respect to theposition of the amplifying medium inside the cavity. Thisdifference decreases when the tube moves closer to output 2.From the curve in Fig. 5, one might think that if the cavity lengthwere sufficiently increased, the frequency shift could be canceledand one would obtain the same line shapes at the two outputs ofthe cavity. In this case the output effect disappears, butunfortunately the laser becomes multimode.

These curves clearly show the complexity of the phenomenathat take place inside a laser. Such complexities make it difficult topredict the kind of asymmetry that one may expect from a givencavity. As already mentioned, different parameters affect popula-tion and saturation inhomogeneities. The asymmetry type is setby the predominant inhomogeneities.

Let us also stress the fact that the present study did not takeinto account many other parameters such as atomic collision, gas

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0 20 40 60 80 100190

200

240

250

260

270

Output 2

Output 1

Fre

quen

cy d

iffer

ence

(ω

0 -

ω1,

2 )

(kH

z)

Amplifying tube position z3 (mm)

Fig. 4. Variation with respect to the tube position of the quantity (o0�o1,2) at the

two outputs.

0 20 40 60 80 100 12045

50

55

60

65

70

75

80

experimental results

numerical results

Fre

quen

cy d

iffer

ence

(ω

1 -

ω2)

(kH

z)

Amplifying tube position z3 (mm)

Fig. 5. Variation with respect to the tube position of the calculated Lamb-dip

frequency difference between the two outputs.

-1.0 -0.5 0.0 0.5 1.0-1

0

1

2

3Output 2

Output 1

( W

a

2 - W

p2 ) (

a.u)

Reduced frequency x

Fig. 3. Variation with respect to the reduced frequency of the quantity (Wa2�Wp

2).

A. Aissani et al. / Optics & Laser Technology 42 (2010) 439–446 443

pressure or discharge current. However, even though theseparameters most probably affect the two kinds of inhomogene-ities, at a quantified level, the presented results are self-explanatory with respect to the physical mechanisms involvedin output-dependent spectral-line shaping.

4. Experimental study

The experimental study was carried out on a He–Ne laser(632.8 nm). The geometrical parameters of the cavity are thoseused in the calculus made in Section 3. The electrical current is setto 15 mA and the gas pressure to 0.6 Torr, with an isotopic purityof Ne20

�99.99%. As explained in the introduction this allows us toeliminate the possible asymmetry which can be induced by thepresence in the gas of isotopes others than the Ne20.

These values are carefully chosen [13,26,39] to have apronounced Lamb-dip. However, as the Doppler width is about1.5 GHz and the free spectral range (FSR) of 357 MHz, the laser ismultimode with an output power of about 1 mW. To force thelaser to a single longitudinal mode oscillation, we have inserted acell filled with air. The misalignment of the cell reduces the gainwidth by increasing the loss level in the cavity. In this case theoutput power is considerably reduced (�70mW).

Let us now focus on the two parts constituting the experi-mental study.

The first part allows us to show the difference of theasymmetry of the line shape at the two outputs and the secondto measure the frequency shift between the respective Lamb-dips.

The experimental setup corresponding to the first part is givenin Fig. 6.

The beam of one of the two outputs is focused on the detector(Dt1 or 2) linked to an electronic amplifier (A). After amplification,the signal is sent to the Y input channel of the scope. The X inputchannel receives the low voltage signal applied to the piezo-electric translator PZT. We can thus monitor and record the outputpower versus the frequency.

We have first recorded the line shape at the two outputs. Theobtained results are presented in Fig. 7. For the sake of clarity andto highlight the line asymmetry, we give a zoom of Fig. 7a and caround the Lamb-dip (see Fig. 7b and d).

The recordings in Fig. 7, show a high frequency asymmetry.These results are in agreement with the numerical ones. Thepopulation inhomogeneity overcomes the saturation one. Theasymmetry is, as in the numerical modeling, higher at the output2. This can be explained by the simple fact that the beam is largerat this output according to the well-known expressions of theGaussian mode beam size. The calculus made on the basis of thewell-known non-perturbed Gaussian beam expressions, showsthat the mode size at the position z2 is approximately 1.5 timeslarger than at z1. As a result, the number of active atoms andtherefore the population inhomogeneity are larger at the positionz2. This explains why the high frequency kind asymmetry is morepronounced at the output 2.

For the second part, we have used the beat-frequencytechnique. This later needs the use of two lasers: a referencelaser (RL) which is servo-controlled on his resonance frequencyo0 and a studied laser (SL) having its two outputs 1 and 2 servo-controlled on their Lamb-dip frequencies oLD1 and oLD2, respec-tively. Note that the amplifying tube of the reference laser isarranged such as the line shape at his output is symmetric

ARTICLE IN PRESS

Oscilloscope

Laser Cavity

M

PZT Supply

Dt1

L

M

M1 M2

PZT Output 1 Output 2

L

Dt2

+ HV

G C

X Y

Fig. 6. Experimental setup used for line shape recordings: (G: amplifying medium, C: air cell, M: mirror, PZT: piezoelectric translator, L: lens, Dt1,2: detectors).

Fig. 7. Line shape at the output 1 (a) with its zoomed view (b). Line shape at the output 2 (c) with its zoomed view (d).

A. Aissani et al. / Optics & Laser Technology 42 (2010) 439–446444

ARTICLE IN PRESS

M

M

M

Laser (SL)

BS

SpectrumAnalyser

L

Dt2

Synchronous detection

Reference laser (RL)

A

+ HV

Output 1 Output 2 BS

LFG

G C

Dt1

FrequencyMeter

Recorder

L

Computer

Fig. 8. Experimental setup used for frequency shift measurements: (BS: beam splitter, SL: studied laser, RL: reference laser, G: amplifying medium. C: air cell, A: electronic

amplifier, LFG: Low frequency generator, Dt1,2: detectors, L: lens).

A. Aissani et al. / Optics & Laser Technology 42 (2010) 439–446 445

[16,26,27]. In these conditions, his Lamb-dip frequency iscoincident with the resonance frequency.

To minimize external perturbations effects, we have made aserial of N ( ¼ 100) measurements on each of the two outputs andwaited enough time to reach the thermal equilibrium of the twolasers.

The detectors used for the output 1 and 2 deliver a signalwhich frequency is proportional to the difference of thefrequencies of the two beams delivered by the lasers SL and RL.The signal after amplification by A is sent to a spectrum analyser,or to a digital frequency meter which displays the value (oLDi�o0)every one second. The data are stored in a computer linked to thefrequency meter. A software programme allows us to obtain themean value ðoLDi �o0 Þ over the N measurements made on theoutput i( ¼ 1,2), the maximum and the minimum values and thefrequency stability. This later corresponds to Allan’s standarddeviation [40]. In the present experiment the relative Allan’sstandard deviation varies between 10�10 and10�12.

Note that the difference between the mean values of theoutputs 2 and 1 gives the frequency shift (oLD2�oLD1) betweenthe respective outputs.

Fig. 8 describes the experimental arrangement used forfrequency shift measurements of the two outputs. In this figure,we can see that three mirrors M have been added to make themeasurements on the output 1. This provides us a convenientmethod to switch immediately to the second output after theacquisition of the measurements on the first one.

In addition, we can see in Figs. 6 and 8 that the detectors areslightly tilted to eliminate feedback effects which might affect theasymmetry.

As reported in Fig. 5, the experimental results show similarbehavior compared to the numerical modeling. The experimentalpoints are well fitted by the numerical ones. Indeed the decreaseof the frequency difference is practically linear, as the tube ismoved close to output 2.

The slight difference between the numerical and experimentalresults is attributed to perturbation of the laser when theamplifying tube is moved. However, the present experimentshows clearly the predicted output effect.

5. Conclusion

In this work, we have studied the line shape in a laser with twooutputs. The theoretical model shows a difference in theasymmetry between the emission lines at the two outputs ofthe laser. This asymmetry is essentially due to the variation of thetransverse and longitudinal parameters with respect to frequencyand to the geometric parameters of the cavity. These differentparameters greatly affect population and saturation inhomogene-ities and compete to privilege one of the two kinds ofinhomogeneities. If the population inhomogeneity is dominant,the line shape shows maximum output intensity at highfrequencies. On the other hand, saturation inhomogeneities giverise to a maximum output at low frequencies. In some favorablecases, the two effects compensate, giving rise to a symmetric lineshape.

As a result, the asymmetry induces a frequency shift betweenthe two outputs. This effect is usually called ‘‘output effect’’.

We have proposed a model, which gives for the first time anexplanation and an evaluation of the well-known output effect,for a conventional laser.

Our report focused on both experiments and theory and theinterface between these two aspects of the study is rathersatisfactory.

It is worth noting that this work is of great interest inmetrology. For purpose, one can study a metrological laser byincluding an absorbing medium in the cavity (generally methaneor iodine). In this case and contrary to a classical laser where theline shape shows a dip, the corresponding one for a metrologicallaser shows a narrow peak, usually named saturated absorptionpeak. Using the maximum of this peak one can make an accuratemeasurement of the frequency shift. This will be carried out in ournext work.

References

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