Properties of a laser cavity containing an absorbing ring

Properties of a laser cavity containingan absorbing ring

Abdelkrim Hasnaoui1 and Kamel Ait-Ameur2,*1Laboratoire d’Electronique Quantique, Faculté de Physique, Université des Sciences et de la Technologie

Houari Boumédiène, B.P. n°32, El Alia, 16111 Algiers, Algeria2Centre de Recherche sur les Ions, les Matériaux et la Photonique, Unité Mixte de Recherche 6252,

Commissariat à l’Energie Atomique, Centre National de la Recherche Scientifique,Université de Caen, Ecole Nationale Supérieure des Ingénieurs de Caen,

Boulevard Maréchal Juin, F14050 Caen, France

*Corresponding author: [email protected]

Received 14 April 2010; revised 13 June 2010; accepted 18 June 2010;posted 23 June 2010 (Doc. ID 127009); published 19 July 2010

This paper considers the transverse optical properties of an absorbing ring when it is lighted by a sym-metrical Laguerre–Gauss beam TEMp0. It is demonstrated that the insertion of an opaque ring havingadequate size inside a diaphragmed laser cavity is able to improve greatly (rate of about 100%) thediscrimination between the TEM00 and the TEM10 modes, while keeping the diffraction losses unchangedor even decreased. © 2010 Optical Society of AmericaOCIS codes: 050.1940, 050.1220, 140.3410.

1. Introduction

The spatial distribution of the light generated by alaser is generally imposed by the cavity eigenmodes.The fundamental mode TEM00, which corresponds tothe one having the lowest losses, is Gaussian inshape in the case of a spherical mirror [1,2].

The fundamental mode characteristics of a lasercavity can be fashioned by inserting an additionalcomponent inside the resonator, which can be

– a diaphragm for the fundamental mode TEM00selection [1–4];

– a phase aperture for improving the transversemode discrimination (TMD) [5], the laser dynamics[6], or the laser brightness [7];

– a phase aberration for the thermal aberrationcorrection in solid-state lasers [8,9];

– an opaque disk in order to force the fundamen-tal laser mode to be TEM10 like in shape [10]; or

– an annular diaphragm for the generation of aBessel–Gauss beam [11].

From the above we note that having a devicethat could be used to influence the geometricproperties of laser light would be a challengesince its effects on the fundamental mode ofan optical cavity would have to be taken intoconsideration. This is why in this paper we areinterested in considering the influence of an intra-cavity opaque ring on the fundamental mode charac-teristics of an apertured cavity. We will focus ourstudy on the case of the familiar plano–concave cav-ity containing a circular hard aperture against theconcave mirror, and an opaque ring set against theplane mirror. The cavity output is assumed to beon the concave mirror side.

In fact, the absorbing ring should be considered asa kind of optical component with regard to its proper-ties. Indeed, the absorbing ring is the basic elementfor producing transverse superresolution, which con-sists of a diffractive managing of light distribution,allowing improvement of the quality of imagingsystems [12–14].

0003-6935/10/214034-10$15.00/0© 2010 Optical Society of America

4034 APPLIED OPTICS / Vol. 49, No. 21 / 20 July 2010

For determining the fundamental mode of the cav-ity containing a diaphragm and an absorbing ring,we will follow the methodology already implementedfor the case of a phase aperture inside an aperturedcavity [15] or for the case of an opaque disk [10]. Thismethodology consists of considering, first, the single-pass diffraction properties of the absorbing ringwhen the incident beam is a symmetrical Laguerre–Gauss beam TEMp0; this is the subject of Section 2.Second, we will consider the ring properties resultingfrom multipass diffraction, i.e., when the ring is in-serted inside a laser cavity; this is the subject ofSections 3 and 4, where we put the emphasis onTMD properties, which have received little attentionin the literature.

2. Opaque Ring

As pointed out in Section 1, the modal properties of alaser cavity including a particular device can be bet-ter understood if one investigates first its single-passproperties when it is illuminated by a set of beamscorresponding to the transverse modes of the barecavity. Here, one considers a particular case becausethe involved transverse modes are symmetricalLaguerre–Gauss modes TEMp0. The two single-passproperties of interest are the transmission of the de-vice and the angular divergence in the far field of thediffracted TEMp0 beam. This methodology has beenalready applied for the case of a phase aperture [15],a stop [10], or a diaphragm [16,17] inserted inside acavity already containing an apertured mirror.

The ring geometry is characterized by an inner(outer) radius ρA (ρB), and a width h, as shown inFig. 1. The normally incident field, of wavelengthλ ¼ 1:06 μm, upon the absorbing ring is a symmetri-cal collimated Laguerre–Gauss beam of order pwhose unit-amplitude distribution is given by

EinðρÞ ¼ Lp

�2ρ2W2

�exp

�−

ρ2W2

�; ð1Þ

where W is the width of the incident beam. TheLaguerre polynomial is equal to unity for theTEM00, to ð1 − 2ρ2=W2Þ for the TEM10 beam, andto ð1 − 4ρ2=W2 þ 2ρ4=W4Þ for the TEM20 beam. ATEMp0 beam is made up of a central main lobe sur-rounded by p concentric rings of light. It is conveni-ent to express the geometric characteristics of thering through the following normalized parameters:YA ¼ ρA=W, YB ¼ ρB=W, and Δ ¼ h=λ.

Hereafter, we will characterize the optical proper-ties of the ring by considering, as a function of YA, itstransmission properties and the diffracted fieldwhen the incident beam is a TEMp0 beam. In parti-cular the diffracted field will be characterized by itsangular divergence.

A. Transmission of a TEMp0 Beam throughan Opaque Ring

It is convenient to introduce the intensity distribu-tion of the TEMp0 incident beam from Eq. (1),

namely, Ip0 ¼ jEinðρÞj2. The transmission Tp0 of thecentered ring for an input TEMp0 beam is definedas the ratio of transmitted and incident powers:

Tp0 ¼R ρA0 Ip0ðρÞρdρþ

R∞

ρB Ip0ðρÞρdρR∞

0 Ip0ðρÞρdρ: ð2Þ

The calculation of Tp0 is based on an integral of func-tions expressed as the product of an exponential anda polynomial that can be found in Ref. [18]:

Tp0 ¼ 1þ Fp0ðYAÞ − Fp0ðYBÞ; ð3Þ

where the function Fp0ðYÞ is expressed as follows:

F00 ¼ 1 − exp½−2Y2�; ð4Þ

F10 ¼ 1 − ð1þ 4Y4Þ exp½−2Y2�; ð5Þ

F20 ¼ 1 − ð1þ 8Y4− 8Y6 þ 4Y8Þ exp½−2Y2�: ð6Þ

The variation of transmission Tp0 versus YA forΔ ¼ 150 and W ¼ 1mm is shown in Fig. 2. It is seenthat the hierarchy of the set T00, T10, T20 changes asthe ring diameter is varied. One can then expect in-teresting modal properties when the ring will be setinside a laser cavity, as will be shown in Sections 3and 4. One can note that curve Tp0 versus curveYA in Fig. 2 displays relative maxima and minima.One finds that Tp0 has a maximum very close tounity when the ring radius corresponds to a zeroof the incident beam intensity.

As usual [17], it is useful to characterize the ring bythe discrimination properties factor defined asFR ¼ T00=T10. This discrimination factor has to beconsidered as a single-pass quantity thatmust be dis-tinguishedwith the discrimination factor of the cavityinto which is inserted a ring. This difference is due tothe cavity resonance effects, which tend to heightenthe single-pass properties of the inserted device. Theresult is shown inFig. 3,which displays the variations

Fig. 1. Opaque ring of internal (external) radius ρA (ρB).

20 July 2010 / Vol. 49, No. 21 / APPLIED OPTICS 4035

of FR versus YA forΔ ¼ 150. For the smallest value ofYA, it is seen thatFR < 1, whichmeans that TEM10 isbetter transmitted than theTEM00 beam. In contrast,one has FR > 1 when YA > 1, meaning that TEM00 isbetter transmitted than the TEM10 beam. The mini-mum (maximum) of FR occurs for YA ¼ 0:55 (YA ¼1:27). TheminimumofFR does not occur exactlywhenthe ring radius corresponds to the zero of the incidentbeam intensity. If it was the case, one should observethe minimum of FR not for YA ¼ 0:55, but rather forYA ¼ ð1=2Þ1=2 ¼ 0:707. This can be easily understoodif we take into account that the ring has a width h sothat the absorbing area is characterized by a normal-ized radius varying from0.55 to 0.7 forΔ ¼ 150. If onereduces the normalized ring width toΔ ¼ 50, one ob-serves that theminimumand themaximumget closerto unity, and this occurs for YA ¼ 0:61 and YA ¼ 1:33,respectively, as shown in Fig. 4.

B. Angular Divergence in the Far-Field Region

It has been shown that diffraction of a TEMp0 beamthrough a circular diaphragm is characterized by re-lative variations in the angular divergence that de-pend on aperture radius and on the incident beamorder. This enables the optimization of TMD insidean apertured resonator [17]. The knowledge of theangular divergence θp of the TEMp0 beam that dif-fracts on the device under study is essential. Indeed,it has been already shown that, when the device isan opaque disk [10] or a phase aperture (circularphase step) [15], for instance, the hierarchy of theset θ1; θ2; θ3;… can be different from that of the pureTEM00;TEM10;TEM20;… beams expressed asθ00 < θ10 < θ20:::. Note that the divergence θp standsfor the diffracted TEMp0 beam, and θ0p for the pureTEMp0 beams. The latter are characterized by ratiosθ10=θ00 ¼ 1:64 and θ20=θ00 ¼ 2:12 [19].

In the case where the hierarchy of divergence be-comes, for instance, θ1 < θ0 < θ2 < θ3… due to dif-fraction through the device , one can expect that

the fundamental mode of the cavity comprising adiaphragm and the device could be TEM10 like inshape.

It is worth remembering that, by definition, thefundamental mode of a laser made up of sphericalmirrors, a selecting aperture, and a diffractive opti-cal element is the first one reaching the oscillationthreshold, i.e., the mode having the lowest losses.Consequently, the fundamental mode is not necessa-rily Gaussian in shape, and, depending upon the dif-fraction properties of the inserted device , it can beTEMp0 like in shape with p > 1. This point will beconsidered in the following.

The divergence θp of the diffracted TEMp0 upon theopaque ring is determined [20] from the effectivewidthWe of the diffracted beam in the far-field regionat a distance D ¼ 30m from the ring:

θp ¼ We

D: ð7Þ

The effective widthWe of a non-Gaussian beam is de-fined as the radius of a circle that receives the samepower than the circle with a radius equal to theGaussian beam width (i.e., 86%). The variations ofratio θp=θp0 as a function of YA, the normalized ringradius, are shown in Fig. 5 for p ¼ 0, 1, and 2. It isseen that the curves in Fig. 5 show a maximum witha position that is further shifted toward the highestvalues of YA as p is increased. The position of themaximum of θp=θp0 for p ¼ 1 (p ¼ 2) corresponds tothe opaque ring positioned exactly on the bright ring(second bright ring), namely, YA ≈ 1:23 (YA ≈ 1:8). Be-cause we have it in mind to consider the simulta-neous presence of an opaque ring and a diaphragminside a cavity, it is important to evaluate, fromFig. 5, a possible change in the divergence hierarchyby considering the variations of ratios θ1=θ0 andθ2=θ0 versus YA. The results are shown in Fig. 6,where the horizontal dashed lines correspond toratios θ10=θ00 ¼ 1:64 and θ20=θ00 ¼ 2:12, whichcharacterize the pure TEMp0 beams. It is seen that,

Fig. 2. Variations of the ring transmission as a function ofnormalized ring radius YA ¼ ρA=W when the incident beam is asymmetrical Laguerre–Gauss mode TEMp0 of order p and of widthW ¼ 1mm, for Δ ¼ 150.

Fig. 3. Variations of the single-pass discrimination factorT00=T10

as a function of the normalized ring YA radius for Δ ¼ 150 andW ¼ 1mm.


for YA close to 0.7 (corresponding to the dark ring forp ¼ 1), the ratio θ1=θ0 is reduced to 1.2 instead of 1.64when there is no diffraction. As it is shown in Fig. 5,this case has to be considered as a quasi-zero-fieldoccluding and, consequently, no more lateral spreadoccurs for the TEM10 beam, while the TEM00 suffersa divergence increase with a rate of about 30%. Inaddition, it is important to note that, for this parti-cular operating point, the ring transmission is max-imum for the TEM10 beam and greater than that ofthe TEM00. It results that this operating point (YAclose to 0.7) could be the best condition to force a cav-ity to have a fundamental mode that is TEM10 like inshape, as it will be shown in Section 3. Another inter-esting feature appears in Fig. 6 since, for YA in therange from 1.2 to 1.3, one observes that θ2 ≈ θ1 withθ1 > 1:64θ00. This property implies that selection of asingle transverse mode by an intracavity aperturecould be improved from the TMD ability, as will beshown Section 3.

3. Laser Cavity with a Ring and a Diaphragm

So far, we have discussed the single-pass diffractionproperties of an opaque ring by considering its trans-mission when the incident beam is a TEM00, aTEM10, and a TEM20 beam and the resulting far-fielddivergence. Now, let us look for the opaque ring prop-erties resulting from multipass diffraction. For that,we will consider a plano–concave resonator, as shownin Fig. 7, made up of

i. an opaque ring of radius ρA and width h setagainst the plane mirror, and

ii. a diaphragm of radius ρD set against theconcave mirror.

Let us introduce the geometric parameterg ¼ ð1 − d=RÞ, which is kept in the stable region(0 < g < 1) by varying R, the radius of curvature ofthe concave mirror, for a constant resonator lengthd ¼ 600mm. Two other geometric parameters of in-terest are the spot sizes characterizing the fundamen-

talmode TEM00 of the bare cavity, i.e., without ring ordiaphragm, namely, the Gaussian mode radii Wp, onthe plane mirror, and Wc, on the concave mirror:

W2p ¼ λd

π

�g

1 − g

�1=2

; ð8Þ

W2c ¼ W2

p

g: ð9Þ

It is clear that the resonantmodes of the cavity shownin Fig. 7 are no longer the pure TEMp0 due to the dif-fractive disturbance occurring on the ring and the dia-phragm. This beam disturbance will be characterizedby two dimensionless parameters YR and YD ex-pressed, respectively, as

YR ¼ ρAWp

; YD ¼ ρDWc

: ð10Þ

The numerical calculation of the resonant field ofthe cavity including an opaque ring and a diaphragmis based on the field’s expansion on the basis of theLaguerre–Gauss transverse modes TEMp0 of thebare cavity. This method has been already describedelsewhere for the case of a plano–concave cavity in-cluding a hard aperture on one mirror, and a phaseaperture on the other mirror [7], and is briefly pre-sented in Appendix A. It is important to characterizethe discrimination performance of the resonatoragainst higher-order transverse modes by introdu-cing the TMD factor, defined as

FC ¼ jΓ0j2jΓ1j2

; ð11Þ

where jΓ0j (jΓ1j) is the eigenvalue modulus asso-ciated with the first (second) transverse mode. Notethat the power associated with a mode changes

Fig. 4. Variations of the single-pass discrimination factorT00=T10

as a function of the normalized ring YA radius forΔ ¼ 150 and 50,and W ¼ 1mm.

Fig. 5. Variations of divergence ratio θp=θp0 versus YA, where θpstands for the diffracted TEMp0 beamupon the absorbing ring, andθp0 for the pure TEMp0 beam, for Δ ¼ 150 and W ¼ 1mm.


by the factor jΓj2 after a cavity round trip (seeAppendix A). A large value of FC indicates that ahigh discrimination against higher-order transversemodes is achieved. One finds in the literature manyreports on the use of amplitude or phase masks in-side a laser cavity in order to impose a particular in-tensity distribution just by setting lines of zeroes ofintensity. Unfortunately, little attention is devotedto the discrimination against the other transversemodes, which certainly will oscillate once the pump-ing power exceeds more or less the threshold. This iswhy it is important to consider the behavior of theTMD factor with the different parameters. Anotherquantity of importance is the beam propagation fac-tor M2, which characterizes the output beam fromthe cavity (see Appendix A). The power losses perround trip due to diffraction and associated withthe fundamental mode are defined as LFM ¼1 − jΓ0j2.

We consider the variations ofM2, LFM, and FC as afunction of YD for two particular values of YR, thenormalized radius of the ring. The first one is YR ¼0:7, for which the fundamental mode is supposed tobe TEM10 like in shape, and the second one YR ¼ 1:3is the value for which the fundamental mode isexpected to be Gaussian with an improved discrimi-nation factor, as explained in Section 2. The calcula-tions of M2, LFM, and FC are made for g ¼ 0:5(half-confocal configuration), for which the discrimi-nation factor has been found maximum [4].

The results are shown in Fig. 8 for YR ¼ 0:7 andthe following should be recognized. First, one ob-serves that the fundamental mode is TEM00 like inshape for YD < 1:72, with a degraded beam propaga-tion factor comprise between 2 and 2.6. Second, oneobserves that the value ofM2 jumps to a higher valueat YD ¼ 1:72, and the fundamental mode is observedto be TEM10 like in shape. However, the selection ofthis TEM10 like fundamental mode is characterizedby very little TMD since FC ≈ 1 for YD > 1:72. Thedramatic change in the M2 factor at YD ¼ 1:72 dueto a change in the transverse intensity distribution

is also visible through a break in the slope of thecurve LFM versus YD (arrow in Fig. 8). We will comeacross the selection of the TEM10 mode again by an

Fig. 6. Variation of divergence ratios θ1=θ0 and θ2=θ0 versus YA,for Δ ¼ 150 and W ¼ 1mm.

Fig. 7. Sketch of the plano–concave cavity of length d ¼ 600mmmade up of an absorbing ring against the plane mirror, and adiaphragm against the concave mirror.

Fig. 8. Variations of beam propagation factor M2, fundamentalmode loss LFM, and TMD for a normalized ring radius YR ¼ 0:7,as a function of the normalized diaphragm radius YD, for Δ ¼20 and g ¼ 0:5. For an ideal pure TEM00 (TEM10) mode, the beamquality factor is M2 ¼ 1 (M2 ¼ 3).


absorbing ring in Section 4, where the transverseproperties of the resonator containing only an ab-sorbing ring, i.e., without the diaphragm.

For YR ¼ 1:3, the fundamental mode is TEM00 likein shape for YD < 2:25 with M2 close to 1.5, and isTEM10 like in shape for YD ≥ 2:25 with a beam qual-ityM2 of about 4.5 (Fig. 9) instead of 3, as if the modewas a pure TEM10 mode. This degradation in thequality of the beam is due to diffraction occurringon the diaphragm edge. Let us now check if the in-troduction of an absorbing ring inside an aperturedresonator makes possible to increase the discrimina-tion factor FC and to reduce the round-trip losses LFMwhile the fundamental mode is TEM00 like in shape,as suggested at the end of Subsection 2.B. For that, it

is convenient to plot the curve FC versus LFM forYR ¼ 1:3 and YD variable. The results are shownin Fig. 10. It is seen that, if one accepts a loss levelLFM ¼ 20%, it corresponds to a TMD factor FC equalto 28, without the opaque ring, and to 60 when anopaque ring with YR ¼ 1:3 is inserted. Inversely,the same TMD factor FC ¼ 60 is possible with lesslosses (20% instead 30%) by introducing the absorb-ing ring. Undoubtedly, the insertion of an opaquering makes it possible to get a TEM00 fundamentalmode with a higher discrimination together with lesslosses, in comparison with what can be obtained withonly the diaphragm. In addition, it is observed that,in the two cases, the beam propagation factor is closeto 1.5. One can point out the use of more sophisti-cated diffractive optics to achieve high modal discri-mination. For instance, the resonator design can bebased on the combination of a diffractive mirror anda diffuser plate [21], or a phase step discontinuity[22]. As a consequence, it is worth noting that thefabrication of an absorbing ring is probably moresimple and less costly than any phase diffractive op-tical element. In addition, its performing ability forimproving the TMDmakes the absorbing ring a com-ponent that allows the design of performing resona-tors from the point of view of TMD.

4. Laser Cavity with a Ring

In Section 3 we found that the fundamental mode ofthe cavity including an absorbing ring on the planemirror and a diaphragm on the concave mirror couldbeTEM10 like inshape if theopeningof thediaphragmis correctly chosen but, unfortunately, with a poordiscrimination against other transverse modes. Theinitial idea of TEM10 mode selection by the ring–diaphragm pair is that the absorbing ring forces theresonant mode to have a zero of intensity. When thesize of this dark circle corresponds to that of theTEM10 cavity eigenmode, then the fundamentalmodeof the cavity becomes TEM10 like in shape. It is worthnoting that the TEM10 mode has only one dark ring,while a TEMp0 is made up of p dark rings. Now the

Fig. 9. Variations of beam propagation factor M2, fundamentalmode loss LFM and TMD for a normalized ring radius YR ¼ 1:3,as a function of the normalized diaphragm radius YD, for Δ ¼20 and g ¼ 0:5. For an ideal pure TEM00 (TEM10) mode, the beamquality factor is M2 ¼ 1 (M2 ¼ 3).

Fig. 10. TMD FC versus fundamental mode losses LFM; solidcurve, without absorbing ring; dotted curve, with absorbing ringof normalized radius YR ¼ 1:3 and normalized widthΔ ¼ 20. Notethat LFM is varied by changing the diaphragm radius.


question raised is, “Is anabsorbing ringable to select afundamentalmode that is TEMp0 like in shape just byimposing only one dark ring?”The answer is yes. Thishas been checked for the first five TEMp0 modes,whose zeros of intensity are given in Table 1. In fact,since the ring has a certain width h, it is very impor-tant to take care to the overlap of the particular darkring that is desired. For instance, if one wants to forcethe fundamental mode to be TEM20 like in shape, onecan set the absorbing ring on the first zero positionedat normalized radial position 0.54 following Table 1.For YR ¼ 0:5 andΔ ¼ 20, the normalized radial posi-tion of the absorbing zone of the ring varies from0.5 to0.54 so that the first zero of TEM20 will be overlapped,and the fundamental mode is effectively a TEM20mode,asshowninFig.11,withM2 ¼ 5:009, indicatingthat we are dealing with a pure mode. However, forYR ¼ 0:53and the samevalue ofΔ, thenormalized ra-dial position of the absorbing zone of the ring varyingfrom 0.53 to 0.57 overlaps well the first zero of theTEM20 mode, but the selected fundamental mode isa high-order transverse mode with p ¼ 12, as shownin Fig. 11. As a consequence, one can expect thatthe selection of a high-order single mode by usingan intracavity opaque ring is not so easy and can beaccompanied by unstable patterns. The latter couldbe the consequence of small changes in beam sizeand, thus, inYR, due, for instance, toa time-dependentthermal lensing. This effect or any another oneresponsible for variations of YR could give rise inthe laser output to jumps from one to another high-order transverse mode.

Up to now, we have found that the selection of aTEMp0 mode by using an opaque ring is certainlyweakly absorbing, but also weakly discriminatingagainst higher-order transverse modes. It is impor-tant to note that the discussed discrimination prop-erties of the resonator are that of the cold cavity, thatis, without any active medium. If one remembersthat a laser, made up of an amplifying medium insidea cavity, is a nonlinear optical system, then it is notsurprising that selection of a single high-order modeby means of wires is operating experimentally[23–26].

5. Conclusion

In this paper, we consider the influence of an opaquering on the transverse properties [losses, TMD, beampropagation factor] of a laser cavity. We have found,depending on the ring sizes, the following:

i. The discrimination between TEM00 and TEM10modes is significantly improved, by a rate of about100%, with respect to the cavity including the usualdiaphragm. It is also possible for the cavity made upof a ring and a diaphragm to have the same TMD asfor the apertured cavity, but with fewer losses. Inboth cases, the beam propagation factor M2 is closeto 1.5.

ii. The fundamental mode of the cavity that in-cludes just an opaque ring can be TEMp0 like inshape with small losses but, unfortunately, with avery low TMD.

The main advantage of using absorbing rings to in-fluence the transverse properties of a laser cavity isthe relative simplicity of fabrication in comparisonwith phase diffractive optical elements. It is impor-tant to note that the properties of a cavity that in-cludes an absorbing ring have been determinedwithout amplifying medium, i.e., for a cold cavity.If this should not be the case, i.e., for a real laserresonator, nonlinear effects have to be taken intoaccount and that will make the modeling difficult.Depending on the considered gain medium, thetransverse modes of the real laser cavity, i.e., withan active medium, move more or less away from thatof the cold cavity. At least, our resonator analysis isall the more valid as the laser is close to thethreshold.

Appendix A

1. Round-Trip Operator

The numerical calculation of the resonant field in theresonator including the aperture and the absorbingring is based on its expansion on the basis of theeigenfunctions of the bare cavity, that is, withoutany diffracting objects. The eigenfunctions are theLaguerre–Gauss functions written for cylindricalsymmetry. The study of the resonant field involvesa decomposition into its two progressive components:a forward beam, propagating in the direction z > 0,

Fig. 11. Far-field pattern of the fundamental mode of the cavityincluding only an absorbing ring characterized by a normalizedradius YR.

Table 1. Roots of Laguerre Polynomials: Lp ðρ=W Þ ¼ 0

pValues of Ratio ρ=W for the Zeros

of Intensity of TEMp0 Mode

1 0.7071062 0.541195 1.3065623 0.455946 1.071046 1.7734074 0.401589 0.934280 1.506090 2.1673795 0.363015 0.840041 1.340975 1.882260 2.51040


and a backward beam in the direction z < 0. TheLaguerre–Gauss functions make up an orthonorma-lized basis, which are written, for the forward beamas

Gfpðρ; zÞ ¼�2π

�1=2 1

WðzÞLp

�2ρ2W2

�exp

�−

ρ2W2

�

× exp�þi

�kρ2

2RcðzÞ− ð2pþ 1ÞϕðzÞ

��; ðA1Þ

and for the backward beam as

Gbpðρ; zÞ ¼�2π

�1=2 1

WðzÞLp

�2ρ2W2

�exp

�−

ρ2W2

�

× exp�−i

�kρ2

2RcðzÞ− ð2pþ 1ÞϕðzÞ

��; ðA2Þ

where k ¼ 2π=λ. Hereafter, the subscripts f and b de-note, respectively, forward and backward quantities.The Gaussian mode of the nonapertured cavity ischaracterized by its beam diameter 2WðzÞ and its ra-dius of curvature Rc at point z. These quantities ofreference, as well as the phase shift ϕ, are z depen-dent and obey the following formulas:

W2ðzÞ ¼ W20½1þ ðz=z0Þ2�; ðA3Þ

RcðzÞ ¼ z½1þ ðz0=zÞ2�; ðA4Þ

ϕðzÞ ¼ arctanðz=z0Þ; ðA5Þ

where z0 ¼ πW20=λ is the Rayleigh range and W0 is

the beam-waist radius expressed by W20 ¼ ðλd=πÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

g=ð1 − gÞpfor our plano–concave cavity. LpðXÞ is

the Laguerre polynomial of order p.The forward and backward fields are assumed to

be linearly polarized and are expressed as linearcombinations of the basis functions:

Ef ðρ; zÞ ¼ exp½iðkz − ωtÞ�Xp

f pGfpðρ; zÞ; ðA6Þ

Ebðρ; zÞ ¼ expfi½kð2d − zÞ − ωt�gXp

bpGbpðρ; zÞ: ðA7Þ

We are studying the stationary field for t ¼ 0 andthen expð−iωtÞ ¼ 1. The objective of the calculation isto find the coefficients f p and bp that are ρ and z in-dependent. The determination of these coefficientsinvolves a matrixM, which represents the round-tripoperator and expresses the change of the forwardcoefficients after a round trip in the cavity:

f 0p ¼Xm

Mpm⋅f m; ðA8Þ

where Mpm is the typical element of a matrix M. Thecalculation of matrix M can be easily adapted from

Refs. [7,27] for the cavity that includes a hard aper-ture and an opaque ring:

Mpm ¼ −rprc exp½2ikd�Xn

RCpnDCnm

× exp½−2iðnþmþ 1ÞϕðdÞ�; ðA9Þwhere rp (rc) is the reflectance (field ratio) of theplane (concave) mirror. The overlapping integralsRCpn and DCnm are given by

RCpn ¼Z2Y2

A

0

exp½−X �LpðXÞLnðXÞ

þZ∞

2Y2B

exp½−X �LpðXÞLnðXÞ; ðA10Þ

DCnm ¼Z2Y2

D

0

exp½−X�LnðXÞLmðXÞ: ðA11Þ

The resonance condition states that, after a roundtrip, the relation f 0p ¼ Γ⋅f p holds for all p. This condi-tion means that the eigenmodes of the apertured re-sonator are represented by the eigenvectors u of thematrixM. Each of them is characterized by a complexeigenvalueΓ such thatMu ¼ Γu. The eigenvector ofMhaving the largest eigenvalue jΓ0j2 corresponds to thefundamental mode. The second eigenvector ofM hav-ing the second greatest eigenvalue jΓ1j2 correspondsto the second transverse mode. Determination of ei-genvectors of M of size (80 × 80) is done numericallyusing a FORTRAN 77 routine based on InternationalMathematics and Statistics Library subroutines. It isimportant to note that themethod of Laguerre–Gaussfunctions is powerful because the coefficients f p andbp are ρ and z independent. This makes possiblethe complete mapping of the spatial distribution ofthe resonant field with a very fast computing time be-cause it involves only summations when applyingEqs. (A6) and (A7). This advantage has allowed stu-dies on the spatial distribution of the fundamentalmode inside a cavity, including apertures and phaseobjects [4,15,27,28]. Most of the computing time isconsumed in the numerical calculation of overlappingintegrals, such asEq. (A11),which canbe reduced by afactor of about 200 thanks the use of a recurrence for-mula for computing the Cpm [29]. This has allowedfinding optimizedparameters of a cavitywith twodia-phragms by using stochastic methods, such as simu-lated annealing, that require some tens of millions ofoverlapping integrals [17].

2. M2 Factor

TheM2 factor can describe beam propagation as wellas beam quality, that is, how localized the beam is.For instance, if we are interested in forcing the fun-damental mode of the cavity to be a symmetrical


Laguerre–Gauss TEMp0 mode, we can judge thequality of the transverse mode selected by comparingits M2 factor to that of a pure TEMp0 beam. Everyinstance of a beam that has sharp discontinuities inits transverse intensity profile can lead to an infiniteresult in the case where the beam width is defined onthe basis of a second-order intensity moment [30]. Inour case, since there are two kinds of discontinuityinside the cavity, due to the diaphragm and the ab-sorbing ring, one has to take care when calculatingthe M2 factor of the output beam. Fortunately, thereare some possibilities that can overcome this dif-ficulty by using different definitions and methods:(i) the generalized truncated second-order moment[31,32], (ii) the use of a factor defined by an86.5% power-content radius [33], and (iii) the self-convergent beam-width approach [34].

Let us consider the electrical field Edðρ; zÞ asso-ciated with any kind of symmetrical beam (diffractedor not) represented as a coherent superposition ofLaguerre–Gauss beams having the same frequencyweighted by coefficients Dp:

Edðρ; zÞ ¼Xp

DpGfpðρ; zÞ: ðA12Þ

The M2 factor is expressed very simply as follows[35]:

M2 ¼��X

p

ð2pþ 1ÞjDpj2�

2

− 4

�Xp

Xq

pðD�pDqÞrδp;qþ1

�2�1=2

; ðA13Þ

where superscript r stands for the real part of thequantity and the asterisk means the complex conju-gate. Note that an incoherent superposition ofLaguerre–Gauss modes arises when a stable-cavitylaser oscillates simultaneously in several Laguerre–Gauss modes with different frequencies. In this case,the beam propagation factor reduces [36] to M2 ¼P

pð2pþ 1ÞjDpj2. In Ref. [37], the calculation of theM2 factor of a Laguerre–Gauss TEMp0 beam dif-fracted upon a circular aperture has beenmade usingEq. (A13). It has been found that the M2 values arenot diverging and, in addition, that the obtained re-sults are similar to those obtained with the concept ofa generalized M2 factor [32]. The determination ofthe M2 factor of the output beam of the cavity thatincludes an absorbing ring and a diaphragm hasbeen done by implementing Eq. (A13), where Dp

characterizes the output beam coefficients whose ex-pression can be found in Ref. [28].

References1. A. G. Fox and T. Li, “Resonant modes in a maser interferom-

eter,” Bell Syst. Tech. J. 40, 453–488 (1961).2. H. Kogelnik and T. Li, “Laser beams and resonator,”Appl. Opt.

5, 1550–1567 (1966).

3. K. Ait-Ameur and G. Stephan, “Effective beam truncation ofthe fundamental mode in an apertured cavity,” Opt. Lett. 18,938–940 (1993).

4. K. Ait-Ameur, “Influence of the longitudinal position of anaperture inside a cavity on the transverse mode discrimina-tion,” Appl. Opt. 32, 7366–7372 (1993).

5. K. Ait-Ameur, M. Brunel, and F. Sanchez, “High transversemode discrimination in apertured resonators using diffractivebinary optics,” Opt. Commun. 184, 73–78 (2000).

6. N. Passilly, M. Fromager, and K. Ait-Ameur, “Improvement ofthe self-Q-switching behavior of a Cr:LiSAF laser using a bin-ary diffractive optics,” Appl. Opt. 43, 5047–5059 (2004).

7. R. de Saint Denis, N. Passilly, and K. Ait-Ameur, “Laser beambrightness of apertured optical resonators,” Opt. Commun.264, 193–202 (2006).

8. S. Makki and J. Leger, “Solid-state laser resonators withdiffractive optic thermal aberration correction,” IEEE J.Quantum Electron. 35, 1075–1085 (1999).

9. J. Bourderionnet, N. Huot, A. Brignon, and J. P. Huignard,“Spatial mode control of a diode-pumped Nd:YAG laser byuse of an intracavity holographic phase plate,” Opt. Lett.25, 1579–1581 (2000).

10. R. de Saint Denis, N. Passilly, M. Fromager, E. Cagniot, and K.Ait-Ameur, “Diffraction properties of opaque disks outside andinside a laser cavity,” Opt. Commun. 281, 4758 (2008).

11. I. A. Litvin and A. Forbes, “Bessel-Gauss resonator withinternal amplitude filter,” Opt. Commun. 281, 2385–2392(2008).

12. M.Martinez-Corral, P. Andrés, C. J. Zapata-Rodriguez, andM.Kowalczyk, “Three-dimensional superresolution by annularfilters,” Opt. Commun. 165, 267–278 (1999).

13. M. Martinez-Corral, M. T. Caballero, E. H. K. Stelzer, and J.Swoger, “Tailoring the axial shape of the point spread functionusing the Toraldo concept,” Opt. Express 10, 98–103 (2002).

14. V. Paeder, T. Scharf, P. Ruffieux, H.-P. Herzig, R. Voelkel, andK. J. Weible, “Microlenses with annular amplitude and phasemasks,” J. Eur. Opt. Soc. Rapid Publ. 2, 07005 (2007).

15. K. Ait-Ameur, “Effects of a phase aperture on the fundamentalmode of a hard-apertured cavity,” J. Mod. Opt. 49, 1157–1168(2002).

16. K. Ait-Ameur, F. Sanchez, and M. Brunel, “The transfer ofTEM00 and TEM01 beams through a hard-aperture,” J.Mod. Opt. 47, 1203–1211 (2000).

17. E.Cagniot,Z.Derrar-Kaddour,M.Fromager,andK.Aït-Ameur,“Improving both transverse mode discrimination and diffrac-tion losses in a plano-concave cavity,” Opt. Commun. 281,4449–4454 (2008).

18. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Seriesand Products, 7th ed. (Elsevier, 2007), p. 106.

19. D. R. Hall and P. E. Jackson, The Physics and Technology ofLaser Resonators (Institute of Physics, 1992), p. 137.

20. A. E. Siegman, Lasers (University Science, 1986), Chap. 17.21. U. D. Zeitner and F. Wyroski, “High modal discrimination for

laser resonators with Gaussian output beam,” J. Mod. Opt. 46,1309–1314 (1999).

22. A. A.Napartovitch, N.N. Elkin, V.N. Troschieva,D. V. Vysotski,and J. R. Leger, “Simplified intracavity phase plates for increas-inglaser-modediscrimination,”Appl.Opt.38,3025–3029(1999).

23. W. W. Rigrod, “Isolation of axi-symmetrical optical-resonatormodes,” Appl. Phys. Lett. 2, 51–53 (1963).

24. K.M.Abramski,H.J.Baker,A.D.Colley,andR.R.Hall, “Single-modeselectionusingcoherent imagingwithinaslabwaveguideCO2 laser,” Appl. Phys. Lett. 60, 2469–2471 (1992).

25. D. Chen, Z. Wang, and J. R. Leger, “Measurement of the modalproperties of a diffracted-optic graded-phase resonator,” Opt.Lett. 20, 663–665 (1995).

26. M. Ciofini, A. Labate, R. Meucci, and P. Y. Wang, “Experimen-tal evidence of selection and stabilization of spatial patterns in


a CO2 laser by means of spatial perturbations,”Opt. Commun.154, 307–312 (1998).

27. K. Aït-Ameur, H. Ladjouze, and G. Stéphan, “Diffraction ef-fects in a resonant cavity with two non-equivalent apertures,”Appl. Opt. 31, 397–405 (1992).

28. R. de Saint Denis, N. Passilly, and K. Aït-Ameur, “Laser beambrightness of apertured optical resonators,” Opt. Commun.264, 193–202 (2006).

29. Z. Derrar-Kaddour, A. Taleb, K. Aït-Ameur, G. Martel, andE. Cagniot, “Alternative model for computing intensity pat-terns through apertured ABCD systems,” Opt. Commun.281, 1384–1395 (2008).

30. P.-A. Bélanger, Y. Champagne, and C. Paré, “Beam propaga-tion factor of diffracted laser beams,” Opt. Commun. 105,233–242 (1994).

31. R. Martinez-Herrero and P. M. Mejias, “Second-order spatialcharacterization of hard-edge diffracted beams,” Opt. Lett. 18,1669–1671 (1993).

32. Z. Mei and D. Zhao, “The generalized beam propagation factorof truncated standard and elegant Laguerre-Gaussianbeams,” J. Opt. A Pure Appl. Opt. 6, 1005–1011.

33. S. Amano and T. Mochizuki, “Propagation characteristicsof diffracted M2 beam,” Appl. Opt. 41, 6325–6331(2002).

34. S. Amarande, A. Giesen, and H. Hügel, “Propagation analysisof self-convergent beam width and characterization ofhard-edge diffracted beams,” Appl. Opt. 39, 3914–3924(2000).

35. S. Vicalvi, R. Borghi, M. Santarsiero, and F. Gori, “Shape-invariance error for axially symmetric light beams,” IEEEJ. Quantum Electron. 34, 2109–2116 (1998).

36. A. E. Siegman, “New developments in laser resonators,” Proc.SPIE 1224, 2–14 (1990).

37. N. Passilly, G. Martel, and K. Aït-Ameur, “Beam propagationfactor of truncated Laguerre-Gauss beams,” J. Mod. Opt. 51,2279–2286 (2004).


Properties of a laser cavity containing an absorbing ring

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