Optics Communications 285 (2012) 5268–5275

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Optics Communications

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

Selection of a LGp0-shaped fundamental mode in a laser cavity: Phase versusamplitude masks

Abdelkrim Hasnaoui a, Thomas Godin b, Emmanuel Cagniot b, Michael Fromager b,Andrew Forbes d,e, Kamel Aıt-Ameur b,c,n

a Faculte de Physique, Universite des Sciences et de la Technologie Houari Boumedi�ene, B.P. no 32, El Alia, 16111 Algiers, Algeriab Centre de Recherche sur les Ions, les Materiaux et la Photonique, Unite Mixte de Recherche de Recherche 6252, Commissariat �a l’Energie Atomique, Centre National de la Recherche

Scientifique, Universite de Caen, Ecole Nationale Superieure des Ingenieurs de Caen, Boulevard Marechal Juin, F14050 Caen, Francec Centre de Developpement des Techniques Avancees, Division Milieux Ionises et Lasers, PO Box 17 Baba Hassan, 16303 Algiers, Algeriad Council for Scientific and Industrial Research, National Laser Centre, PO Box 395, Pretoria 0001, South Africae School of Physics, University of KwaZulu-Natal, Private Bag X54001, Durban 4000, South Africa

a r t i c l e i n f o

Article history:

Received 1 March 2012

Received in revised form

4 June 2012

Accepted 13 June 2012Available online 25 August 2012

Keywords:

High order transverse modes

Laguerre-Gauss modes

Transverse mode selection absorbing mask

18/$ - see front matter & 2012 Elsevier B.V. A

x.doi.org/10.1016/j.optcom.2012.06.089

esponding author at: Universite de Caen, Cen

eriaux et la Photonique, Centre National d

ixte de Recherche 6252, 14050 Caen, France

ail address: kamel.aitameur@ensicaen.fr (K. A

a b s t r a c t

Laser beams of a single high-order transverse mode have been of interest to the laser community for

several years now. In order to achieve such a mode as the fundamental mode of the cavity, mode

selecting elements in the form of a phase or amplitude mask are often placed inside the resonator. Such

elements have the role to impose one or several zeros of intensity of the desired mode. In this paper, we

consider the use of the most simple phase (amplitude) mask which is a transparent p-plate (absorbing

ring) set inside a diaphragmed laser cavity for selecting a pure LGp0 mode of radial order, p. We analyse,

for each type of mask, the origin of the transverse mode selection, and contrary to what one might

expect we find that it is not necessary the absorbing mask that results in the highest losses.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

Most laser applications not only need a high power level butalso require a high intensity. This is why the research efforts inlaser technology have been centred during the last decades on thegeneration of diffraction-limited laser beams, i.e., having a beampropagation factor M2

¼1. Such a beam is Gaussian-shaped, andhas been extensively considered in the literature. However,forcing a laser to oscillate on a single high-order transverse modeis today a new challenge that is of great interest. The latter can beeither an improvement of laser power extraction [1,2] by forcingthe oscillation on a Laguerre–Gaussian LGpl mode (of radial orderp and azimuthal order l), or the obtaining of very small focalvolume [3] when a symmetrical Laguerre–Gauss LGp0 is focussedafter to be reshaped by an adequate diffractive optical element(DOE). This property of super-resolution concerns many applica-tions (nonlinear fluorescence microscopy, 3-D laser prototyping,optical tweezers, direct writing lithography, y) where a smallfocal volume and high spatial resolution are required.

ll rights reserved.

tre de Recherche sur les Ions,

e la Recherche Scientifique,

.

ıt-Ameur).

In the present paper, we will focus our attention on forcing thefundamental mode of a laser cavity to be LGp0 like in shape butsingle mode, i.e., no mixing of several transverse modes. Let usrecall that a LGp0 beam is made up of a central lobe surrounded byp concentric rings of light and p concentric rings of zeros ofintensity. Thus, the idea for forcing the fundamental mode TEM00

of the laser cavity to be LGp0 like in shape lies in imposing theposition of the p zeros of intensity by setting an adequate phase[1,2] or amplitude [4–7] mask whose geometry follows closelythe location of the Laguerre polynomial zeros.

The two diffracting pupils which will be introduced inside thelaser cavity in order to force one zero of intensity are (i) a phasemask called as p-plate, and (ii) an amplitude mask taking theform of an absorbing ring.

The phase mask, called as p-plate, is a very simple DOE madeup of a two-level phase plate with a discontinuity of magnitude p.This device introduces a p-phase shift in the central portion of theincident beam. The p-plate is known to have interesting proper-ties since it allows for instance the reshaping of a Gaussianbeam [8]. The transmittance tPI (electrical field ratio) of a p-plateof radius r0 is given by

tPIðrÞ ¼�1 for rrr0

þ1 for r4r0,

(ð1Þ

where r is the radial coordinate.

Table 1Laguerre–Gauss polynomials Lp(X) of order p with the reduced variable defined as

X¼2r2/W2

P Beam Lp M2p

0 TEM00 1 1

1 TEM10 1�X 3

2 TEM20 X2�4Xþ2 5

3 TEM30 �X3þ9X2

�18Xþ6 7

A. Hasnaoui et al. / Optics Communications 285 (2012) 5268–5275 5269

The amplitude mask which is considered here is an absorbingring which is characterised by an inner (outer) radius rA (rB). Itstransmission tR is then defined as

tRðrÞ ¼0 for rArrrrB

þ1 elsewhere

(ð2Þ

The objective of this work is to consider the insertion of apupil, characterised by Eqs. (1) or (2), inside a cavity which is inaddition apertured by a circular diaphragm in order to force itsfundamental TEM00 mode to be LGp0 like in shape. The latter is asymmetrical Laguerre–Gauss beam of radial order p having anintensity profile given by

IpðrÞ ¼ I0 � L2pð2r

2=W2Þ � exp½�2r2=W2

�, ð3Þ

where Lpis the Laguerre polynomial of order p which are given inTable 1. A LGp0 mode is made up of a central peak surrounded by p

rings of light separated by p zeros of intensity given by the zerosof Lp. As it will be shown later, the same pupil (p-plate orabsorbing ring) can be used to select different values of modeorder p provided that its size matches a zero of the LGp0.

Usually, the selection of the LG00 mode as the fundamentalmode is achieved by inserting a hard aperture inside the laserresonator. The losses introduced in this way are minimum for theLG00 mode because its lateral extent, i.e., its angular divergence, isminimum.

In this paper, we will show that the fundamental TEM00 modeof the cavity in which a p-plate or an absorbing ring is insertedcan be LGp0 in shape. We will study in Section 2 the influence ofthe p-plate on the divergence hierarchy of the LGp0 basis.Similarly, we will consider in Section 3 the single-pass propertiesof an absorbing ring in order also to understand the mechanism oftransverse mode selection. The two single-pass properties ofinterest are the transmission of the pupil and the angulardivergence in the far-field of the diffracted LGp0 beam. Finally,in Section 4 we will determine the fundamental mode of thecavity made up on one side of a circular diaphragm, and the pupil(p-plate or ring) in the other side. It will be shown that theselection, of a LGp0-shaped fundamental mode using an absorbingring is characterised by a loss level which is smaller than the oneobtained when the pupil is a transparent p-plate.

Z2D

T1 T2

TEMp0

Beam-waistW0=1mm

D

Far-fieldregion

Fig. 1. Sketch for considering the diffracting properties of a cascade of two

p-plates separated by a distance 2D¼10 mm. The diffraction pattern is deter-

mined in the far-field region at distance Z¼30 m. It is assumed that the beam-

waist of the LGp0 incident mode is located at distance D0¼20 mm.

2. Effect of a p-plate on the divergence hierarchyof the LGp0 basis

The transverse mode discrimination due to a hard or a softaperture is linked to the divergence hierarchy y0, y1, y2,... of thetransverse modes LG00, LG10, LG20,... For the cold cavity made up ofonly two mirrors this divergence hierarchy becomesy0oy1oy2,... so that the truncation, i.e., the losses, introducedby the internal aperture is minimum for the LG00 mode whichbecomes the fundamental TEM00 mode of the apertured cavity[9,10]. Our objective is to understand more deeply the effect ofthe p-plate, and why the fundamental mode TEM00 of the cavitycan be LGp0 like in shape. In this purpose we have to consider first

the single-pass diffracting properties of a p-plate for an incidentLGp0 beam. More precisely, we have to consider the diffractingproperties of a cascade of two p-plates separated by a distance 2D

as shown in Fig. 1. The reason of considering two cascadingp-plates is, as explained in Section 4, the intra-cavity p-plate willbe set at a certain distance D from the concave mirror. We willfocus our attention on the perturbation of the divergence hier-archy which will explain why, for particular radius r0 of thep-plate, the fundamental TEM00 mode of the cavity could be LGp0

like in shape for p¼1, 2 and 3. Note that we limit our study tovalues of p ranging from 1 to 3, but the technique can extended tohigher values. Prior to proceeding, let us consider the secondsingle-pass property of the p-plate which is its transmission whenilluminated by a LGp0 beam. It is a reasonable approximation toconsider the p-plate as a transparent device. Indeed, even weaccount for the losses due to the Fresnel reflection, the p-platetransmission should be independent of mode order p and conse-quently does not play any particular role in the transverse modediscrimination.

To determine the transverse distribution of light intensityId (r, Z) in the far-field, at a distance Z¼30 m from the secondp-plate, the diffraction of the TEMp0 beam incident on p-plates T1

and T2 (Fig. 1) is modelled by the usual Fresnel–Kirchhoffdiffraction integral. The latter has been expressed in a sum ofHankel transforms having known analytical expressions allowinga fast computation. This technique has been already described indetail in Ref. [11].The quantity of interest which has to bedetermined is the divergence of the diffracted beam ye:

ye ¼We

Z, ð4Þ

where We is the effective width of the diffracted beam which isbased on the second-moment of the intensity distribution inplane Z, and is given by

W2e ¼

2R1

0 Idðr,ZÞr3drR10 Idðr,ZÞrdr

ð5Þ

It is useful to recall that diffraction of a Gaussian beam uponthe phase discontinuity of a p-plate distorts the far-field intensitypattern in such way that this can be used for achieving laser beamshaping [8]. It is consequently surprising, after the incident LGp0

beam has been diffracted upon the cascade of p-plates T1 and T2,to observe a far field pattern pretty close to the incident one, asshown in Fig. 2. The similarity between the double diffractedbeam and the incident LGp0 beam occurs only if the distance D issufficiently small so that the reshaping from the first p-plate doesnot occur yet. The double diffraction through the p-plates acts asif the effect of the second p-plate compensates that of the firstone, and consequently restoring the shape of the beam, howeverwith an increasing divergence as shown in Fig. 3. It is seen inFig. 3 by arrows A, B and C that the beam having the smallest

0.0

0.2

0.4

0.6

0.8

1.0

Nor

mal

ised

inte

nsity

ρ/W-3 -2 -1 0 1 2 3

-100 -80 -60 -40 -20 0 20 40 60 80 1000.0

0.2

0.4

0.6

0.8

1.0

Nor

mal

ised

inte

nsity

ρ/W

Fig. 2. (a) Intensity profile of the LG20beam incident on the cascade T1 and T2 of

p-plates, (b) Intensity profile in the far-field of the cascade T1–T2, when the size of

the p-plate correspond to the arrow B in Fig. 3.

0.00.0

5.0x10-4

1.0x10-3

1.5x10-3

2.0x10-3

2.5x10-3

3.0x10-3

CB

A

LG30LG20LG10LG00

Div

erge

nce

θ e (r

ad)

ρ0/W0.5 1.0 1.5 2.0 2.5 3.0

Fig. 3. Variations of the divergence ye of the diffracted LG00, LG10, LG20 and LG30

Laguerre–Gauss beams of width W normally incident upon a p-plate of radius r0.

A. Hasnaoui et al. / Optics Communications 285 (2012) 5268–52755270

divergence can be either the LG30, LG20 or LG10 beam dependingon the radius r0 of the p-plate. In fact, these minima occur whenthe p-plate radius corresponds to the first zero of intensity of theincident LGp0 beam. It can be also observed in Fig. 3 that the

divergence ye oscillates with the ratio r0/W, and is minimumwhen the size of the p-plate corresponds to any zero of the LGp0

intensity pattern. These results demonstrate that the diffractionof a set of LGp0 beams through a cascade of two p-platessufficiently close gives rise to a new set of LGp0 beams with anew hierarchy of divergence depending on the ratio r0/W.Consequently, one can easily understand why a p-plate inside alaser cavity combined with some spatial selection process(diaphragm for instance) is able to force the fundamental TEM00

mode to be LGp0 like in shape for p¼1, 2 and 3. For that one has toset the phase discontinuity of the p-plate on the first or thesecond zero of intensity of the LGp0 beam we want to select, andthe result is that the mode having the minimum of losses, i.e, thefundamental TEM00 mode, will be not necessarily the usual LG00

mode (Gaussian mode) but can be a LGp0 mode with p¼1, 2 or 3.Now, we have to confirm that the mode having the lowest

round-trip losses (i.e., fundamental TEM00 mode), of a cavityincluding a p-plate and a diaphragm can be LG10, LG20 or LG30

like in shape. This is addressed in the Section 4. In the nextSection we will consider the single pass properties of an absorb-ing ring enlighten by a LGp0 beam.

3. Effect of an absorbing ring on the divergence hierarchy ofthe LGp0 basis

In a similar way as done for the p-plate we can characterisethe absorbing ring by the far-field divergence ye when theincident beam is a LGp0 mode. As indicated above, the ringgeometry is characterised by its inner (outer) radius rA (rB),mean radius rR¼(rBþrA)/2 and width h¼(rB�rA).

We have considered the case of a ring having a widthh¼20 mm, and an incident beam characterised by W¼1 mm.We have found that the divergence of the diffracted LGp0 beamsis not affected by the ring because the ratio h/W is only 2%.However, if one increases the width of the absorbing ring toh¼150 mm, one sees that the divergence value associated witheach diffracted LGp0 beam is changed, but the initial hierarchydivergence is not affected. This behaviour is radically distinctfrom that of the p-plate displayed in Fig. 3. Now, it remains toexamine the variations of Tp the ring transmission, of a LGp0 beam,when its radius rR is varied. The transmission Tp is defined as apower ratio given as follows:

Tp ¼

R rA

0 IpðrÞrdrþR1rB

IpðrÞrdrR10 IpðrÞrdr

ð6Þ

It can be recognised in Fig. 4 that transmission Tp oscillateswith ratio rR/W, and is maximum when the ring size correspondsto any zero of the LGp0 beam intensity pattern which are given byTable 2. It is also seen in Fig. 4 that the hierarchy of the set T0, T1,T2 and T3 changes as the ring size is varied. As it will be shownlater, this property allows to force the fundamental mode TEM00

of a laser cavity to be LGp0-shaped provided that the ring is setone zero of intensity.

4. The fundamental mode of a cavity including a p-plate andan aperture

In the two previous sections we have considered the single-passdiffraction properties of a p-plate and absorbing ring. Now, the nextstep is to consider the p-plate and the absorbing ring and theirproperties resulting from multiple-pass diffraction when they areinserted inside a cavity whose one mirror is diaphragmed.

Let us consider in Fig. 5a the sketch of the plano-concavecavity of length L including a circular aperture set against the

0.0

0.975

0.980

0.985

0.990

0.995

1.000

LG30LG20LG10

LG00

Tp

ρR/W0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Fig. 4. Variation of the absorbing ring transmission Tp as a function its normalised

radius rR/W when the incident beam is a symmetrical Laguerre–Gauss LGp0 of

order p and width W¼1 mm. The ring width is h¼20 mm.

Table 2Roots of Laguerre polynomials.

p Values of ratio r/W for the zeros of intensity of LGp0 mode

1 0.707106

2 0.541195 1.306562

3 0.455946 1.071046 1.773407

0 LZPI=(L-D)

Z

Mp Mc

output

f1p f2p

b1p b2p

1 2

Diaphragm π-plate

0 L

Z

Mp Mc

output

fp

bp

Diaphragm Absorbing ring

Fig. 5. Sketch of the plano-concave cavity including a diaphragm against the plane

mirror and (a) a p-plate set at a distance D¼5 mm from the concave mirror of

radius of curvature R¼150 mm, (b) the absorbing ring is set against the concave

mirror.

A. Hasnaoui et al. / Optics Communications 285 (2012) 5268–5275 5271

plane mirror and a p-plate set at distance D¼5 mm from theconcave mirror of radius of curvature R¼150 mm (the reason forplacing the p-plate a small distance from the mirror can be foundin Ref. [12]).

In Fig. 5b the absorbing ring of radius rR is assumed to be setagainst the concave mirror because it is the position for which apriori the truncation losses are minimum. This can be understoodby taking notice of double truncation occurring on the forwardand backward beams, for the absorbing ring somewhere else thanthe mirror positions. In contrast, if the ring is against the concavemirror, it is only the backward beam which is truncated. Thatmakes clear the minimum of fundamental mode losses.

Intuitively, it could have been expected, as many peoplebelieve, that the fundamental mode losses are higher when alossy component (i.e., absorbing ring) is inserted inside the lasercavity in comparison with the case where a transparent device (p-plate) is inserted. The situation is far from being so simple since, asshown in Sections 1 and 2, that the necessary mechanism of lossesto achieve the transverse mode discrimination is due to thediaphragm in Fig. 5a, and both to the diaphragm and ring inFig. 5b. Our objective now is to provide an objective answer tothe question which cavities in Fig. 5 is more lossy to select aLGp0-shaped fundamental mode. This is why our aim in this Sectionis to compare the performance of each cavities in Fig. 5 bydetermining, for a given Transverse Mode Discrimination (TMD)factor, the losses of the fundamental mode when the ring and the p-plate sizes correspond to the first zero of the first-three LGp0 modes.

The determination of the fundamental mode of the cavitiesshown in Fig. 5 is made by using the mode expansion methodwhich is based on the Laguerre–Gauss functions. The details aregiven in Appendix 1. Our aim now is to determine the perfor-mance of transverse mode selection achieved by the p-plate orabsorbing ring from one side, and a diaphragm in the cavity otherside. The important point will be to compare the losses resultingfrom the insertion of each mask for a given transverse mode

discrimination (TMD) factor. The latter is expressed as a functionof the two-first eigenvalue modulus 9G09 and 9G19 associated withthe two-first eigenmodes TEM00 and TEM10, respectively. InAppendix 2, we discuss the definition of the TMD factor, notedas F in the following. Indeed, one finds in literature differentdefinitions which unfortunately can overestimate or underesti-mate the performance of the resonator in term of transversemode selection.

The diaphragm (p-plate) is characterised by its normalisedradius YC¼rc/W0 (YPI¼r0/W(ZPI)), where W0 is the radius of thebeam-waist located on the plane mirror and given by

W02¼

lL

pg

1�g

� �1=2

ð7Þ

In a similar way, one introduces the normalised ring radiusYR¼rR/W(L), where rR is the mean radius of the absorbing ring ofwidth h¼20 mm.

A p-plate (or an absorbing ring) having a fixed radiusr0¼174 mm (rR¼180 mm) is inserted inside the cavity at positionZPI (Z¼L). The ratios YPI and YR are varied by changing theparameter g by cavity length L adjustment. The reason for thatis we wish a fundamental mode which is LGp0 like in shape, with p

adjustable, by resorting to the same p-plate or absorbing ring.Now, let us return to our main objective which has beenintroduced at the beginning of Section 4, and which consists todetermine which cavity shown in Fig. 5 is the most lossy for agiven TMD factor. The ratios YPI and YR are set equal to the firstzero of the three-first LGp0 modes, i.e., p¼1,2 and 3. The round-trip operator for the cavity including a phase aperture or an

Table 3Comparison of the cavity losses including an absorbing ring, or a p-plate, on the

concave mirror, and a diaphragm on the plane mirror. The normalised radius of

the p-plate and ring are equal (YPI¼YR¼0.707) so that the fundamental mode is

LG10�shaped. The concave radius of curvature is R¼150 mm.

L (mm) Yc F LFM (%) M2 9G09 9G19

p-plate 93.1 2.9 1.019 22.9 3.011 0.878 0.869

Absorbing ring 93.1 3.9 1.021 0.3% 3.005 0.998 0.988

Table 4Comparison of the losses of the cavity including an absorbing ring, or a p-plate, on

the concave mirror, and a diaphragm on the plane mirror. The normalised radius

of the p-plate and ring are equal (YPI¼YR¼0.54) so that the fundamental mode is

LG20�shaped. The concave radius of curvature is R¼150 mm.

L (mm) Yc F LFM (%) M2 9G09 9G19

p-plate 124 4.1 1.038 10.4 5.002 0.946 0.929

Absorbing ring 124 4.1 1.017 0.18 5.000 0.999 0.990

Table 5Comparison of the losses of the cavity including an absorbing ring, or a p-plate, on

the concave mirror, and a diaphragm on the plane mirror. The normalised radius

of the p-plate and ring are equal (YPI¼YR¼0.45) so that the fundamental mode is

LG30�shaped. The concave radius of curvature is R¼150 mm.

L (mm) Yc F LFM (%) M2 9G09 9G19

p-plate 137 3.1 1.0027 20.1 7.06 0.893 0.892

Absorbing ring 137 3.4 1.0027 0.16% 7.008 0.999 0.997

0.0

0.2

0.4

0.6

0.8

1.0

Nor

mal

ised

Inte

nsity

r (mm)

0.0

0.2

0.4

0.6

0.8

1.0

Nor

mal

ised

Inte

nsity

r (mm)

-80 -60 -40 -20 0 20 40 60 80

-80 -60 -40 -20 0 20 40 60 80

-150 -100 -50 0 50 100 1500.0

0.2

0.4

0.6

0.8

1.0

Nor

mal

ised

Inte

nsity

r (mm)

Fig. 6. Radial intensity distribution of the fundamental mode TEM00 (blue curve)

and LGp0 modes (red curve) for the parameters of (a) Table 3, (b) Table 4,

(c) Table 5 The calculations have been made up at distance Z¼10 m from the

cavity. (For interpretation of the references to color in this figure legend, the

reader is referred to the web version of this article.)

A. Hasnaoui et al. / Optics Communications 285 (2012) 5268–52755272

absorbing ring is given in Appendix. The fundamental mode lossesLFM and eigenvalues modulus 9G09 and 9G19 for each cavity shownin Fig. 5, for a given TMD factor, are determined and compared inTables 3–5. Note that, the transverse mode discrimination factor F

is fixed by adjusting the diaphragm radius, i.e., Yc, so that its valueis the same as much as possible for the two cavities of Fig. 5. Inorder to check that the fundamental mode TEM00 of the cavityincluding a p-plate (or an absorbing ring) and a diaphragm can beLGp0 like in shape we can consider the far-field intensity pattern butalso the output beam quality factor M2 calculated from thecoefficients of expansion of the laser output field [13]. For referencepurpose, one can note that for a pure LGp0 beam one should haveM2¼2pþ1.Tables 3–5 show clearly as expected that the fundamental

mode TEM00 of an apertured cavity can be LG10, LG20 or LG30 like inshape if one introduces a p-plate or an absorbing ring having a sizecorresponding to the first zero of intensity of the desired mode.

In Fig. 6 are given the intensity distributions of the funda-mental mode TEM00 (blue curves) in the far-field region togetherwith the intensity distribution of the pure LGp0(red curves) for thecavity given in Fig. 5a. It is seen that the fundamental mode TEM00

selected with the parameters given in Tables 3–5 is very close to apure Laguerre–Gauss mode for p¼1. However, it moves away froma pure LG mode as p order increases. It is important to note that thefundamental mode obtained with an absorbing ring is very close toa pure LG mode whatever the order p. However, what is lessexpected is that one finds that the cavity giving rise to the largestlosses is not the one with the absorbing ring but rather the one withthe transparent p-plate. At first sight one can imagine that the p-plate, and the absorbing ring play exactly the same role, which is toimpose a zero of intensity in the fundamental mode. This is correctbut, as demonstrated in Sections 2 and 3, the transparent p-plate isable to turn the divergence hierarchy of the LGp0 basis upside downwhile the absorbing ring does not change the divergence hierarchyof the LGp0’s. This means that the mechanism of discrimination isdifferent in the two cases. Indeed, for the cavity including a p-plateit is the diaphragm which is responsible for the fundamental modeTEM00 selection. It is necessary clipping introduces a relatively highloss level, while it is the ring itself which essentially achieves thetransverse mode selection. Since its width h¼20 mm is relativelysmall, the losses LFM are also very small. In addition, one can noticefrom a practical point of view, that it is more simple to fabricate anabsorbing ring deposited on a transparent substrate than etchingthe latter with a depth of magnitude l/2 with about 10 nm accuracyfor the p-plate.

A. Hasnaoui et al. / Optics Communications 285 (2012) 5268–5275 5273

5. Conclusion

We have considered the insertion of a phase or amplitudemask inside a laser resonator in order to force its fundamentalTEM00 mode to be LGp0 like in shape, with p¼1, 2 and 3. Themasks we have considered are the simplest since the amplitude(phase) mask is an absorbing ring (p-plate). Although the twokinds of mask impose a zero of intensity to the oscillating mode,the mechanism of transverse mode discrimination is different.The effect of the transparent p-plate is to turn the divergencehierarchy of the LGp0 basis upside down, resulting in the intra-cavity diaphragm selecting as the fundamental TEM00 mode, theLGp0 mode having the lowest divergence. The LGp0 mode selectedis that having its first zero corresponding to the phase disconti-nuity of the p-plate. In fact, the fundamental mode is so clippedthat the losses are relatively high, while it is the ring itself whichessentially achieves the transverse mode selection. Since its widthis relatively small, the losses LFM are also small. Contrarily to whatcould be expected intuitively, on the basis that one mask istransparent and the other is absorbing, we have demonstratedthat the absorbing ring is more efficient than the p-plate to selecta fundamental TEM00 mode which is LGp0 like in shape. Inaddition, we have noticed that the quality of mode purity ishigher for the cavity with an absorbing ring than with a p-plate.

Acknowledgements

The authors acknowledge the support of the Conseil Regional

Basse Normandie (France), the Direction Generale de la Recherche

Scientifique et du Developpement Technologique (Algeria), theNational Research Fundation (South Africa), and the Partenariat

Hubert Curien (France) under the grant PROTEA No 25886XC.

Appendix A

The determination of the resonant field of the cavities shownin Fig. 5 is based on its decomposition into its two progressivecomponents: a forward (backward) beam propagating in thepositive (negative) Z direction. The origin of the axial coordinateZ is defined by the plane mirror position which is hard-aperturedby a circular diaphragm of radius rC. In Fig. 5a, a p-plate of radiusr0 is set at a distance D¼5 mm from the concave mirror of radiusof curvature R¼150 mm while the absorbing ring is assumed tobe set against the concave mirror (Fig. 5b). The reflectivity(intensity ratio) of the plane and concave mirrors is notedR1 and R2, respectively.

The numerical calculation of the resonant field is based on itsexpansion on the basis of the eigenfunctions (eigenmodes) of thebare cavity, i.e., made up of only mirrors Mp and Mc. In thefollowing, we will assume a perfect axial symmetry. The ortho-normalised basis is formed by 80 Laguerre–Gauss functions whichare written, for the forward beam as:

Gf pðr,ZÞ ¼ffiffiffi2p

q1

WðZÞ Lp2r2

WðZÞ2

� �exp � r2

WðZÞ2

� �exp þ i kr2

2Rc ðZÞ�ð2pþ1ÞfðZÞ

h in o,

ðA1Þ

and for the backward beam as:

Gbpðr,ZÞ ¼ffiffiffi2p

q1

WðZÞ Lp2r2

WðZÞ2

� �exp � r2

WðZÞ2

� �exp �i kr2

2Rc ðZÞ�ð2pþ1ÞfðZÞ

h in o,

ðA2Þ

where k¼2p/l. Hereafter, the subscripts f and b denote, respec-tively, forward and backward quantities. The Gaussian mode ofthe nonapertured cavity is characterized by its beam diameter2W(Z) and its radius of curvature Rc at point Z. These quantities of

reference, as well as the phase shift f, are Z dependent and obeythe well known following formulas:

W2ðZÞ ¼W2

0½1þðZ=ZRÞ2�, ðA3Þ

RcðZÞ ¼ Z½1þðZR=ZÞ2�, ðA4Þ

fðZÞ ¼ arctanðZ=ZRÞ, ðA5Þ

where ZR ¼ pW02=l is the Rayleigh range, and W0 is the beam-

waist radius expressed by W02¼ ðlL=pÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig=ð1�gÞ

pfor our plano-

concave cavity. Lp(X) is the Laguerre polynomial of order p.

(a)

Cavity with a diaphragm and a p-plate (Fig. 5a):The forward and backward fields are assumed to be linearlypolarized and are expressed as linear combinations of thebasis functions on both sides of the p-plate:

Ef jðr,ZÞ ¼ exp½iðkZ�otÞ�X

p

f jpGf pðr,ZÞ, ðA6Þ

Ebjðr,ZÞ ¼ expfi½kð2L�ZÞ�ot�gX

p

bjpGbpðr,ZÞ: ðA7Þ

The index j is 1 in the region 1, defined by 0oZoZPI and 2 inthe region 2 for which ZPIoZoL, where ZPI¼(L�D) is theposition of the p-plate. We are studying the stationary fieldfor t¼0 and then exp(� iwt)¼1.Note that the functions of the basis satisfy the orthonorma-lization condition:

2pZ 1

0Gf pðr,zÞGn

f mðr,zÞrdr¼ dpm ðA8Þ

2pZ 1

0Gbpðr,zÞGn

bmðr,zÞrdr¼ dpm, ðA9Þ

where the symbol n denotes the complex conjugate of thequantity. The computation of the forward and backwardfields requires the determination of the four (rand z inde-pendent) coefficients f1p, f2p, b1p and b2p of the field expan-sion for 0oZPIoL.The different coefficients are related to each other by theboundary conditions at the p-plate and aperture plane, andtheir determination involves the matrix M, which representsthe round-trip operator and expresses the change of theforward coefficients after a round-trip in the aperturedcavity:

f jp0 ¼

Xm

Mpmf jm ðA10Þ

The typical element Mlq of the matrix M is written as

Mlq ¼ ðR1R2Þ1=2exp½�i2fD�

Xp

CAqp

Xn

CDlnCA

np

�exp½�i4pðfD�fAÞ�exp½�iðnþqÞfA�, ðA11Þ

where

fD ¼ arctanlL

pW20

!ðA12Þ

fA ¼ arctanlZPI

pW20

!ðA13Þ

CApm ¼

Z 10

tPIðXÞexpð�XÞLpðXÞLmðXÞdX ðA14Þ

A. Hasnaoui et al. / Optics Communications 285 (2012) 5268–52755274

CDpn ¼

Z 2Y2C

0expð�YÞLpðYÞLmðYÞdY ðA15Þ

With X ¼2r2

W2ðZPIÞ

, Y ¼2r2

W20

, YC ¼rc

W0, ðA16Þ

The round-trip operator M holds the information aboutreflection at the mirrors and about amplitude (phase) clip-ping at the edge of the diaphragm (p-plate). By definition ofthe resonance condition, the relation f jp

0 ¼Gf jpholds for all p

after a round-trip. This means that the eigenmodes of thephase/amplitude apertured cavity are represented by theeigenvectors u of the matrix M, and each of them charac-terised by a complex eigenvalue G such that

Mu¼Gu: ðA17Þ

The eigenvector of M having the largest eigenvalue 9GFM9 isby definition the fundamental mode of the cavity whosepower loss per round trip is given by

LFM ¼ 1� GFMj j2

ðA18Þ

(b)

100

1000

F2

Cavity with a diaphragm and an absorbing ring (Fig. 5b):For this case, as shown in Fig. 5b, the determination of theresonant field requires the knowledge of only two coeffi-cients, the fp’s and bp’s. As previously, the change of theforward coefficients after a round-trip in the cavity includinga diaphragm on the plane mirror, and an absorbing ring canbe described by Eq. (A10) but the typical element Mpm isgiven by

Mpm ¼ ðR1R2Þ1=2X

n

CDpn � CR

nmexp½�2iðnþmþ1ÞfD�, ðA19Þ

where,

CRnm ¼

Z 2Y2A

0expð�ZÞLpðZÞLmðZÞdZþ

Z 12Y2

B

expð�ZÞLpðZÞLmðZÞdZ

ðA20Þ

and

Z ¼2r2

W2ðLÞ

, YA ¼rA

WðLÞ, YB ¼

rB

WðLÞ, ðA21Þ

The numerical computation of overlapping integrals CApm, CD

pn

and CRnm is done using a double precision FORTRAN 77

routine based on the integration subroutine DQDAG fromIMSL. The numerical calculation of the eigenvectors andeigenvalues of the complex matrix M is made using thedouble precision routine DEVCCG. The compiler used is fromABSOFT company.

MD

0 10 20 30 40 50 601

10F1

T

LFM (%)

Fig. 7. Variations of the two possible Transverse Mode Discrimination factors

F1 and F2 as a function of LFM, the fundamental mode losses, which are varied by

changing the aperture diameter.

Appendix 2. Transverse mode discrimination factor-What is itgood for?

Basically, a laser oscillator is made up at least from two mainoptical devices which are the amplifying medium, and the opticalresonator. Although, the amplifying medium plays a role in thebuilding of the resonant field, one can consider at first sight thatthe spatial distribution of the laser light is mainly imposed by thecavity eigenmodes which are assumed to be Laguerre–Gauss LGp0

modes because of the assumed cylindrical symmetry. Note thatthe fundamental mode of the cavity is Gaussian in shape, and has

a beam propagation factor M2¼1. In the design of lasers, it is

important to obtain oscillation in the fundamental mode TEM00

only since the presence of higher-order modes degrades the beamquality. Usually, the suppression of higher-order transversemodes is achieved by inserting in the laser cavity a hard or a softaperture. The latter causes diffraction losses increasing with ordermode p. As a consequence, discrimination ability of the aperturedcavity is studied by only considering the discrimination betweenthe two-first modes, i.e., TEM00 and TEM10 modes. The latter areeigenmodes of the apertured cavity, and are expressed as a linearcombination of the eigenfunctions of the non-apertured resonatoras reminded in the Appendix.

The propagation after a round-trip inside the apertured reso-nator can be represented with a matrix M which holds theinformation about reflection at the mirrors and about clippingat the aperture edge [14]. Its eigenvectors u represent theeigenmodes of the apertured cavity, and each of them is char-acterised by a complex eigenvalue G such that Mu¼Gu. The fieldmodulus (power) associated to a mode changes by the factor 9G9(9G92) after a round-trip. The eigenvector of M having the largest(second largest) eigenvalue G0 (G1) corresponds to the TEM00

(TEM10) mode of the apertured cavity. The TEM00 mode iseffectively selected provided that the losses of the second-ordermode, namely the TEM10 mode, are greater than the laser gain. Itis then convenient to express the discrimination performance ofthe apertured cavity against higher-order mode through a factordenoted F and called Transverse Mode Discrimination (TMD)factor. This number F implies that a large value indicates that ahigh discrimination against higher-order transverse modes isachieved. The TMD factor depends only on eigenvalues 9G09 and9G19, but we find in literature the use of different formulations.The first one F1 is expressed as the ratio of TEM10 and TEM00 losses[15–17],

F1 ¼1� G1j j

2

1� G0j j2

ðA22Þ

The second one F2 results from the ratio of squared eigenvalues9G09 and 9G19 [18],

F2 ¼G0j j

2

G1j j2

ðA23Þ

In the two above definitions, it is usually admitted that a large(small) value of F1 and F2 is the result of a high (low)

Table 6Values of 9G09

2, 9G192, F1, F2 for the two extreme cases: (opened aperture,

LFM¼8�10�3%) and (closed aperture, LFM¼23.6%).

LFM (%) G02

��� ��� G12

��� ��� F1 F2

8�10�3 0.9999 0.9627 433 1.04

23.6 0.7638 2.041�10�2 4.1 37.4

A. Hasnaoui et al. / Optics Communications 285 (2012) 5268–5275 5275

discrimination against higher-order transverse modes. Unfortu-nately, it is possible to come to contradictory conclusions, namelybad or good transverse mode discrimination, depending on whichF factor definition is used as it will be shown in the following.

In order to compare the two TMD factors F1 and F2, we willconsider a plano-concave cavity with a circular aperture againstthe plane mirror. The numerical calculation is performed forl¼1064 nm and a cavity length L¼600 mm. The geometricalparameter g¼(1�L/R) is equal to 0.5 (half-confocal configura-tion), where R is the radius of curvature of the concave mirror.In the following we will consider the variations of the funda-mental mode loss per round-trip LFM¼1�9G09

2 and the two TMDfactors F1 and F2 when the aperture diameter is varied. Forconvenience we will plot the variations of the TMD factors versusthe fundamental mode losses LFM. The result is shown in Fig. 7and the following should be recognised:

(i)

TMD factor F1 stands out against the behaviour of F2 whenthe fundamental mode losses are varied.

(ii)

When the aperture is largely opened, i.e., LFMo1%, thetruncation is negligible, and one can consequently expectthat there is no longer discrimination between transversemodes. As a consequence, one should get a discriminationfactor close to unity. We can note that only F2 fulfils thisrequirement.

(iii)

When the aperture is closed, i.e., LFM is increased, one canexpect also an increase of the Transverse Mode Discrimina-tion because of increase of lateral extent with order mode p.Unfortunately, it is not what is observed with F1 factor. Onceagain, F2 has the expected behaviour since it increases withLFM, i.e., when the aperture is closed.

Except around LFME10%, at which the two TMD factors havethe same magnitude, we can notice that when F1 factor expressesthat the TMD is good (bad), the factor F2 expresses that it is bad(good).

It is therefore of prime importance to discuss which TMDfactor (F1 or F2) is suitable for describing the TMD properties of a

resonator. Although discrimination against higher-order trans-verse modes in a laser cavity is a workhorse of laser physicsamong some decades, we believe, to the best of our knowledge,that such a comparative study of TMD formulations has not yetconsidered. Now, it remains to understand why factor F1

expresses wrongly a good TMD when the diaphragm is opened,and a bad TMD when it is closed. Table 6 summarises, for the twoextreme cases (opened aperture, LFM¼8�10�3%) and (closedaperture, LFM¼23.6%), the values of 9G09

2, 9G192, F1, F2. It is seen

from Table 6, that it is the subtraction operation that appears inthe definition of F1 which is at the origin of the problem when thesquared eigenvalues are close to unity (9G09

2E9G192) or when the

second squared eigenvalue is smaller than the first one(9G09

2b9G19

2).One can conclude that among TMD factors F1 and F2which are

used in the literature, only the factor F2¼9G092/9G19

2 is suitablefor describing correctly the Transverse Mode Discriminationability of a laser cavity. Note that the above considerations onthe definition of the transverse mode discrimination concern onlythe angular-symmetrical modes.

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