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mprovement of the self-Q-switching behavior of ar:LiSrAlF6 laser by use of binary diffractive optics

icolas Passilly, Michael Fromager, and Kamel Aıt-Ameur

It has been shown experimentally and theoretically that Q-switching behavior is possible in a flash-lamp-pumped Cr-doped LiSrAlF6 �Cr3�:LiSAF� laser that consists only of two mirrors, a laser crystal, anda diaphragm. We demonstrate that insertion into a laser of a binary diffractive optical element canspeed up the dynamics of the self-Q-switched laser such that the output pulse is shortened �from 60 to33 ns� and its energy is increased �from 36 to 54 mJ�. The self-Q-switching behavior of the laser has theability to produce a laser pulse with a duration that one can adjust continuously from 60 to 700 ns justby opening the diaphragm. © 2004 Optical Society of America

OCIS codes: 140.3540, 050.1380.

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. Introduction

mode of laser operation that is extensively usedor the generation of high pulse power of some tensf nanoseconds is known as Q switching. Thisechnique can be either active �Pockels cell,cousto-optic modulator� or passive �saturable ab-orber�.1 Both techniques are based on establish-ent of a high loss level before the pulse is generated

o store energy in the amplifying medium by opticalumping because laser action is prohibited. When aow loss level is restored, the stored energy is releasedn the form of a short pulse of light. Active and pas-ive Q-switching techniques require a supplementaryevice inside the laser cavity. Another possibility foreneration of high pulse power is to use a self-Q-witching technique that gives time-dependent lossesith no more components than the two mirrors, the

aser medium, and a hard aperture. Such behavioras been observed experimentally in a flash-lamp-umped Cr:LiSrAlF6 �Cr3�:LiSAF� laser2,3 and theo-etically modeled.4 The mechanism for intensity-ependent losses is based on the combination of aard-aperture and a time-dependent nonlinear lensing

The authors are with the Centre Interdisciplinaire de Rechercheons Lasers, Unite Mixte de Recherche 6637 Institut Superieure dea Matiere et du Rayonnement, 6 Boulevard Marechal Juin,14050 Caen, France. K. Aıt-Ameur’s e-mail address isamel.aitameur@ismra.fr.Received 2 December 2003; revised manuscript received 21 May

004; accepted 15 June 2004.0003-6935�04�265047-13$15.00�0© 2004 Optical Society of America

10

ffect coming from the active medium. This effect isue to direct coupling between the refractive index ofhe laser material and the excited-ion population.he variation in refractive index is assumed to beroportional to the excited-ion population of the up-er level of the laser transition. The correspondingonstant of proportionality K has been experimen-ally determined for ruby �Cr3�:Al2O3� from the out-ut frequency chirp5,6 or from the dynamics of theaser’s far-field divergence.7 Because of the satu-ation mechanism, the Gaussian laser beam in-uces a radial variation of the population inversion,esulting in a lensing effect in the amplifyingedium. This lensing effect is time dependent be-

ause the inversion of population changes withime. The time changes cause the geometricalharacteristics of the laser beam to change; also,hen a hard aperture is inserted into the cavity the

oss level will become a function of time. Let usecall that a converging lensing effect in ruby haslready been experimentally demonstrated.7 Athe time when self-Q-switching in a Cr3�:LiSAFaser was theoretically modeled,4 the type of lensingconverging or diverging� was experimentally un-nown. As there was no reason for assuming aifferent type of lensing in Cr3�:LiSAF from that inr3�:Al2O3, we considered positive lensing in Cr3�:iSAF with proportionality constant K assumed toe proportional to the concentration of active ions.Recently an experiment was performed to charac-

erize the lensing effect caused by coupling betweenhe refractive index and the inversion of population inr3�:LiSAF.8 Contrary to what was expected, a di-

September 2004 � Vol. 43, No. 26 � APPLIED OPTICS 5047

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erging lensing effect was found. As a consequence,e consider again in this paper the modeling of the

elf-Q-switching behavior of a Cr3�:LiSAF laser butith a more-accurate analysis. In particular, we

ake into account the time evolution during the pulsef the mode volume and the M2 factor. After that,e show that the insertion of a binary diffractive

ptics inside the cavity can improve the performancef the pulse that is generated.In Section 2 we give the basics of the theoreticalodeling of the lensing effect, the amplifying me-

ium, diffraction losses, mode volume, and the beamropagation factor �M2 factor�. In Section 3 we con-ider the performance of a self-Q-switching laser thatncludes a hard aperture by determining the energynd duration of the laser pulse. The time evolutionf the mode volume and the M2 factor will also beonsidered. In Section 4 we show that the insertionf a phase aperture inside the hard-apertured laseran improve the dynamics of the self-Q-switching be-avior.

. Modeling of the Laser Cavity

he laser is made up of a plano–concave cavity ofength L including a laser rod and a circular aperture.wo extreme positions can be considered both for themplifying medium and the hard aperture: againsthe plane or against the concave mirror. It has beenhown already that the best position for the laser rods against the concave mirror, and, because we areealing with a negative lensing in the active medium,he best position for the hard aperture is against theoncave mirror4 as shown in Fig. 1�a�. As is dis-ussed below, the transverse profile of the excitedopulation gives rise to a lensing medium whosequivalent diverging lens, of focal length f �t�, is lo-ated against the concave mirror as shown in Fig.�b�.

. Modeling of the Lensing Effect

he laser rod of length Lrod is assumed to be uni-ormly pumped by the flash lamp, and the transverseature of the problem is taken into account; we con-ider that the radial variation of the inversion ofopulation density is due only to the saturationechanism caused by the laser beam. That beam is

ssumed to be Gaussian in shape and to have a widthhat is constant along the rod axis and equal to Wc�t�,he width of the TEM00 mode in the plane of theoncave mirror.

The evolution of the spatiotemporal behavior ofhe inversion of population is different near theaser rod axis �region 1� and at its periphery �region�. Indeed, region 1 contributes to the laser oscil-ation, and consequently the inversion density, de-oted Nd��, t�, has a transverse profile that is

mposed by the saturation mechanism. In region, at the periphery of the oscillating mode, stimu-ated emission is inhibited and the inversion den-ity, denoted Ni, is constant in space and in timeuring the laser pulse emission. Therefore the in-

048 APPLIED OPTICS � Vol. 43, No. 26 � 10 September 2004

ersion density profile in the laser rod can be ex-ressed as

Nd��, t� � Ni � �Ni � N�t��exp��2�2�Wc2�t��, (1)

here N�t� represents the value of the inversion den-ity on the rod axis �� � 0�. It is shown below thathe temporal evolution of on-axis inversion N is de-ermined from the laser rate equations. A variationn��, t� of the refractive index of the laser material is

nduced when the chromium ions are excited, andhis variation is assumed to be proportional to theractional inversion density5–8:

n��, t� � KNd��, t�

NT, (2)

here K is a constant of proportionality that charac-erizes the laser material and NT is the total densityf Cr3� ions. The value of K for Cr3�:LiSAF wasecently experimentally measured from the timeariation of the laser far-field divergence8:

K � 5 � 10�4. (3)

ote that a similar experiment7 with a ruby laserielded a value Kruby � �6 10�6.The beam propagation inside the laser rod was

escribed in the framework of the ABCD formalism,

ig. 1. �a� Sketch of the apertured laser: 1, plane mirror ofeflectivity R1; 2, circular aperture of radius �H; 3, laser rod ofength Lrod; 4, concave mirror of radius of curvature R and ofeflectivity R2. �b�Geometry of the equivalent cavity. The radialariation of the excited population gives rise to a lensing mediumhose equivalent thin lens of focal length f is assumed to be locatedt the entrance of the laser rod.

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nd after some algebra4 it was found that the laserod is equivalent to a thin lens of focal lens f �t�, giveny

f �t� ��2Wc

2�t�K��t�Lrod

. (4)

unction ��t� is positive and increases during themission of a laser spike. It takes the form

��t� �Ni � N�t�

NT. (5)

ote that Eq. �4� shows that the lensing effect undertudy is a converging effect in ruby, whereas it isiverging in Cr3�:LiSAF. Geometrical parameter gf the cavity is time dependent and is given by

g�t� � 1 � L� 1f �t�

�1R� , (6)

here R is the concave mirror’s radius of curvature.he spot size on the concave mirror is time dependentnd is expressed as

Wc2�t� �

�L � 1

g�t��1 � g�t���1�2

. (7)

. Modeling the Amplifying Medium

t is assumed that on-axis �for � � 0� values of inver-ion density N�t� and ��t� are related by rate equa-ions written within the framework of the plane-waveheory. It is important to note that these equationsnclude the saturation mechanism but for a uniformransverse intensity profile. However, the trans-erse nature of the saturation mechanism is takennto account when we assume that the inversion den-ity profile follows Eq. �1�.The Cr3�:LiSAF is a four-level system for which

ate equations take the form9

�N�t

� �N��c �N�

� Wp�NT � N�, (8)

��

�t� ��N�cε1 �

�

tR� �

N�

ε2, (9)

here, c is the speed of light in the laser rod, tR�2�L � �n0 � 1�Lrod���n0c�� is the photon round-tripime, n0 ��1.4� is the refractive index of the laser rod,��3.2 10�20 cm2� is the effective stimulated cross

ection, � ��67 �s� is the fluorescence lifetime of thepper level of the laser transition, � is the photon losser round trip including outcoupling and diffractionosses, ε1 ��2Lrod��ctR�� is a factor expressing that thective medium does not fill up the whole cavity, ε2 isfactor expressing that spontaneous emission par-

icipates to some degree in laser oscillation, Wp is theumping rate, NT ��1.3 1020 cm�3� is the totalensity of active ions, ��t� is the number of photonser unit of mode volume, and N�t� is the number ofnverted ions per unit of crystal volume.

10

. Modeling of Diffraction Effects

oss term �, which appears in Eq. �9�, can be ex-ressed as the sum of two terms:

� � �ln�R1 R2� � �d, (10)

here R1 and R2 are the reflectivities of the cavityirrors and �d is the diffraction loss term that is due

o an intracavity aperture. In what follows, we re-all the basics for the determination of the fundamen-al mode, TEM00, of the apertured cavity. Thisode is expressed as a linear combination ofaguerre–Gauss functions that are eigenfunctions ofhe nonapertured resonator.10

The study of the resonant field involves its decom-osition into its two progressive components: a for-ard beam �in the positive z direction� and aackward beam �in the negative z direction�. Therigin of axial coordinate z is defined by the plane-irror position of reflectivity R1. The circular hard

perture of radius �H is set against the concave mir-or of reflectivity R2 with radius of curvature R. Theagnitude of the beam disturbance is measured by

he beam truncation ratio:

YH � �H�Wc. (11)

he numerical calculation of the resonant field in thepertured resonator is based on the field’s expansionn the basis of the eigenfunctions of the bare cavity,.e., without any aperture. Note that because theavity is assumed to be axially symmetric we areealing with beams that have axial symmetry. Therthonormalized basis is formed by 80 Laguerre–auss functions. The coefficients of this expansionermit calculation of the field anywhere inside andutside, with a considerable economy of computa-ional time in comparison with methods based on theepeated use of the Fresnel–Kirchhoff integral. De-criptions of the calculations are given in Appendix A.The fields propagate in the forward �z � 0� and

ackward �z � 0� directions and are expressed as ainear combinations of the basis functions

Ef��, z� � exp�i�kz � �t�� �p

fp Gfp��, z�, (12)

Eb��, z� � exp�i�k�2L � z� � �t�� �p

bp Gbp��, z�.

(13)

omputation of the forward and backward fields re-uires the determination of two �z-independent� co-fficients, fp and bp, which are related to each other byhe boundary conditions at the mirrors and the ap-rture planes. Their determination involves matrix, which expresses the change of the forward coeffi-

ients after a round trip in the apertured cavity �seeppendix A�:

fp� ��m

Mpmfm. (14)

September 2004 � Vol. 43, No. 26 � APPLIED OPTICS 5049

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he eigenvector of M that has the largest eigenvalue0 corresponds to fundamental mode TEM00, so fp� �

0fp holds for all p. In what follows, �02 will beiewed as the aperture’s round-trip transmission,nd thus the diffraction loss term is given by

�d � �ln��02�. (15)

t is important to note that �02 has to be an increas-ng function during laser spike emission to speed uphoton density ��t�. This increased speed, in turn,esults in a rapid decrease of inversion density N�t�nd thus to a reduction of f , leading to an increasef �02 and so on. This phenomenon can be vieweds a loss avalanche by analogy with the behavior of aaturable absorber and justifies our designating theaser behavior self-Q switching. The challenge nows to find the better choice for all parameters to gethe largest loss avalanche that yields the shortestulse that has the largest energy possible. The bestositions for the laser rod and the hard aperture haveeen already found4: The laser rod should begainst the concave mirror and not depend on theign of K. However, the hard aperture should beositioned against the concave �plane� mirror for K ��K � 0�.The key to performing self-Q switching is the dy-

amics of variation of the aperture’s round-trip trans-ission �02 as a function of focal length f . Figureshows such variation for K � 0 and K � 0 and has

rompted the conclusion that self-Q switching is notossible for K � 0.4 The main argument for such aonclusion was that variations of �02 become signif-cant only for the smallest values of f , i.e., after theaser pulse is over. The laser pulse is then gener-ted with a quasi-constant loss level, and its charac-eristics of pulse duration and energy correspondore to a spiky emission than to a Q-switched pulse.

n this paper it is shown that performing self-Qwitching is also possible for K� 0. In addition, it ishown that the dynamics of variation of � 2 versus

ig. 2. Variations of �02, the aperture round-trip transmission asfunction of f , for K � 0 �negative lensing� and K � 0 �positive

ensing�.

0

050 APPLIED OPTICS � Vol. 43, No. 26 � 10 September 2004

f can be improved by insertion of a phase aperturen front of the plane mirror, yielding a more powerfululse for the same pump energy.So far we have seen that diffraction acts on the loss

evel of the laser cavity, but it is important to notehat diffraction also changes the geometry of the fun-amental mode, which is no longer Gaussian in anpertured resonator. There are two important pa-ameters of the apertured cavity: the M2 factor ofhe output beam and the fundamental mode volumem. The two quantities contribute to the value ofrightness B of the laser pulse11:

B�t� � 2P�t�

�2�M2�t��2, (16)

here P�t� is the instantaneous power of the laserutput. It is given by

P�t� �T��t�Vm�t�EL

tR, (17)

here T is the power transmission of the output mir-or, EL is the photon laser energy, and Vm�t� is theundamental mode volume of the apertured cavitysee Appendix A�. It has already been shown thatntracavity diffraction causes a widening of the beamn the resonator12 and hence an increase of funda-

ental mode volume Vm. Another quantity of ref-rence is volume VG of the fundamental mode of theavity �Gaussian mode� without the aperture. It isefined from Gaussian beam area S�z� � W2�z��2 asollows:

VG � �0

L

S� z�dz. (18)

aking into account Eq. �A3� of Appendix A, we fi-ally find that

VG ��L2

2 �� g1 � g�

1�2

�13 �1 � g

g �1�2� . (19)

ote that calculations of power P�t� in Ref. 4 wereade with VG as the fundamental mode volume. It

s then interesting to make the comparison with vari-tions of Vm and VG of the Gaussian mode volume ofhe nonapertured cavity as a function of geometricalarameter g. The results are shown in Fig. 3, wherehe variation of Vm as a function of g is plotted for twoalues of aperture radius �H. We shall see in Sec-ion 3 that, during the emission of a laser pulse, theariation in focal length f is so large that geometri-al parameter g can vary in a wide range, for in-tance, from 0.06 to 0.8. As a consequence, theesultant huge variation in the fundamental modeolume causes a deformation of the pulse shape thate discuss below.Another parameter that describes the transverse

roperties of the laser beam is the beam propagationactor, usually referred to as the M2 factor. The M2

actor is simply numerically calculated from the co-

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fficients of the field expansion on the basis of theaguerre–Gauss functions as given in Appendix A.deally, the fundamental mode of the nonaperturedavity is Gaussian with beam propagation factor

2 � 1. As the fundamental mode of the aperturedavity is not Gaussian, it is interesting to consider the

2 factor of the beam emerging from the plane mir-or, i.e., Mp

2, and that from the apertured concaveirror, i.e., Mc

2. Figure 4 shows, for two values ofhe aperture radius, the variations of Mp

2 and Mc2 as

function of f . The interesting result is that thewo possible outputs of the laser cavity are not equiv-lent because the corresponding M2 factors are notnly different but vary in different ways as parameteris increased. Note that, during the laser pulse

mission, parameter g is an increasing function ofime.

To make clear the assumptions made in the mod-ling of the self-Q-switched Cr3�:LiSAF laser, let usummarize the three levels of hypothesis used:

�i� The interaction between light and an amplify-ng medium is described in the framework of planeaves for determining the on-axis quantities��t� and�t�.�ii� We model the lensing effect by assuming that

he laser beam is Gaussian inside the rod.�iii� Diffraction effects that occur on the aperture

dge are taken into account for determination ofound-trip transmission �02, mode volume Vm, andhe M2 factor.

s was shown above, the laser model is a combinedodel of these three levels of transversality. It is a

uitable compromise between precision and con-umption of CPU time.Let us now consider the self-consistent character of

he problem because the laser rod is equivalent to aens whose focal length f �t� depends on beam size

c�t�, which in turn depends on the value of f �t�. Forolving this problem, the two variables f �t� and Wc�t�re decoupled. For that, at t � 0, the time of the

ig. 3. Variations of Gaussian mode volume VG �dotted curve�nd of fundamental mode volume Vm for two values of radius �H �.85 mm �solid curve� and �H � 1.1 mm �dashed curve� as a func-ion of geometrical parameter g.

10

eginning of the pump pulse, geometrical parameteris initialized to its value obtained from Eq. �6� inhich f�1 � 0. After that, Eqs. �8� and �9� are nu-erically solved by use of a six-order Runge–Kutta

outine that computes variables N and � at discreteimes tj. Decoupling of the variables is achieved byalculation of focal length f, from Eq. �4� at time tj, byse of the values of Wc and � determined at therevious step, that is, at time tj�1.

. Apertured Laser Dynamics

he numerical calculations are made for the samealues of parameters used in Ref. 4 and in the exper-ments,2,3 except for the value of K for the Cr:LiSAFaser, which is known today8 and is 5 10�4. Let usow summarize these parameters. We consider these of a Cr3�:LiSAF rod of length Lrod � 55 mm,hich is 1.5% chromium doped and transversallyumped by a pulsed flash lamp that gives rise toniform excitation. The pump pulse is a square ofuration tp � 60 �s. As usual, the pump rate isssumed to be proportional to electrical energy Epischarged into the flash lamp:

W � � E . (20)

ig. 4. Variations of the beam quality factor, as a function of g, forwo values of radius aperture �H: �a� Mp

2, characterizing theeam emerging from the plane mirror, and �b� Mc

2, correspondingo the beam emerging from the apertured concave mirror.

p p p

September 2004 � Vol. 43, No. 26 � APPLIED OPTICS 5051

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p is the pumping coefficient, which represents theotal efficiency of the pumping process.13 In all cal-ulations we used �p � 47 J�1 s�1, and pump energyp has the highest value possible that will allow theeneration of a single pulse. In other words, the restf the spiky train is suppressed. This choice for theaser operation was made to be close to the experi-

ental conditions.2,3

The geometrical characteristics of the cavity are� 1.4 m and Rc � 1.5 m. The plane output mirror

as a reflectivity R1 � 0.5, and R2 � 1 for the concaveirror. The laser wavelength is � � 830 nm.Since the size of the beam on the concave mirror

aries with time during the laser emission �owing tohe variation of f �, beam truncation ratio YH�t� �H�Wc�t� is also a function of time. To know whetherhe laser has a strong diaphragm, one must know thealue of YH before the laser pulse begins to grow.his value is denoted YH

i, where the superscript indicates that the value is initial, i.e., a value thatorresponds to the cold cavity � f 3 ��. In what fol-ows, there will be other time-dependent parametershose initial values are identified by superscripts i.Figure 5 shows the variations of pulse energy and

ig. 5. Characteristics of the laser pulse: �a� variations of theulse width �FWHM� as a function of YH

i, �b� variations of the pulsenergy as a function of YH

i for K � 10�4 �circles�, for K � 5 10�4

squares�, and without the lensing effect �triangles�.

052 APPLIED OPTICS � Vol. 43, No. 26 � 10 September 2004

ulse width �FWHM� as a function of YHi. Before

roceeding, it is important to note that for all points ofig. 5 the energy is adjusted such that the laser outputontains only one pulse, at the limit of emergence ofhe second spike. As YH

i is reduced, the aperture sizes reduced and consequently the pump energy in-reases, as shown in Fig. 6. However, the importantroperty illustrated by Fig. 6 is that the pump energys almost the same with or without the lensing effect.

Let us return to Fig. 5, which shows that the lasers not self-Q switched for K � 10�4 because it wasound that the energy and the duration of the pulsesre almost the same with or without the lensing ef-ect. It should be noted that K� 10�4 was the valuesed in Ref. 4 and then the conclusion was that self--switching behavior is possible only for K � 0, i.e.,ositive lensing as in ruby. However, as we statedn Section 2, the lensing effect in Cr3�:LiSAF wasxperimentally identified8 and negative lensing wasound with K � 5 10�4. For this value of K, Fig.

shows that the more the aperture is closed, theetter the laser is Q switched. Indeed, one can ob-erve that pulse duration �pulse energy� goes from05 to 77 ns �4.4 to 30 mJ� when YH

i varies from 1.2o 0.6. At the same time the pump energy variesrom 13 to 31 J. For K � 0, i.e., without lensing, onean observe that the pump energy is multiplied by aactor of �2.4 whereas the peak power is multipliedy a factor of �1.6, when YH

i is reduced from 1.2 to.6. In other words, as the aperture is closed thencrease in the pump energy is used only for compen-ating for the increase in the laser threshold. How-ver, when K � 5 10�4, one will find for the sameonditions that the peak power is multiplied by aactor of �60. As a consequence, these results dem-nstrate that the lensing effect that is due to theoupling between the refractive index and the inver-ion of population in Cr3�:LiSAF combined with aard aperture improves the efficiency of the laser.n other words, aperturing the laser gives rise to

ig. 6. Variations of electrical pumping energy Ep supplied to theash lamp as a function of initial beam truncation ratio YH

i with-ut the lensing effect �triangles� and for K � 10�4 to K � 5 10�4

squares�.

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ignificant enhancement of its output, in contrast tohat could be expected in most lasers.Figure 7�a� illustrates a typical self-Q-switched

ulse shape together with the variation of beam prop-gation factor Mp

2 that characterizes the outputeam from the plane mirror. It can be seen that Mp

2

ecreases continuously during the pulse generationecause beam truncation ratio YH is also a decreasingunction of time. This is so because of the consider-ble variation of the geometrical parameter shown inig. 7�b�. Such a variation in g is a particularity of

he self-Q-switching behavior under study. Theinimum value of g that corresponds to the cold cav-

ty � f3 �� before the laser pulse builds up is equal tomin � 1–1.4�1.5 � 0.066. The maximum value of ghat is reached at the end of the laser pulse dependsn the initial beam truncation ratio YH

i. We haveound that gmax � 0.72 for YH

i � 0.6 and gmax � 0.28or YH

i � 1.0. A time-varying M2 factor in nanosec-nd pulses has been already observed in Q-switchedasers.14,15

Figure 7�b� shows also that fundamental modeolume Vm decreases as the pulse develops. Thisecrease is responsible for the asymmetry of theaser pulse shown in Fig. 7�a� because P�t� is pro-

ig. 7. Time evolution for YHi � 0.8 and Ep � 20.8 J of �a�

nstantaneous output power �solid curve� and Mp2 �dashed curve�

nd �b� geometrical parameter g �solid curve� and fundamentalode volume Vm �dashed curve�.

10

ortional to the product ��t�Vm�t� according to Eq.17�. Indeed, it can be seen that the rise time islightly larger than the fall time, unlike in active Qwitching,16 for which it is rather the fall time of theaser pulse that is larger than the rise time.

It has already been shown4 that self-Q switchings based on the mechanism of a loss avalanche thatccurs more easily for K � 0 than for K � 0 for aiven value of K. Furthermore, Fig. 5 shows thathe Cr3�:LiSAF laser is self-Q-switched for K � 5 0�4 but not for K � 10�4. Consequently, one canxpect that the behavior of the self-Q-switchedr3�:LiSAF laser can be significantly improved ifne speeds up the variation of �02 as a function ofocal length f. To do so, we consider in Section 4he insertion of a phase aperture inside the hard-pertured laser that facilitates the laser’s beingelf-Q switched.

. Hard-Apertured Laser Dynamics with a Phaseperture

nsertion of a diffractive optical element inside aaser cavity is generally done for one of three maineasons: �i� tailoring of intracavity beams17–20, �ii�orrection of thermal aberration in solid-state la-ers,21,22 and �iii� enhancement of transverse modeiscrimination.23,24 In general, these diffractiveptic elements have continuous relief and conse-uently are expensive components. However, bi-ary diffractive optical elements, which are lessxpensive, can also have interesting properties.inary optics are used for beam tailoring25–27 and

or enhancement of the transverse mode discrimi-ation28 and of the fundamental mode volume29 ofn apertured cavity.The phase aperture is made from a transparent

late with refractive index n that has a relief of depth� ��4�n � 1� and diameter 2�PI. As in the hard-perture case, the phase-aperturing effects can beeasured by the ratio

YPI � �PI�W0, (21)

here W0 is the size of the unperturbed Gaussianeam on the plane mirror. Note that the phase ap-rture introduces a �2 phase shift during one trans-ission of the electrical field and, thus, a -phase shift

n the central region of the beam after reflection on thelane mirror. Here we are interested not directly inhe ability of the phase aperture to modify the spatialistribution of the laser intensity but rather in thehase aperture’s consequence for the laser dynamics,s is shown below. It has been demonstrated thatound-trip losses, and consequently the value of �02,re highly sensitive to the parameter YPI in a plano–oncave cavity whose concave mirror is hard aperturedhile a �2 aperture is set against the plane mirror.29

t is worth remembering that losses occur not on thehase aperture but on the edge of the hard aperture.owever, diffraction effects on the edge of the phaseiscontinuity result in a widening or a shrinking of theeam that is incident upon the hard aperture. The

September 2004 � Vol. 43, No. 26 � APPLIED OPTICS 5053

rltbl

vt

ds

msmaadt

Y

FtaWttcvw

wsmYotsIteY

sesp

FaYPI � 3.4, �b� YH � 0.6 and YPI � 1.9.

Fpe

5

esultant spatial filtering on the hard-aperture edgeseads to more-or-less large variations in the value of�02. Our purpose is to use this property to speed uphe variation of �02 as a function of focal length fecause to do so is the key to inducing the necessaryoss avalanche that will yield self-Q-switching behavior.

Figure 8�a� shows that a judicious choice of initialalue YPI

i � 3.4 makes possible the improvement ofwo properties of the apertured cavity:

�i� The initial value, i.e., for f�1 � 0, of �02 isecreased, and that allows more population to betored in the upper level.�ii� The variation of �02 as f varies becomesore steep. Note that, although the resultant

peed-up of the �02 variations versus f is not dra-atic, it will lead to an improvement in the loss av-

lanche because of the nonlinear behavior of themplifying medium. It is important to note that YPIecreases as the pulse develops. YPI varies from 3.4o 1.6 in that case.

Note that, in contrast, a less appropriate value ofi can slow down the variation of � 2, as shown in

PI 0

054 APPLIED OPTICS � Vol. 43, No. 26 � 10 September 2004

ig. 8�b�. It remains now to quantify by how muchhe performance of the laser pulse is improved afterphase aperture is placed against the plane mirror.ith regard to output-beam quality, Fig. 9 shows

hat Mp2 is moderately degraded by the diffraction

hat occurs on the phase aperture. In addition, wean see from Fig. 9 that the fundamental modeolume is increased with respect to that obtainedithout the phase aperture, as shown in Fig. 7�b�.A comparison of the pulse characteristics obtainedith and without a phase aperture inside the laser is

ummarized in Table 1. The calculations wereade for several values of YH

i, and for each of themPI

i has a value that gives rise to the best steepnessf the curve for �02 versus f . However, we haveaken care to keep the value of Mp

2 as close as pos-ible to that obtained with only the hard aperture.ndeed, it has already been observed that fundamen-al mode TEM00 of the cavity including a phase ap-rture can look like a TEM10 or a TEM20 mode whenH

i is less than 1.3.29

Table 1 shows that the effect of the phase aperturelightly increases the pump energy and the pulsenergy and decreases the pulse duration. For in-tance, for YH

i � 0.8 the rate of increase is 6% for theump energy and 250% for the peak power. Conse-

ig. 9. Same as Fig. 7, except that the laser cavity includes ahase aperture �YPI

i � 3.4� against the plane mirror. The pumpnergy is Ep � 22 J.

ig. 8. Time evolution of �02 with a phase aperture �solid curves�nd without a phase aperture �dashed curves�: �a� YH

i � 0.8 andi i i

qat

spwssctetsokh

5

WomtdedpcdiottsteuppnwIid0uabQ

tse�o3mBtav

A

Ffcamrac

1

Tof0T0z

ttwiwtf

a

wdtiitp

wt�L

uently, a given peak power level can be achieved forlower pump energy when an adequate phase aper-

ure is inserted.Undoubtedly, the presence of a phase aperture in-

ide the cavity of a Cr3�:LiSAF laser is able to im-rove the phase aperture’s self-Q-switching behaviorhen its diameter is correctly chosen. Note that

imilar results are expected if the phase aperture iset in the plane of the hard aperture against a con-ave mirror. However, we can expect better resultso be obtained if we optimize the diffractive opticallement by finding the better-adapted phase aberra-ion profile that leads to the steepest curve �02 ver-us f . However, the beam quality factor of theutput beam should be not forgotten because it is wellnown that the M2 factor of a laser beam can beighly degraded by traversing a phase aberration.30

. Conclusions

e have investigated the self-Q-switching behaviorf a flash-lamp-pumped Cr3�:LiSAF laser. Theechanism for intensity-dependent loss is based on

he combination of a diaphragm and a time-ependent nonlinear lensing effect. The lensingffect is due to coupling between the refractive in-ex of the amplifying medium and the excited-ionopulation. This coupling is characterized by aonstant K, and the laser behavior is quite different,epending on whether K is positive �diverging lens-ng� or negative �converging lensing�. In a previ-us paper,4 Fromager and Aıt-Ameur concludedhat self-Q switching is not possible for K � 0. Inhis paper we have demonstrated that self-Qwitching occurs if K � 0 is sufficiently high. Forhese simulations the value K � 5 10�4 that wasxperimentally measured for Cr3�:LiSAF8 wassed, and the results show that self-Q-switchingulses can be generated. For instance, one gets aulse with an energy of 36 mJ and a duration of 60s for a pump energy of 36 J. The cavity lengthas L � 1.4 m, and the rod length Lrod was 55 mm.

t is interesting to note that the pulse durationncreases continuously from 60 to�700 ns when theiaphragm is opened; i.e., YH

i varies from 1.2 to.55. In other words, the laser’s behavior contin-ously changes from free running to Q-switchings the aperture is closed. Such behavior was noteen observed with the usual active or passive-switching techniques.We have shown that the insertion of a phase aper-

ure inside the laser can speed up the dynamics of aelf-Q-switched laser such that output pulse is short-ned �from 60 to 33 ns� and the energy is increasedfrom 36 to 54 mJ�. Note that this improvement isbtained for a pump energy that increases only from6 to 37 J and for an M2 factor that keeps the sameagnitude at the time of the laser pulse maximum.etter results should be obtainable if one optimized

he diffractive optical element by finding the phaseberration profile that yields the steepest curve �02ersus f but keeps a low value for the M2 factor.

10

ppendix A

ormulas and methods used in the calculation of dif-raction effects are described in this appendix. Weonsider a plano–concave cavity and three cases ofperturing: �i� a circular aperture against a concaveirror, �ii� a circular aperture against a concave mir-

or or a plane mirror, and �iii� a phase aperturegainst a plane mirror and a diaphragm against aoncave mirror.

. Round-Trip Operator

he study of a resonant field involves decompositionf the field into its two progressive components: aorward beam, which propagates in the direction z �, and a backward beam for the opposite direction.he position of the plane mirror defines the origin z�of the axial coordinate. The cavity axis defines the

ero of the transverse coordinate �.The numerical calculation of the resonant field in

he apertured resonator is based on its expansion onhe basis of the eigenfunctions of the bare cavity, i.e.,ithout any aperture. Note that because the cavity

s assumed to be axially symmetric we are dealingith beams that have axial symmetry. The or-

honormalized basis is formed by 80 Laguerre–Gaussunctions, which are written for the forward beam as

Gfp��, z� � �2 �

1�2 1W

Lp�2�2

W 2�exp�� �2

W 2�� exp��i� k�2

2 Rc� �2p � 1� �� (A1)

nd for the backward beam as

Gbp��, z� � �2 �

1�2 1W

Lp�2�2

W 2�exp�� �2

W 2�� exp��i� k�2

2 Rc� �2p � 1� �� , (A2)

here k � 2 ��. Hereafter, the subscripts f and benote forward and backward quantities, respec-ively. The Gaussian mode of the nonapertured cav-ty is characterized by its beam diameter 2W�z� andts radius of curvature Rc at point z. These quanti-ies of reference, as well as phase shift , are z de-endent and obey the following formulas:

W 2� z� � W02�1 � � z�z0�

2�, (A3)

Rc� z� � z�1 � � z0�z�2�, (A4)

� z� � arctan� z�z0�, (A5)

here z0 � W02�� is the Rayleigh range and W0 is

he beam-waist radius, expressed by W02� ��d� ��g�

1 � g��1�2 for our plano–concave cavity. Lp�X� is aaguerre polynomial of order p.The forward and backward fields are assumed to be

September 2004 � Vol. 43, No. 26 � APPLIED OPTICS 5055

ln

Wt

o

Tq�toeia

AAii

Tp

ii

NspcTo

w�i

RNwti

Frr

w

Er

5

inearly polarized and are expressed as linear combi-ations of the basis functions

Ef��, z� � exp�i�kz � �t�� �p

fp Gfp��, z�, (A6)

Eb��, z� � exp�i�k�2L � z� � �t�� �p

bp Gbp��, z�.

(A7)

e are studying the stationary field for t � 0 andhen exp��i�t� � 1.

Note that the functions of the basis satisfy therthonormalization condition

2 �0

�

Gfp��, z�Gfm*��, z��d� � �pm, (A8)

2 �0

�

Gbp��, z�Gbm*��, z��d� � �pm. (A9)

he symbol * denotes the complex conjugate of theuantity. The aim of the computation is to find the�- and z-independent� coefficients fp and bp. To findhe matrix M that represents the round-trip operator,ne has to find out how the coefficients are related toach other at the phase–amplitude planes. Let usntroduce �H���, the radial transmittance of the hardperture of radius �H:

�H��� � �1 � � �H

0 � � �H. (A10)

. Hard Aperture Against the Concave Mirrort z � L, coefficients bp can be deduced from the

nversion of Eq. �A7�, which is obtained in the follow-ng steps:

�i� Equation �A7� at z � L is written as

Eb��, L� � exp�ikL� �p

bp Gbp��, L�. (A11)

he boundary condition imposed by the diaphragm atlane z � L is expressed as

E ��, L� � � ���R E ��, L�. (A12)

Table 1. Pulse Characteristics of the Self-Q-Switched

Pulse Characteristic 0.55 �3.7� 0.6 �3.

Pulse energy with PA �mJ� 53.4 37.4Pulse energy without PA �mJ� 35.8 29.5FWHM with PA �ns� 33.5 41.7FWHM without PA �ns� 57.4 77.3Peak power with PA �kW� 1138 634Peak power without PA �kW� 501 305Ep with PA �J� 37.1 32.3Ep without PA �J� 36.5 31.6

Note: The numbers in parentheses are YPIi.

b H 2 f

056 APPLIED OPTICS � Vol. 43, No. 26 � 10 September 2004

�ii� Equation �A11� is multiplied by Gbp*��, L� andntegrated over plane z � L. We finally obtain, tak-ng into account Eq. �A12�,

bp � 2 R2�m

fm �0

�A

Gbp*��, L�Gfm��, L��d�.

(A13)

ote that, as the concave mirror is an equiphaseurface, phase term exp�ik�L � �2�2R��, which ap-ears in the product Gbp*Gfm, is constant over theoncave mirror’s surface and is equal to exp�ikL�.he relationship between the values of bp and fm isbtained from Eq. �A13�:

bp � R2�m

CpmHfm exp��2i�p � m � 1� L�,

(A14)

here L is the value of phase shift , given by Eq.A5�, at plane z � L and Cpm

H is given by the follow-ng overlapping integral:

CpmH � �

0

2YH2

exp��X�Lp�X�Lm�X�dX. (A15)

educed variable X is an abbreviation for 2�2�W2.ow let us express the relationship between the for-ard and backward coefficients that is obtained from

he inversion of Eq. �A6� at plane z � 0. The results

fp � R1 exp�2ikL�bp. (A16)

inally, if one links Eqs. �A14� and �A16�, one finds aelation between the forward coefficients after aound trip in the resonator:

fp� ��m

Mpm fm, (A17)

here matrix element Mpm is defined by

Mpm � R1R2 CpmH exp�2i�kL � �p � m � 1� L��.

(A18)

quation �A17� allows us to define matrix M thatepresents the round-trip operator. This matrix

:LiSAF Laser with and without a Phase Aperture �PA�

lue of Beam Truncation Ratio YHi

0.7 �3.5� 0.8 �3.4� 0.9 �3.0� 1.0 �2.8�

29.8 23.9 16.3 11.519.3 15.2 12.8 9.791.6 140.2 248.2 421.2

149.1 235.2 261.9 317.0258 135 54 23105 53 39 2526.3 22.0 18.4 16.324.7 20.8 18.6 16.8

Cr3�

Va

6�

hadfmec

Tw

Ntss0bscc

BTt

Er

Afi

Tt

Ib

it

w

tvatgttmitt

CLcapa�amsm trrp

Ttwt

Ttdb�

olds all the information about amplitude truncationt the edge of the aperture. As the resonance con-ition states, that means that relation fp� � �fp holdsor all p. In other words, one can see that eigen-

odes of the apertured cavity correspond to the eig-nvectors u of matrix M. Each of them isharacterized by a complex eigenvalue � such that

Mu � �u. (A19)

he eigenvector of M that has the largest eigenvalue�0 corresponds to the fundamental mode TEM00hose power loss per round trip is given by

LFM � 1 � �02. (A20)

ote that the first eigenvector is numerically ob-ained by an iterative application of matrix M on atarting vector3 that has a single component corre-ponding to the Gaussian mode, i.e., column vector �1,, 0, . . .�. This numerical method converges quicklyecause the determination of the first eigenvector ofquare matrix M of size �80 80� does not require theomputation of the 79 remaining eigenvectors be-ause its eigenvalue is by definition the largest.

. Hard Aperture against the Plane Mirrorhe aperture imposes the following boundary condi-

ion:

Ef��, 0� � �H���R1 Eb��, 0�. (A21)

quation �A6� can be inverted to give, at z � 0, theelationship between the values of fm and bp:

fm � R1 exp�2ikL� �p

bp CpmH. (A22)

t z� L, the relation between backward and forwardelds is given by

Eb��, L� � R2 Eb��, L�. (A23)

he inversion of Eq. �A7� yields for the relation be-ween bp and fp

bp � R2 exp�2ikL�exp��2i�2p � 1� L� fp. (A24)

f one links Eqs. �A22� and �A24�, one gets a relationetween the backward coefficients after a round trip

10

n the cavity that permits one to define matrix M withypical element Mpm such that

Mpm � R1R2 CpmH exp�2i�kL � �2p � 1� L��,

(A25)

hich gives

bp� ��m

Mpmbm. (A26)

A comparison of Eqs. �A18� and �A25� shows thathe modulus of the round-trip operator has the samealue whether the aperture is against the plane orgainst the concave mirror. However, the phaseerm is different, and the consequence is that for aiven value of beam truncation ratio YH the aper-ured cavity has the same loss level and discrimina-ion factor between the first two transverse modes noatter which mirror is apertured.12 The difference

n diffraction effects appears in the spatial distribu-ion of intensity inside12,32 and outside33 the aper-ured cavity.

. Hard-Apertured Cavity with a Phase Apertureet us consider the plano–concave cavity whose con-ave mirror is hard apertured and in which a phaseperture is inserted against the plane mirror. Thehase aperture is made from a transparent plate withrefractive index n and has a relief of depth e �

�4�n � 1� and of diameter 2�PI. As for the hard-perture case, the phase-aperturing effects can beeasured by the ratio YPI � �PI�W0, where W0 is the

ize of the unperturbed Gaussian beam on the planeirror. Note that the phase aperture introduces a�2 phase shift during one transmission of the elec-rical field and thus a phase shift in the centralegion of the beam after reflection on the plane mir-or. The phase aperture is characterized by its com-lex transmittance, given by

�PI��� � ��i � � �PI

�1 � � �PI. (A27)

he boundary condition imposed by the hard aper-ure on the concave mirror is given by Eq. �A12�,hereas that associated to the phase aperture is writ-

en as follows:

Ef��, 0� � R1 �PI2���Eb��, 0�. (A28)

he relationship between the values of bp and fm ishe same as that given in Eq. �A14�. Now let usetermine the relation between the forward andackward coefficients. For that, the inversion of Eq.A6� at z � 0, with Eq. �A28� into account, leads to

fn � 2 R1 exp�2ikL�

� ���0

�PI

�p

bp Gbp��, 0�Gfn*��, 0��d�

� ��

�

�p

bp Gbp��, 0�Gfn*��, 0��d�� . (A29)

PI

September 2004 � Vol. 43, No. 26 � APPLIED OPTICS 5057

L

Iii

Tei

2

Iacfmwaai

w�h

I8t

cm

Epc

3

TrTdebw

�f

H

H

H

4

TLci

wqgcaGAmmdg

R

5

et us introduce the following abbreviation:

CpnPI � �

0

2YPI2

exp��X�Lp�X�Ln�X�dX. (A30)

f YPI3 �, then CpnPI3 �pn; then the second integral

n Eq. �A29� can be expressed as a function of the firstntegral. Finally, one finds that

fn � R1 exp�2ikL� �p

bp��pn � 2CpnPI�. (A31)

he link between Eqs. �A14� and �A31� allows one toxpress round-trip operator M, whose typical elements written as

Mpm � R1R2 exp�2i�kL � L�� �n��pn

� 2CpnPI�Cnm

H exp��2i�n � m� L�. (A32)

. Fundamental Mode Volume

t has already been shown that intracavity phase–mplitude diffraction causes a widening of the intra-avity beam17,18 and hence an increase of theundamental mode volume,12,29 denoted Vm. The

ode volume is computed on the basis of effectiveidth Weff of the fundamental mode and its variationlong the cavity axis. To get the variation of Weff asfunction of z, one needs intensity distribution I��, z�

nside the cavity, which is given by

I��, z� � Ef��, z� � Eb��, z�2, (A33)

here Ef and Eb are calculated from Eqs. �A6� andA7�. The effective width is deduced from the top-at criterion, which is expressed as

�0

Weff

I��, z��d� � 0.86 �0

�

I��, z��d�. (A34)

ntensity distribution I��, z� has been computed for00 values of �, and the equality in Eq. �A34� is de-ermined within a tolerance of 2%.

The fundamental mode volume of the aperturedavity is then deduced by analogy with the Gaussianode volume defined by Eq. �18�:

Vm �

2 �k�1

100

Weff2� zk�z. (A35)

valuation of Eq. �A35� is made for 100 longitudinalositions zk, and z � L��99� is the longitudinal in-rement.

. Output Beams

he cavity has two outputs, one from the plane mir-or side and the second from the concave mirror side.o characterize the two output beams, one has toetermine, for each output, the coefficients of the fieldxpansion from the knowledge of bp and fp and theoundary condition imposed by the output mirror,hich can be apertured or not. Let us denote by 1h

p

058 APPLIED OPTICS � Vol. 43, No. 26 � 10 September 2004

2hp� the coefficients for the output beam emergingrom the plane �concave� mirror:

ard aperture against the concave mirror

1hp � 1 � R1 bp, (A36)

2hp � 1 � R2�q

CqpHfq exp�2i�p � q� L�. (A37)

ard aperture against the plane mirror

1hp � 1 � R1�q

CqpHbq, (A38)

2hp � 1 � R2 fp. (A39)

ard aperture with a phase aperture

1hp � 1 � R1�q��qp � �i � 1�Cqp

PI�bq, (A40)

2hp � 1 � R2�q

CqpHfq exp�2i�p � q� L�. (A41)

. M2 Factor

he output beam is a coherent superposition ofaguerre–Gauss beams weighted by Dp, the coeffi-ient of the expansion. In that case the Mout

2 factors expressed simply as follows34:

Mout2 � ���

p�2p � 1�Dp2�2

� 4��p�

qp�Dp*Dq�

r�p,q�1�2�1�2

, (A42)

here the superscript r stands for the real part of theuantity and the asterisk means the complex conju-ate. Following the output beam that we consider,oefficient Dp represents 1hp or 2hp. Equation �A42�pplies to a coherent superposition of Laguerre–auss modes, all of which have the same frequency.n incoherent superposition of Laguerre–Gaussodes arises when a stable-cavity laser oscillates si-ultaneously in several Laguerre–Gauss modes with

ifferent frequencies. In this case the beam propa-ation factor is reduced35 to Mout

2 � ¥p �2p � 1�Dp2.

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10

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