Entropic propagation of Gaussian and super-Gaussian-like beams

15 February 2000

Ž .Optics Communications 175 2000 1–12www.elsevier.comrlocateroptcom

Entropic propagation of Gaussian and super-Gaussian-like beams

E. Alberdi a, M. Lehman a,b,), R. Torroba c, M. Garavaglia c

a ´ ´Laboratorio de Optica, Depto de Fisica, UniÕersidad Nacional del Sur, Bahia Blanca, Argentinab Sofilab SACV, Mexico DF., Mexico´

c ( )Centro de InÕestigaciones Opticas CIOp and UniÕersidad Nacional de La Plata, La Plata, Argentina

Received 24 November 1998; received in revised form 23 November 1999; accepted 30 November 1999

Abstract

An entropic formulation to describe the free propagation of Gaussian beams, in a similar way to the thermodynamicŽ . Ž . Ž .theory is developed. We consider two basic applications: 1 an extension to super-Gaussian-like SGL beams, and 2 the

Ž .effect of lenses convergent and divergent on the propagation of Gaussian beams. We are interested in such applicationsbecause the SGL profiles are obtained through the convolution product using rectangle and Gaussian functions and so, theycan be related to the Gaussian beams. The propagation in the Fresnel and Fraunhofer regions are studied, obtaining the lawsfor the optical entropy of the system. Also, we include some properties and a brief discussion about the condition underwhich the beam can be considered as an isolated system. For both applications, the evolution of the characteristic width isderived from the entropic postulates. q 2000 Published by Elsevier Science B.V. All rights reserved.

1. Introduction

The concept of information plays an essential rolein modern science and engineering. It is of greatimportance in communications, that is transmissionof messages by acoustical, mechanical, electrical oroptical signal means, but also in the fields of com-puters, automation and heredity.

It is clear that the optical information is carried bythe electromagnetic beams originated from coherentor incoherent sources. In the case of coherent orquasi-coherent beams, which are important for themany applications that they give rise to, differentprofiles of intensity distribution have been proposed

w xand studied 1–8 . The super-Gaussian beams repre-

) Corresponding author. E-mail: [email protected]

sent an area of interest and attraction on which manyw xauthors have focussed in the last years 9–14 . They

are important because higher energy that in theGaussian case are obtained, and they are emitted bya wide class of laser systems including lasers with

w xhigh gain and saturated active media 15,16 .An approximation that uses Gaussian beams,

Ž .called super-Gaussian-like SGL beam has recentlybeen developed for the study of approximatedsuper-Gaussian coherent-beam propagation in the

w xFraunhofer region 17 . The field profile is obtainedas a convolution between a rectangle and a Gaussianfunctions. The relationship between the characteristicwidth influences on the final intensity distribution. Itis related to the super-Gaussianity index and thewidth of the super-Gaussian profile to which ap-proaches. The general theoretical problem for obtain-ing the beam profiles with uniform energy distribu-

0030-4018r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0030-4018 99 00732-4

( )E. Alberdi et al.rOptics Communications 175 2000 1–122

tion can also be considered as another application ofthe present study. Such applications include laserbeams with high-irradiance distributions to be usedin material processing, laser material interaction

w xstudies and lithography 18,19 .Furthermore, we can relate some optics applica-

tions with the concept of spatial entropy, in a similarway to that developed by other authors. Classically,the thermodynamic entropy function is defined as a

w xmeasurement of disorder in the system 20–22 .However, some interesting optical applications of theentropy function exist, such as light propagationw x w x w x23–25 , speckle 26 and image processing 27,28 .The entropy concept has also been used to obtain a

w xmeasurement of optical coherence 29 , a foundationof the angular change in Gaussian and super-Gaus-sian beams occurring when they are reflected from a

w xplane surface 30 and related with polarization stud-w xies 31 .

w xFrom the statistical point of view 32,33 , theoptical entropy for the element i, j of a system hasbeen defined as:

SS i , j syI i , j ln I i , j , 1Ž . Ž . Ž . Ž .Ž .where I i, j is the normalized intensity level de-

tected by a sensor in the i, j position for a matricialŽ .distribution of cells photodiodes . It is useful to

compare in this case the photodiodes distribution in aCCD camera with the cells defined by some authorsto introduce the microstate and macrostate concepts,which are important for a basic understanding of the

w xphysical meaning of entropy function 22 . In ourcase, each microstate of the system will be character-ized by the possible gray levels that can take eachphotodiode, and the general macrostate will definethe characteristic parameters of this system.

In this work, we use the entropy concept to studythe propagation of a coherent Gaussian and SGL-beams, and to characterize in this way changes in thewidth along the propagation on the optical axis. Thephenomenon is similar to an irreversible polytropicprocess in the thermodynamics sense, which is inprinciple easy to deal with, but a local entropy canbe defined if we want to reduce it as a more simpleprocess that successively take metastable states inthe evolution system. The possibility of a simpleexperimental setup, besides the simple numericalimplementation permits us to relates the different

concepts as entropy, width of beam and super-Gaus-sian index. Also, we have to take into account theeffects of lenses placed in the trajectory of beam, toobtain in this way the change in the characteristicwidth from the entropic formalism that we are devel-oped previously. Here, only these applications arestudied, without considering other constraints such asthe case of wave-guides, resonant cavities, mirrors orinteraction with some type of surface. Later applica-tions involving these elements and the interactionwith more complex structures are enumerated in theconclusions.

2. Mathematical basis

For the Gaussian beam case, which is representedusing the scalar diffraction theory, the spatial distri-

w xbution of the electromagnetic field is given by 7 :

Gauss r ;W zŽ .g

2U W r0 0s exp y 2W z W zŽ . Ž .g g

2r=exp yj k y j k zq j z z , 2Ž . Ž .

2 R zŽ .

U being a constant factor, r 2 sx 2 qy2 is the0

transversal position to the propagation direction alongthe z axis, k is the wavenumber and the characteris-tic beam parameters are:

° 2zW z sW 1q ,Ž . )g 0 ž /z0

2z~ 0 3Ž .R z sz 1q ,Ž . ž /z

zy1z z s tan .Ž .¢ z0

Ž .It is known that W z is the transversal width,gŽ . Ž .whose initial value at zs0 is W 0 sW , R z isg 0Ž .the wavefront radius of curvature, z z is the phase

retardation and z the Rayleigh range. We will0

assume the validity of the maximum principle for the

( )E. Alberdi et al.rOptics Communications 175 2000 1–12 3

total entropy to study the propagation of SGL-beams,relating the entropy function with the main parame-ters of the beam in each case. Fig. 1a shows aGaussian profile of the type used for our calcula-tions. The formalism will be developed based onpostulates and corollaries that will allow us to studythe problems of interest using the optical entropyfunction. Here, only free propagation is considered;then, because the beam has no interaction with an-other systems, it can be considered as an isolatedsystem in the thermodynamics sense.

2.1. Entropic formalism for the propagation ofGaussian beams

Postulate 1. There exists an adimensional func-( )tion SS r , called optical entropy of the beam, de-

Ž . Ž .Fig. 1. a Two-dimensional Gaussian profile, and b the corre-spondent entropy function.

fined as a function of the spatial coordinates, whichtakes an extremal Õalue on the corresponding coor-dinate to the propagation direction of the beam. Theoptical entropy function, at a transÕersal plane tothe optical axis, can be defined as:

I r I rŽ . Ž .SS r sy ln , 4Ž . Ž .

I I0 0

where I is a normalization factor, which can be0

either the total intensity of beam at the exit of source(or the intensity for the central component on the

)propagation direction . Here, we are interested inbeams with reflection symmetry or cylindrical sym-metry around the optical axis. Fig. 1b shows thecorresponding entropy function for the Gaussian

( )profile of Fig. 1a calculated using definition 4 .

Postulate 2. The propagation of a coherentmonochromatic beam into a homogeneous medium,isotropic, stationary and without losses is equiÕalentto an irreÕersible polytropic process, in a similarway to the thermodynamics case.

If we study a directional beam originated from acoherent source, it can be statistically representedthrough a photon gas that leaves the recipient at veryhigh pressure and it is expanding towards the infinitespace. If we use as normalization factor the value ofthe central component I in the propagation direc-0

tion, the beam can be studied as an isolated system.Fig. 2a shows the evolution of the transversal inten-sity distribution along the propagation direction for acoherent monochromatic Gaussian beam.

We introduce the polytropic processes for aGaussian beam in relation to the corresponding ther-modynamic process for a Gaussian beam as shownin Fig. 2b, and later to include the behavior ofSGL-beams. In this figure can be observed the rela-

Žtionship between the mean intensity proportional to.the radiation pressure and the characteristic width of

beam while is propagating along the z direction. Inthis way, a similarity exists between these facts and

Ž .the curves P–V pressure-volume of thermodynam-ics, considering different values for the Rayleighrange. We can obtain a similar relation to that usedin thermodynamics. With the mean intensity of a

Ž .beam proportional to the pressure radiation and the


Ž .Fig. 2. a Intensity distribution for a Gaussian profile along theŽ .propagation direction, b mean intensity as a function of the

width for different Rayleigh ranges.

Ž .characteristic width proportional to the volume ,where the 99% of the energy is concentrated. Wefinally have:

p² :I x 2 D sK , K : constant value, 5Ž . Ž . Ž .that characterize such processes. For our case, arelation with an isothermal process can be obtainedbecause ps1. This is an approximation because themean intensity is calculated into the finite intervalw x w xyD,D and not into the total interval y`,` .

Corollary 1. For a beam characterized by thepreÕious postulate, it is Õerified that the total entropyin the r-space is increased in the free propagationalong the optical axis. The total entropy is defined asa function of the spatial intensity distribution, andfor our case is:

q` I r I rŽ . Ž .TSS r sy ln dr , 6Ž . Ž .H

I Iy` 0 0

which under certain conditions can take extremalÕalues.

We will always distinguish between two terms:Ž .1 entropy, applied to the entropy distribution alongthe transversal coordinate according with Postulate 1,

Ž .and 2 total entropy, applied to the integral of thesame function on this coordinate, in accordance withCorollary 1.

Then, in accordance with the maximum entropyprinciple for the thermodynamics case, for the initialŽ . Ž .i and final f states, or positions along the trajec-

w xtory of the system, it is verified that 22 :

ST r yST r )0 . 7Ž . Ž . Ž .f i

The entropy evolution with the propagation in aGaussian beam is illustrated in Fig. 3a and thecorresponding to total entropy is shown in Fig. 3b.

Ž . Ž .Fig. 3. a Entropy function along the propagation direction, btotal entropy for different position on the propagation direction.


We should point out the important effect of thenormalization factor I in the previous definition. If0

ŽI is the total energy of the beam which remains0.constant along the propagation then, for the calcula-

tion of total entropy function we need to take intoaccount the source contribution. But, for our case,we consider that the normalization factor refers tothe intensity for the traversal coordinate to the propa-

gation direction. That is, we know the propagationdirection of the beam and so, this component willhave entropy equal to zero and the remaining oneswill be referred to it. In the last case the beampropagation begins at zs0, with an initial entropygiven by the source, which increases for valuesz)0. In such case it is possible to consider the beamas an isolated system.

Ž . Ž .Fig. 4. Initial and final states of a Gaussian beam: a laser spot for two different positions from the source. b Intensity profile for bothŽ .cases. c The corresponding entropy function.


The quantification and direction of such variationare obtained considering two states of the systemŽ .initial and final . Fig. 4 shows two different statesof the laser spot obtained with a CCD camera andthe entropy function for both cases. We can see forthese cases that the peaks for the entropy functionare related with the lateral expansion of beam. Then,we can relate the distance between them with acharacteristic beam width.

3. Applications to simple systems

In accordance with Postulate 1, we should keep inmind that the entropy function varies with the propa-

Ž .gation direction. As we pointed after Eq. 7 thecentral coordinate xs0 have a minimum entropybecause we reference all rays respect to the central

Ž .ray along the z axis . This means that the necessaryinformation to determine the trajectory of the centralray is 0. But, also we need to use a formal expres-sion to obtain the position for the extremal values ofentropy function, according with Postulate1. If weconsider, as approximation, that each differentialdisplacement of the wave front have a constant valuein the entropy function, we can determine the posi-tion of extremal entropies, which can be expressedthrough the derivative on n direction:ˆ

ES x , z ES x , zŽ . Ž .nP= S x , z sn qn s0Ž .ˆ x zE x E z

8Ž .

where ns n ,n is the unitary vector normal toŽ .ˆ x z

the wave front. As a direct basic application we areinterested to relate the width of beam in propagationalong z-axis and the entropy function, obtaining thefollowing conditions:

° E I x , z E I x , zŽ . Ž .x qz s0

E x I E z I0 0

~ 9Ž .I x , zŽ .

ln q1s0¢ I0

The first condition is true for z™`, which is com-patible with the fact that for this position the beam isa plane wave, and for this case we have a stationary

Ž .state entropy variation equal to 0 . The secondŽ .condition in Eq. 9 is related with the intensity

Ž .decay for exp y1 . We apply this development fortwo interesting cases which involve Gaussian pro-files and permits us the calculation of width forSGL-beams and focussing Gaussian beams by lenses.

3.1. SGL-beams propagation in the free space

Let us consider one-dimensional beam profilesobtained through the convolution between Gaussian

w xand rectangular functions. That is 17 :

w x w x w xSGL x ;W ,W s rect x ,W )Gauss x ;W ,r 0 r 0

10Ž .being W and W the characteristic widths of eachr 0

function. This operation allows to obtain profileswith a uniform energy distribution that, from thetheoretical point of view, can be interesting forapplications in lasers for material processing or toapproach the study of super-Gaussian beams. Thedifferent profiles, obtained through the convolutionproduct, represents an approximation to the distribu-

2 ation exp y xrW as shown in Fig. 5 for differ-Ž .ent super-Gaussianity indexes, given the adequatelyparameters to W and W .r 0

Because it is very complex to analyze and associ-ate the different processes that are considered with

Ž .Fig. 5. Approximation to SGL-beams. a Convolution productŽ .between rectangular and Gaussian functions, b approximations

obtained for super-Gaussian profiles with different index a.


the corresponding propagation of super-Gaussianbeams in thermodynamics, we only present this ap-proximation comparing with ideal gases, and we willleave a more complete analysis for a later article.

Corollary 2. The expansion of a beam is carriedout in the direction of maximum local entropy andthe initial position of extremal that indicates suchexpansion depends on the super-Gaussianity index.

The entropy function for different profiles can beseen in Fig. 6a for the position zs0; also, thesuper-Gaussian indexes, which corresponds to theapproximation of super-Gaussian profiles, are indi-

Ž .cated in each case as1, as2 and as5 . Asexpected, a high correlation exists between the inten-sity distribution and the entropy function, since it isknown from the classical thermodynamics that both

Ž .formulations energetics and entropics are equiva-lent. Fig. 6b shows the derivative for each case,which indicates that the entropy variation is on thetransversal coordinate direction. But evidently, to beable to characterize a beam from the entropy point of

Ž . Ž .Fig. 6. a Entropy function for transversal coordinate and bŽ .derivative for each case in a . We approximate for different

super-Gaussian indexes.

view they are required the initial and final states ofthe system, as the thermodynamic case.

To obtain the functional structure for the propaga-Ž .tion of SGL-beams defined in Eq. 10 , we must

develop the convolution product between the initialprofile and the corresponding Fresnel exponential.The propagation of a Gaussian beam into the Fresnel

Ž .region is given by Eq. 2 . Due to the associativity ofthe convolution product, the propagation of SGL-beams into the Fresnel region can be studied starting

Ž .from the definition given by Eq. 10 , which finallyis obtained solving the integral:

SGL x ,W ,W zŽ .r g

X 2U W x yxqW0 0 rs exp yH ž /W z W zŽ . Ž .yWg gr

2Xx yxŽ .=exp yj k y j k zq j z z dx ,Ž .

2 R zŽ .11Ž .

and then, the solution for the propagation along the zaxis in the Fresnel region can be analytically ob-tained. The intensity distribution at a transversalplane to the propagation direction is given by:

w xŽ .I x ;W ,W zSGL r g

2 Ž .p U W R z0 0s

2 2 42 Ž . Ž .4 R z q k W z( g

2' Ž . Ž .Ž .W y x 2 R z q i k W z2 Ž .r g= erf

22 Ž . Ž . Ž . Ž .W z R z 2 R z y i k W zw x(g g

22' Ž . Ž .Ž .W q x 2 R z q i k W z2 Ž .r g

qerf .22 Ž . Ž . Ž . Ž .W z R z 2 R z y i k W zw x(g g

12Ž .

As an example, the beam propagation represent-ing an approximation to super-Gaussian profile with

Ž .index as8 W s1.95, W s0.26 is shown in Fig.r 0

7 for different positions on the propagation directionŽ .Nfs10, 5, 1 and 0.1 , being the Fresnel number Nfdefined through:

2W q3 W zŽ .r g

Nfs , 13Ž .l z


Fig. 7. Transversal intensity distribution for the propagation ofŽ .super-Gaussian-like profile as8 , characterized with the Fresnel

Ž . Ž . Ž . Ž .number: a Nf s10, b Nf s1, c Nf s0.5, d Nf s0.1. DotŽ .lines indicate the entropy distribution and constant value exp y1 .

being l the wavelength of the monochromatic beam.In each case, the entropy function is shown in ac-

Ž .cording with the second condition of Eq. 9 , whichŽ .have maxima values at exp y1 . This constant value

is related with the entropic width that characterizesthe beam propagation.

Fig. 8a shows the evolution of mean intensity as aŽfunction of the characteristic beam width containing

.99% of the total energy , which can be related withthe case of Fig. 2b for Gaussian beams. Also, it is asimilar result to those obtained for a polytropicthermodynamic process. Here we estimate the widthfor different positions along the trajectory of beamand we calculate the corresponding mean intensity.Then, it can adjust the square points with a curve,using some approximated method, as the b-splinethat we use in the present case. Another way tocharacterize the width of beam, using the entropy

concept is possible with the measurement of distancebetween the bigger peaks in the entropy function,whose results are shown in Fig. 8b for approxima-

Žtions to different super-Gaussian indexes a s.1,2,8,20 . This measure can be variable and it is not

always growing, since until a certain value exists anenergy re-distribution in the propagated beam andthe form of the profile is not conserved. For thecases a)1, there is always a decay of the intensitycurve due to these irregularities in the intensityprofile. However, the width that contains 99% of theenergy, as we used previously, is increasing alongthe propagation. Approximately above the value Nfs0.5 the entropic width maintains its form andshow a continuous increase. Inside the Fraunhofer

Ž .Fig. 8. Two characteristic widths. a Evolution of mean intensityof a super-Gaussian-like beam, as a function of the width contain-ing 99% of the intensity profile, along the propagation directionŽ . Ž .W s1.9, W s0.3 . b Evolution of entropic width, accordingr g

with Fig. 7, for different SGL-beam approximations.


region and considering the convolution properties,the intensity distribution can be written as:

I fŽ .SGL x

2w xs FF rect x ,W )Gauss x ;W zŽ .� 4r g

22w xs FF rect x ,W FF Gauss x ;W z� 4 Ž .� 4r g

14Ž .

� 4being FF is the Fourier transform. The entropyfunction is given by:

S x , z sFF S x , z qFF S x , z 15Ž . Ž . Ž . Ž .SG R G G R

where the subindex R and G indicate rectangular andGaussian functions respectively. The result of Eq.Ž .15 indicates a difference with thermodynamics en-tropy, because in this case the entropy of a system isadditive with respect to the subsystem components.

Ž .Finally, replacing into the second part of Eq. 9 , weobtain the non-linear equation:

22W z sin W fŽ . Ž .g r x2exp f y1 s , 16Ž .x2 W fr x

from which the value f that correspond to thex

extremal entropy can be calculated using a numericalmethod and also, an entropic width in the spatial

Žfrequency domain can be calculated for the far.field .

3.2. Focussing Gaussian beams by thin lenses

So far, we have considered the Gaussian orsuper-Gaussian beam as propagating in free-space,without interactions with other systems. In this sense,the evolution of the system, under these conditions,is an irreversible process according with Postulate 2.Without bonds, it does not return to the initial stateand the total entropy is increasing along the propaga-tion. The latter means the beam width will growalong the propagation direction. What happens if weplace a thin lens in the beam path? In this case, thesystem is compound by the electromagnetic beamand the lens. In the first place, it is clear that if weuse a bi-convex lens, the beam width will decrease atthe lens output up to a certain value depending onthe focal length, and will again expand as in free-propagation. Fig. 9a shows these facts graphically,

Ž .and the widths before indicated with symbol y

Ž . Ž . Ž . Ž . Ž . Ž .Fig. 9. a Focussing of beams by a lens. b Evolution of Gaussian beams magnitude of field : B1 , initial state. B2 , before lens. C andŽ .D after lens for convergent and divergent lens respectively.


Ž .and after indicated with q the lens are related withthe different parameters through:

q Ž .W z

y Ž .sW z

2y y2R z R zL Ly= z q 1y q2 1y q1y yž / ž /( ž / ž /f R f R

17Ž .

also, f is the focal distance, R the curvature ratio, zL

the distance from the lens and:

2yl Ryz s 18Ž .2yž /W zŽ .

The expression into the root is always greater than1 for z™` and then, we can see that the entropywill have an amount regarding the free propagatedbeam. In this sense the total process is clearly ther-modynamically irreversible as we expected. We cansee in Fig. 9b the changes in a Gaussian beam when

Ž . Ž .is propagating between two positions B1 to B2Ž .and also after cross a convergent C and divergent

Ž .D lenses respectively. The entropy variation willŽ .exhibit two ways for the beam parameters a as a

Ž .beam width change, b as a beam curvature changeŽ .see Fig. 10a .

The total entropy can be seen in Fig. 10b for thetwo cases of a beam: in a free propagation and after

Ž .crossing a thin lens beam q lens propagation . Wedefine a total entropy introduced by the lens as a thedifference between the total entropy for free propa-gation and the total entropy for the beam that crossed

Ž Ž ..the lens both according with expression 6 . Forthis last case, it has a minimum width for a Gaussianbeam and then, is propagated with a bigger value ofthe entropy in the free space. As a consequence, theCorollary 1 is verified and the total entropy is in-creased. However, the lens introduce an entropyDS)0 in the final state according to like it is ofwaiting for an irreversible process. We can describethe process in the thermodynamic sense if we imag-ine the lens as a piston, compressing the photon gas,which was expanding and propagating in a givendirection. That is, the lens performs work over theparticles, changing the propagation direction. As aconsequence, each ray component of the beam will

Ž . Ž .Fig. 10. a Diagram to shown the effect of a lens on a ray. bEvolution of total entropy for a Gaussian beam that crosses a lens.

Ž .change the entropy see Fig. 10b , then varying thetotal entropy in the system.

Also, it is easy to calculate the entropy variationsand consequently the entropic beam width. Fig. 11aand Fig. 11b shown the entropy changes, from the

Ž .initial state B1 in Fig. 9b, for the following twopositions when the beam goes through convergent

Ž .and divergent lenses before and after . Obviously,the system returns to the initial state once free ofbonds, that is, once the beam propagates a distancegreater than the focal length. Therefore, it behaves asa free-propagation beam. We can derive a relation-ship between the Gaussian beam width and the en-tropy peak positions. From the extremal principle we

w xcan deduce the relation 30 :

W z WŽ .g 0xs" 1q2 ln 19Ž .)' W zŽ .2 g

Fig. 11c and Fig. 11d show the entropy differencebetween the free propagated beam and states previ-ously considered at a position after a convergent


Ž . Ž .Fig. 11. Change of entropy function, a from the initial state B1Ž . Ž . Ž . Ž . Ž .towards states B2 and C convergent ; b also for states B2

Ž . Ž . Ž . Ž .and D divergent . c and d : Difference between a freepropagated beam and the corresponding lens–beam interaction,for convergent and divergent respectively.

lens. The entropy introduced by the lens for a givenposition along the propagation direction, can be cal-culated from the difference between the entropy

Žbefore and after the lens but also, before the focal.length in that position. Another possibility is consid-

ering the entropy variation for each component withrespect to the free-propagation beam as shown inthese last figures.

4. Conclusions

In this work a formulation based on the entropicrepresentation is studied for the propagation of su-

Ž .per-Gaussian-like beams SGL , which are obtained

as convolution of rectangular and Gaussian profilespropagated along one direction. This operation al-lows to obtain the approximated representation ofsuper-Gaussian beams and, in consequence, it pro-vides a simple method to study the propagation inthe Fresnel and Fraunhofer regions.

Starting from postulates and theorems it is possi-ble to characterize the propagation of such beams,similar to the corresponding thermodynamic process,and to be able to characterize them from the point ofview of the entropy. Some properties of this functionare obtained and their relationship with the beamparameters are studied. A polytropic property thatrelates the mean transversal intensity versus the char-

Žacteristic width which concentrates about 99% of.the total beam energy along the propagation direc-

tion is deduced from the plots, for approximations toprofiles with different super-Gaussian indexes.

We will include more discussion about the en-tropy changes and entropy properties in further arti-cles. Here, we used two different widths to character-ize the evolution of beam along its propagation.Using the bigger points of entropy plot we can

Žcharacterize the beam width which we called en-.tropic width for a position on the z-axis, because

these values are related to a characteristic parameterŽ Ž ..of Gaussian beams exp y1 , which permits us to

compare the intensity distribution with another pro-files, that is, super-Gaussian beams with differentorders. The calculation of total entropy for eachposition along z-axis permits us to characterize theevolution of a beam as a whole, and relate it withthermodynamic processes. Also, from the propertiesfor propagation of Gaussian beams, we can deduce

Ž .the corresponding simple cases: a super-Gaussianbeams, obtained as convolution of rectangular and

Ž .Gaussian profiles, and b the effect of lenses in thepath of Gaussian beams.

Through this analysis, different elements can bealso studied as future applications. The assimilationof many electromagnetic situations to thermody-namic systems can be carried out with the characteri-zation of the entropy function. Wave-guides, reso-nant cavities, surfaces, etc., are some optical ele-ments that can be studied, and innumerable applica-tions can take place in several cases, so much insidethe conventional optics as in non-imaging optics orin all that means state changes in the matter by


means of the use of electromagnetic radiation fortechnological applications.

Acknowledgements

This work was partially supported by Secretarıa´General de Ciencia y Tecnica of the Universidad´

Ž .Nacional del Sur Bahıa Blanca, Argentina and´Ž .Sofilab SACV Mexico DF . We thank the referees´

for their useful suggestions and comments to im-prove the presentation of this paper.

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Entropic propagation of Gaussian and super-Gaussian-like beams

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