INTENSITY MOMENTS OF A LASER BEAM FORMED BY SUPERPOSITION OF HERMITE-GAUSSIAN MODES A. Ya. Bekshaev I. I. Mechnikov Odessa State University

Transcription

1 INTENSITY MOMENTS OF A LASER BEAM FORMED BY SUPERPOSITION OF HERMITE-GAUSSIAN MODES A. Ya. Bekshaev I. I. Mechnikov Odessa State Universit SUMMARY Epressions are obtained for the Wigner function moments of a paraial light beam represented b arbitrar coherent superposition of Hermite-Gaussian beams with plane wave fronts. Possibilities are discussed for application of the obtained results to modeling real laser beams and to design of optical sstems implementing prescribed transformations of a beam transverse structure.

2 INTENSITY MOMENTS OF A LASER BEAM FORMED BY SUPERPOSITION OF HERMITE-GAUSSIAN MODES A. Ya. Bekshaev I. I. Mechnikov Odessa State Universit The spatial-angular intensit distribution (SAID) of radiation is one of the most important characteristics of a light beam. Its detailed description generall requires specifing a great number of parameters, which is practicall inconvenient and, quite often, superfluous. Therefore, various schemes of more rational SAID characterization have been proposed among which two approaches seem to be the most suitable and universal: a sstematic use of the Wigner function moments [, ] and the beam representation as a superposition of the standard Hermite-Gaussian (HG) modes [3]. The first approach is distinguished b theoretical self-consistenc as well as b descriptiveness and immediate measurabilit of relevant parameters, the second one is useful in calculations and provides possibilit to establish direct relations between the SAID parameters and the phsical picture of the laser beam generation. Obviousl, it would be quite suitable to know the correspondence between the beam characteristics emploed in both approaches, and the present report contributes to solution of this problem in important special cases. Results of the below analsis ma be used, in particular, for practical representation of an arbitrar paraial light beam b means of a certain superposition of HG modes. This will enhance the possibilities and the area of applicabilit of the "embedded" laser beam method [,], enhancing and supplementing the known recommendations for beam modeling b an incoherent miture of two Gaussian beams or b an off-ais Gaussian beam []. As usual, we consider a beam propagating along ais z, and the local instantaneous optical field (e.g., electric field strength) is epressed through the slowl varing comple amplitude u [3] E!,, z, t" % Re u!,, z" ep( * i! kz&' t" ) + # $

3 3 where, are Cartesian coordinates in the beam cross section, k and ' are the radiation wave number and frequenc, t stands for the time. For a superposition of HG modes, the normalized transverse distribution of the beam comple amplitude obtains the representation u (, )%, amnumn(, ), a mn, %, () with the distribution of a normalized HG mode [3] m -n &. /. / (. /) umn (, ) %! 0 m! nb! b " Hm 3 Hn ep& -. () 5b 6 3b 3b b 5 6 * 5 6+ Here m 7 0, n 7 0 are the integer mode indices, H m denotes the Hermite polnomial of m-th order, b and b are orthogonal transverse sizes of the Gaussian envelope [3]. Due to completeness of the set of functions (), the epansion () u,, i. e. for ever real paraial beam. eists for ever function! " The moment matri of a paraial beam is defined b equalit []. p p / 3. Q Q / 3 p p p p p pp 3 p p pp p! " Q % % k 3 I,, p, p dd dp dp Q Q 5 6, where Q... Q are 9 matrices, p and,,, spectrum,! "! " p are arguments of the beam angular I p p is the beam Wigner function [] normalized b condition 8 I,, p, p dddp dp % ; integrals are taken within infinite limits of each variable, while the presumed spatial confinement of the beam provides their convergence. The equivalent representation of the moment matri (see, e.g., Refs. [0 ]).. M M / Q M % 3 M M % 5 6 k is also often used

4 Our task is to find the moment matri for the beam described b Eqs. () and (). Using formulas from Appendi A of paper [] as well as recurrent relations for the Hermite polnomials [5], after rather cumbersome transformations we obtain the representation Q %. b / 3 kb! K -L:" kbb! M :- N: " & L::! M ::& N:: " b 3 3 b 3kbb! M :-N:" kb! K - L: " &! M ::- N:: " & L:: 3 b % 3 b, 3 & L:: &! M:: - N:: "! K & L: "! M: & N: " 3 b kb kbb 3 3 b! M:: & N:: " & L::! M: & N: "! K & L: " 3 b kbb kb 5 6 (3) where ; mn m-, n K %, amn ( m-), K %, amn n - ( ), () ; L %, a a ( m-)( m-), L %, amnam, n- ( n -)( n - ), (5) ; ; M %, a -am-, n ( m-)( n -), N %, amnam-, n- ( m-)( n -), (6) one prime (two primes) denote the real (imaginar) part of a comple quantit. Results (3) - (6) give a general ground for the solution of our problem. As could be epected, there is no one-to-one mapping between the two schemes of the beam characterization. For ever HG mode superposition, one can determine the unique moment matri but the inverse procedure choice of a HG superposition with a given moment matri can be realized in multitude of was (of course, if no eplicit restrictions for coefficients a mn are imposed). One can easil see that for arbitrar moment matri possessing not more than ten independent elements, from Eqs. (3) -

5 5 (6) the relations follow which unambiguousl determine ten real quantities forming K, K, L, L, M and N. Their knowledge allows to find, at most, ten coefficients a and, therefore, corresponding representation of epansion () can be unique onl if number of the superposition members is limited or the obe certain interrelations. Nevertheless, in spite of the mentioned ambiguit of modeling an arbitrar moment matri b the superposition of the form (), Eqs. (3) - (6) allow us to determine those parameters of HG modes which are especiall important in this procedure. Now, with several eamples, consider some consequences and was of utilization of results (3) - (6).. Let the beam be represented b a "pure" HG mode (); then in Eqs. (3) - (6) onl one of coefficients a mn equals to unit while all the rest reduce to zeroes. The moment matri becomes diagonal: Q. Q Q / ( m- n-) % % diag kb! m -", kb! n -",,. (7) 5 6 * bk bk + mn 3 Q Q mn Invariants of this moment matri (see []) possess the form < % M % m -; < % M % n -, (8) % - -, Sp! Q " Q Q (m ) (n ) mn det Q mn (m ) (n ) & % (9) Relations (8) and (9) permit one to find a HG beam with the given moment matri. As is seen from the formulas, this is possible not alwas but onl for beams whose moment matri has integer invariants < and <.. Now consider a beam with simple astigmatism [3], in which the transverse distribution of the comple amplitude is smmetric or antismmetric with respect to the coordinate aes (this class embraces, for eample, all aiall smmetric beams that occur in practice most often). HG mode () is a special case of this beam; its smmetr or antismmetr is indicated b evenness or oddness of the corresponding inde. Obviousl, superposition () preserves these properties if all its members have the same parit. Consequentl, for beams with simple astigmatism M % N % 0 and

6 6 the moment matri, as should be epected, acquires the distinctive for such beams form with zeros at crossings of rows and columns with different parities. In this case, the beam is described b two 9 moment matrices Q and Q where, for eample, Q! ". : :: / kb K - L & L. p /!,,, " 3 % k8 I p p d d dpdp % 3p L! K L" p :: : 5 & & 6 3 kb 5 6. The beam invariants satisf the relationships < % det Q %! K & L ", det Q! K L " which, for the pure HG mode, lead to equalities (8) and (9). < % % &, 3. Yet another interesting situation appears when all coefficients of the series () are real. This means that phases of all the superposition members are equal, and since each of them describes a beam with plane wavefront, the resulting beam also possesses a plane front. Reall, then all imaginar parts of amounts () - (6) equal to zeroes and matri (3) turns out to be a block-diagonal matri with Q % Q % 0, which, according to [], is a characteristic sign of the beam waist [6] (a section with plane wavefront).. Tpical cases of realization of the form () superpositions occur upon the multimode generation in laser resonators with degenerate modes of tpe (). Let us restrict ourselves b the situation when b % b % b; then all beams () with identical values of m- n are degenerated. We consider two simplest superpositions that belong to the case m- n %. Herewith, in epansion () onl two coefficients a mn can differ from zero: a 0 and a 0. i а) Let a 0 %, a0 %. Such a beam is described b the comple amplitude distribution

7 7. /. / - r ua (, ) % ( - i)ep ep 3& r i % 3& - = b 0 5 b 6 b 0 5 b 6 (0) (r and = are the polar coordinates in the beam cross section) and represents an elementar eample of a singular beam with the screw wavefront dislocation [7]. In this case, Eqs. () (6) give K the moment matri obtains the form where I is the unit 9 matri and matri. % K %, L % L % 0, M % i, N % 0 and. / kb I J Q% Qa % 3, () & J I 3 5 kb 6 0 J %. / 3 & 0 is the simplest antismmetric б) The net superposition differs from the previous one b the fact that both nonzero coefficients of the epansion () are real: a % a %. Such a "small" 0 0 mathematical difference leads to noticeable phsical consequences. The corresponding comple amplitude distribution. /. / - : : - : ub (, ) % ( - )ep ep 3& % 3& b 0 5 b 6 0 b 5 b 6 () represents a "pure" HG 0 mode in the coordinate frame turned around the ais z b 5>: : % ( - ), : % (& - ), and the moment matri of beam () has a block-diagonal form Here a reference to singular optics and orbital angular momentum [3] would be quite relevant but I did not know about these notions when the paper was prepared.

8 8. / kb P 0 Q% Qb % 3, (3) 0 P 3 5 kb 6 where 0 is zero 9 matri, P %. / This eample is rather demonstrative since the difference in forms of matrices Q a and Q b can be directl attributed to the wave front singularit of the first beam. However, one can easil make sure that the invariants of matrices () and (3) are equal: Q ab %,! " det, 9 Sp Q Q & Q % 0; ab hence, matri Q a can be transformed to form (3) b some smplectic transformation []. As both beams (0) and () are formed b combining the same HG modes, this means that the beam with transverse distribution (0) can be transformed to form () with the help of a certain optical sstem consisting onl of quadratic phase correctors (parabolic lenses and mirrors, generall astigmatic) and transversel homogeneous (free) intervals [8]. This conclusion ma be important for adaptive optics where the correction of phase aberrations causing wavefront dislocations is usuall treated as a complicated problem due to impossibilit to create a correcting mirror of the required shape [7,9]. As can be seen, in this case at least, a beam with the wavefront dislocation can be transformed into a beam with the same qualit and without phase singularities, which, if necessar, can afterwards be corrected b usual means of adaptive optics. In conclusion, we present an eample of such a transformation. The algorithm of arbitrar moment matri transformation to the block-diagonal form (Ref. [], Sec. ) enables to calculate an optical sstem after transmission through which the beam (0) accepts form (). Such a sstem can onl be determined ambiguousl, and the,

9 9 practical choice of a certain option is ultimatel dictated b considerations of convenience. In particular, one can show that where Q b % HQ H!, a H %. / 0 kb & 0 kb 3 & 3&! kb " & 3 0 &! kb " 0 & 5 6 is the transmission matri (generalized ABCD matri [,,6]) of the transforming optical sstem. This matri has zero elements at crossings of rows and columns of different parities, i.e. describes an optical sstem with simple astigmatism. It can be decomposed into two independent submatrices H. /. A B / kb? 3 C D % 3! kb " &, 5 6 3& 5 6 H. /. A B / & kb? 3 C D % 3! kb " &. () 5 6 3& & 5 6 It is eas to understand the phsical meaning of these matrices. Both of them admit the representation H,.& g, z0z0 f / % 3 & f & g 5, 6 corresponding to the "waist to waist" transformation ([6], Sec. 8.); here f % kb, 0 0 z % z % kb are coinciding confocal parameters (Raleigh ranges) of the input and output beams, g % g %, g % g % &. We can readil imagine an optical sstem implementing the transformation H; one of possible options is presented in the figure. 3 This sstem is formed b two identical astigmatic lenses 3 Obviousl, this optical sstem is a variant of the known mode converter transforming between HG and Laguerre-Gaissian modes [].

10 0 placed in the input and output reference planes. The distance between the lenses equals to L % kb, focal lengths of the lenses in plane z amounts to f & % L! & " &, and, in plane z, f! " L % -. This causes that in plane z the beam eperiences stronger focusing than in plane z, and the HG 0 and HG 0 components of (0) obtains different additional phase shifts which compensate the initial phase difference between them. Reall, when transmitting the sstem with matrices (), mode HG 0 gets the multiplier [3]. 3 5 A - & 3 & / B B /. ~ 3 A 6 ~ 6,. and mode HG 0, in turn the multiplier 3 A 5 - & & 3 / B B /. ~ 3 A is the comple radius of the input beam wavefront curvature). It can be easil seen from () that the ratio of these factors eactl equals to i, and thus the phase difference between the HG components of beam (0) vanishes after transmitting the optical sstem. Considered eamples, list of which can be continued, illustrate peculiarities of the two approaches to the beam characterization and demonstrate the efficac of their combined application. All this will facilitate the utilization of the results obtained in specific investigations of laser beams. In the newer literature, this shift is commonl referred to as Gou phase [3].

11 I II z L Figure caption Optical sstem removing the wavefront dislocation: z is the sstem ais; I, II are the input and output reference planes in which the thin astigmatic lenses are situated; and are schematic contours of the longitudinal beam sections in planes z and z, correspondingl.

12 REFERENCES. Anan'ev Yu.A., Bekshaev A.Ya., "Theor of intensit moments for arbitrar light beams", Optics and Spectroscop, 99, V. 76, No, pp Siegman A.E., "Handbook of Laser Beam Propagation and Beam Qualit Formulas Using the Spatial-Frequenc and Intensit-Moment Analsis", Draft version of 7//99 (Also available at: Siegman A.E., "How to (Mabe) Measure Laser Beam Qualit", DPSS Lasers: Applications and Issues (OSA TOPS, V. 7), Washington D.C.: Optical Societ of America, 998, pp. 8-99). 3. Anan ev Yu.A., Laser Resonators and the Beam Divergence Problem, Adam Hilger, New York, 99. Bastiaans M.J., "Wigner distribution function and its application to first-order optics", J. Opt. Soc. Amer., 979, V. 69, pp Janke E., Emde F., Lösch F. Tafeln höherer Funktionen, B.G. Teubner, Stuttgart, Gerrard A., Birch J.M., Introduction to Matri Method in Optics, John Wile & Sons, London, Zel'dovich B.Ya., Pilipetskii N.F., Shkunov V.V., Wave front reversal, "Nauka", Moscow, 985 (in Russian). 8. Sudarshan E.C.G., Mukunda N., Simon R., "Realization of the first order optical sstems using thin lenses", Opt. Acta, 985, V. 3, N 8, pp Vorontsov M.A., Shmal'gauzen V.I., Principles of adaptive optics, "Nauka", Moscow, 985 (in Russian). 0. Bekshaev A.Ya., Popov A.Yu., "Method of light beam orbital angular momentum evaluation b means of space-angle intensit moments", Ukrainian Journal of Phsical Optics, 00, V. 3, No, pp

13 3. Bekshaev A.Ya., Soskin M.S., Vasnetsov M.V., "Optical vorte smmetr breakdown and decomposition of the orbital angular momentum of light beams", J. Opt. Soc. Amer. A, 003, V. 0, No 8, pp Bekshaev A.Ya., Soskin M.S., Vasnetsov M.V., "Transformation of higher-order optical vortices upon focusing b an astigmatic lens", Opt. Commun., 00, V., No -6, pp Soskin M.S., Vasnetsov M.V., "Singular optics", Progress in Optics, 00, V., pp Beijersbergen M.W., Allen L., van der Veen H.E.L.O., Woerdman J.P., "Astigmatic laser mode converters and transfer of orbital angular momentum", Opt. Commun., 993, V. 96, pp. 3-3.