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A detailed study of Mathieu–Gauss beams propagation through an apertured ABCD optical system
Optics Communications 265 (2006) 594–602 www.elsevier.com/locate/optcom
A detailed study of Mathieu–Gauss beams propagation through an apertured ABCD optical system A. Chafiq, Z. Hricha, A. Belafhal
*
Laboratoire de Physique Mole´culaire, De´partement de Physique, Universite´ Chouaı¨b Doukkali, Faculte´ des Sciences, B. P. 20, 24000 El Jadida, Morocco Received 27 November 2005; received in revised form 19 March 2006; accepted 21 March 2006
Abstract Using Collins formula and the expansion of Mathieu beams in terms of Bessel beams we derive the exact propagation equations of Mathieu–Gauss beams through an apertured paraxial ABCD optical system. A comparison between the exact propagation equations and the approximated ones, which are derived by expanding the circ function into a finite sum of Gaussian functions, is presented. The propagation characteristics of zeroth-order Mathieu–Gauss beams in (y–z) and (x–z) planes are analyzed with detailed numerical calculations. It is found that the profile of the propagated Mathieu–Gauss beam is similar to that of Bessel–Gauss beam. Furthermore, the focalization of the Mathieu–Gauss beams through a thin lens is illustrated and analyzed with numerical results. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Collins formula; Mathieu beams; Mathieu–Gauss beams; Bessel–Gauss beams; Aperture function
1. Introduction In recent years, an increasing amount of attention has been paid to the nondiffracting beams in both optics and acoustics. The importance of these beams is due to the fact that they can propagate indefinitely without any change in their transverse shape. Their potential applications in wireless communications, optical interconnections, laser machining, medical imaging and surgery, among others, make them very relevant. Durnin et al. [1,2] have introduced the first kind of these beams called Bessel beams which are solutions of Helmholtz wave equation expressed in circular cylindrical coordinates. Three other nondiffracting solutions of Helmholtz equation namely, cosine [3], Mathieu [4–6] and parabolic [7,8] beams were reported in last few years. The ideal nondiffracting beams have an infinite extent and energy, so they are not physically realizable. For that reason, many authors have been devoted to study approximated versions of these ideal beams. Gori et al. [9] have proposed to modulate them by a Gaussian function and have introduced the most important family of the approximated nondiffracting beams named as Bessel–Gaussian beams. More recently, Gutie´rrez-Vega and Bandres [10] have presented a detailed study of the so-called Helmholtz–Gauss beams which are a general nondiffracting beams modulated by a Gaussian function. Bessel–Gauss and Mathieu–Gauss beams are then regarded as members of these beam families. In Ref. [10], the propagation of the Helmholtz–Gauss beams was performed in a general way by the resolution of Helmholtz wave equation in paraxial approximation. However, only the propagation in free space has been treated. On the other hand, Li et al. [11] have proposed new beams obtained by modulating Bessel beams by flat-topped Gaussian envelopes. The authors showed that these beams can be expressed as a sum of Gori Bessel–Gaussian beams. The main contribution of this work is to give a detailed *
Corresponding author. Tel.: +212 68 50 43 44; fax: +212 23 35 34 54. E-mail address: [email protected] (A. Belafhal).
0030-4018/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.03.048
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analytical study of the propagation of Mathieu–Gauss beams through an apertured ABCD optical system by using the Collins diffraction formula and by expanding these beams in terms of Bessel–Gauss beams. A rigorous and also an approximated calculation of the propagation equations of these beams through an apertured and unapertured ABCD optical systems are derived using the fact that the aperture function can be expanded into a finite sum of Gaussian functions. This last technique has been extensively used by many authors in the last few years in deriving approximated analytical propagation equations and factors of beams propagating through an apertured paraxial optical systems (see for example Refs. [13–20]). This paper is organized as follows: the analytical expression of Mathieu–Gauss field expressed in terms of Bessel–Gauss beams is given in Section 2. In Section 3, the paraxial propagation equations of Mathieu–Gauss beams through an apertured and unapertured ABCD optical systems are derived. Some numerical calculations and analysis are given in Section 4. Finally, we end this paper with a conclusion in Section 5. 2. Mathieu–Gauss beams The expansion of Mathieu beam in terms of Bessel functions basis is given by (details are found in Ref. [12]) 8 1 P ð2nÞ j > uc2n ðq; u; q; z ¼ 0Þ ¼ A2j ðqÞð1Þ cosð2juÞJ 2j ðk t qÞ; > > > j¼0 > > > > 1 > P ð2nþ1Þ 2jþ1 > > cos½ð2j þ 1ÞuJ 2jþ1 ðk t qÞ; > < uc2nþ1 ðq; u; q; z ¼ 0Þ ¼ j¼0 A2jþ1 ðqÞðiÞ 1 P > ð2nþ2Þ > > us2nþ2 ðq; u; q; z ¼ 0Þ ¼ B2jþ2 ðqÞð1Þjþ1 sin½ð2j þ 2ÞuJ 2jþ2 ðk t qÞ; > > > j¼0 > > > 1 > > > us2nþ1 ðq; u; q; z ¼ 0Þ ¼ P Bð2nþ1Þ ðqÞðiÞ2jþ1 sin½ð2j þ 1ÞuJ 2jþ1 ðk t qÞ; : 2jþ1
ð1Þ
j¼0
where ucm and usm are the even and odd modes of Mathieu beams. q, u, z are the cylindrical variables, and kt = k sin /0, kz = k cos /0 are the transverse and longitudinal components of the wave vector ~ k of plane waves that reconstruct the beam [21]. /0 is the semi-angle of the cone on which lie the wave’s vectors associated with the plane waves. q is the ellipticity ð2nÞ ð2nþ1Þ ð2nþ2Þ parameter given by q ¼ h2 k 2t =4, where 2h is the interfocal separation. The coefficients A2j ðqÞ; A2jþ1 ðqÞ; B2jþ2 ðqÞ, and ð2nþ1Þ B2jþ1 ðqÞ are expansion coefficients of Mathieu functions [22], that respect of each ellipticity parameter q a set of recurrence equations and J is Bessel function of the first kind. For the sake of simplicity a constant factor of 2p was omitted in Eq. (1). The insertion of a Gaussian term in the expressions of Eqs. (1) leads to the field distribution of Mathieu–Gauss beams at the plane z = 0 q2 e MGm ðq; u; z ¼ 0Þ ¼ ucm ðq; u; q; z ¼ 0Þ exp 2 ; ð2:aÞ w0 q2 ð2:bÞ MGmo ðq; u; z ¼ 0Þ ¼ usm ðq; u; q; z ¼ 0Þ exp 2 ; w0 where the superscripts e and o design even and odd modes of Mathieu–Gauss beams, and w0 is the waist of the Gaussian beam. The exponential term of Eqs. (2) can be inserted in the summation of ucm and usm because it is independent on the index summation. Subsequently, one can easily deduce that the Mathieu–Gauss beams can be built-up by a superposition of Bessel–Gauss beams. In Fig. 1, the two dimensional behaviors of both zeroth-order Mathieu–Gauss beam obtained from Eqs. (2) and the ideal Mathieu beam are presented. By comparing the reported profiles one can see the reduction of the intensity in the sides of the Mathieu–Gauss beam which is due the Gaussian envelope. 3. Propagation equations of Mathieu–Gauss through ABCD optical systems As it is well known, the propagation of beams through a paraxial optical system can be described by Collins formula [23]. Thus, if we consider an apertured paraxial ABCD optical system illuminated by one of the Mathieu–Gauss beams family of Eqs. (2), the relationship between the input and the output field distributions are given by Z q0 Z 2p ik k e, o e, o ðoutÞ ðinÞ 2 2 MGm ðq2 ; h; zÞ ¼ q1 MGm ðq1 ; u; z ¼ 0Þ exp ikz þ i ðAq1 2q1 q2 cosðu hÞ þ Dq2 Þ dq1 du; 2pB 0 2B 0 ð3Þ
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Fig. 1. Transverse intensity distribution of zeroth-order Mathieu–Gauss and Mathieu beams at plane z = 0 for the parameters kt = 15000 m1 and q = 25: (a) Mathieu–Gauss beam with a Gaussian waist w0 = 2 mm, and (b) ideal nondiffracting Mathieu beam.
oðoutÞ ðq ; h; zÞ is where MGme, oðinÞ ðq1 ; u; z ¼ 0Þ is the field distribution at the point (q1, u) in the input plane z = 0, and MGe, 2 m the field distribution at the point (q2, h) in the output plane located at z distance. A, B, C and D are the elements of the transfer matrix characterizing the optical system. k is the wave vector and q0 is the radius of the input aperture. For the sake of simplicity, we consider only the first family of the above beams i.e., the even modes MGe2n , the treatment of the other Mathieu–Gauss family is the same. Substituting from Eqs. (2) into Eq. (3) and using the following equality [22] 1 X in J n ðzÞein/ . ð4Þ eðiz cos /Þ ¼ 1
Eq. (3) becomes eðoutÞ
MG2n
1 X 2i 2 ðr; h; zÞ ¼ eðikzþiDr =gÞ A2n 2j ðqÞ cosð2jhÞ g j¼0
Z 0
1
2rq qJ 2j ðaqÞJ 2j expðQq2 Þdq; g
ð5Þ
where A Q¼bi ; g g ¼ B=ld ;
ð6:aÞ 2
ð6:bÞ
q0 w0
and where r = q2/q0, q = q1/q0 and a = ktq0 are the non-dimensional variables and b ¼ is the truncation parameter. In these expressions ld ¼ kq20 =2 is the diffraction length related to the aperture radius. From Eq. (5) it is clear that the propagation characteristics of Mathieu–Gauss beams can be easily deduced from those of Bessel–Gauss beams [9,13,24,25]. In the following, we will perform the integral expression of Eq. (5) for the apertured and unapertured optical systems. 3.1. Apertured case 3.1.1. Exact propagation equation of Mathieu–Gauss beams The finite integral of Eq. (5) can be performed in the same way as the method used in Ref. [24] by Overfelt and Kenney So, using the fact that 0 j 6¼ 0; J 2j ð0Þ ¼ ð7Þ 1 j¼0 the axial field distribution of the Mathieu–Gauss beam can be written as 8 1 P k > > j2Q=aj < 1; expðQÞ ð2Q=aÞ J k ðaÞ < i ikz ð2nÞ k¼1 eðoutÞ e A0 ðqÞ MG2n ð0; zÞ ¼ 1 P > Qg > : expða2 =4QÞ expðQÞ ða=2QÞk J k ðaÞ j2Q=aj > 1: k¼0
ð8Þ
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From Eq. (8), we can deduce that the axial intensity of the even family of Mathieu–Gauss beams is the product of the axial ð2nÞ intensity of the zeroth-order Bessel–Gauss beams and the coefficient A0 ðqÞ. We note that if we repeat the above calculations for the odd Mathieu–Gauss family we will find that their axial intensity is equal to zero. To perform the integral of Eq. (5) in the transverse plane one can use the product of Bessel functions formula [22] J m ðazÞJ l ðbzÞ ¼
1 2 2 ðaz=2Þm ðbz=2Þl X ð1Þk ðaz=2Þ2k 2 F 1 ðk; m k; l þ 1; b =a Þ ; k!Cðm þ k þ 1Þ Cðl þ 1Þ k¼0
ð9Þ
where C is the gamma function and 2F1 is the hypergeometric function. So, After some algebraic calculations Eq. (5) can be rewritten as 1 1 1 X X A2n i ð1Þm Qm X 2 2j ðqÞ eðoutÞ 2j ðar=2gÞ cosð2jhÞ MG2n ðr; h; zÞ ¼ eðikzþiDr =gÞ Gk ðj; m; a; g; rÞ; g ð2jÞ! m! j¼0 m¼0 k¼0
ð10Þ
where 8
2r 2 2 ð1Þk ða=2Þ2k > 2 F 1 k;2jk; 2jþ1; ð g Þ =a > < k!ð2jþkÞ!ð2jþmþkþ1Þ
Gk ðj; m; a; g; rÞ ¼ k 2k 2 2r 2 > > : ð1Þ ðr=gÞ2 F 1 k;2jk; 2jþ1; a =ð g Þ k!ð2jþkÞ!ð2jþmþkþ1Þ
2r g =a < 1; 2r g =a > 1:
ð11Þ
If the exponential term involved in the integral of Eq. (5) is not written in its series form, the summation over the index m in Eq. (10) can be substituted by the incomplete gamma function. It is to be noted that Eq. (8) can be obtained from Eq. (10) by taking r = 0. 3.1.2. Approximate propagation equation of Mathieu–Gauss beams To describe approximately the propagation of Mathieu–Gauss beams through an apertured ABCD optical system, we use the important technique in which the finite integral of Eq. (5) is transformed into an infinite one, by using the circ function 1 0 6 q 6 1; circðqÞ ¼ ð12Þ 0 q > 1; and afterwards expanding it into a finite sum of complex Gaussian functions circðqÞ ¼
kX ¼10
Ek expðLk q2 Þ;
ð13Þ
k¼1
where Ek and Lk are the expansion and Gaussian coefficients, respectively which are found by a computer optimization [26]. So, by inserting this aperture function in the integral of Eq. (5) and by using the following integral result [22] 2 Z 1 1 rd r þ d2 xJ l ðrxÞJ l ðdxÞ expðmx2 Þdx ¼ I l exp ð14Þ 2m 2m 4m 0 with jarg mj < p4, Re r > 0 and Re d > 0, where Il is the modified Bessel function of the first kind, and after some algebraic calculations, the output field distribution reads X X ð2nÞ 2 eðoutÞ 2 j MG2n ðr; h; zÞ ¼ ieðikzþiDr =gÞ Ek ð1=S k gÞ exp½ððagÞ þ 4r2 Þ=4g2 S k A2j ðqÞð1Þ cosð2jhÞJ 2j ðiar=S k gÞ; k ¼10
1
k¼1
j¼0
ð15Þ
where S k ¼ Lk þ Q.
ð16Þ
Eq. (15) shows that the propagated Mathieu–Gauss beams have a similar expansion form of the propagated Mathieu beams [12] but with an added exponential term characterizing the paraxial approximation and the modulation by the Gaussian function. Eq. (15) leads to the axial intensity expression IðzÞ ¼
kX ¼10 m ¼10 X k¼1
m¼1
fk ða; b; A; BÞfm ða; b; A; BÞ;
ð17Þ
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where fn ða; b; A; BÞ ¼ En ð1=S n gÞ expða2 =4S n Þ
ð18Þ
with the * representing the complex conjugation, n is an integer index. From Eq. (18) it is shown that the axial intensity depends essentially on the parameter of truncation b, the transfer matrix coefficients A and B implicit in Sk and kt implicit in a. On the other hand, it must be stressed out that the method used above in solving the finite integral of Eq. (5) can be applied to the other family of Helmholtz–Gauss beams. Particularly, Ding and Liu [13] and Zhao et al. [15,16] have used this technique to derive approximated propagation equations of Bessel–Gauss beams and cosine-Gaussian beams, respectively, through apertured ABCD optical systems. 3.2. Unapertured case Let us now consider a Mathieu–Gauss beam propagating through an ideal unapertured ABCD optical system, i.e., the aperture radius is equal to infinity. By using in a second time the equality of Eq. (14), the expression of the output field distribution (Eq. (5)) reduces to h i 1 eðoutÞ 02 0 0 2 02 0 0 MG2n ðr0 ; h0 ; zÞ ¼ expðikz þ iDr =g Þ exp ððag Þ þ 4r Þ=4g ðg iAÞ A þ ig0 1 X ð2nÞ j A2j ðqÞð1Þ cosð2jh0 ÞJ 2j ðia0 r0 =g0 iAÞ; ð19Þ j¼0
where r = q2/w0, a = ktw0 and g 0 = B/zR are the non-dimensional variables in this paragraph. With zR ¼ kw20 =2 is the Rayleigh length related to the Gaussian waist. From Eq. (19) one can deduce the axial intensity 0
IðzÞ ¼
0
1 2
A þ ðg0 Þ
2
2
2
exp½ðag0 Þ =2ððg0 Þ þ A2 Þ.
ð20Þ
It is to be noted that Eq. (20) is in accordance with results of Ref. [10] and it can also be regarded as a generalized form of Eq. (15) in that reference. Furthermore, Eq. (19) can be obtained from Eq. (10) when the aperture radius approaches infinity. 4. Numerical calculation results and discussion To study the propagation properties of Mathieu–Gauss beams through an ABCD optical system, the exact analytical form of Eqs. (8) and (10) can be used. Nevertheless Eq. (10) will lead to numerical summation of hypergeometric functions. To avoid this difficulty the approximated expression of Eq. (15) should be used. In this case we have to truncate the expansion series building-up Mathieu–Gauss beams, this operation can be done without any problem because there are only some coefficients that possess a significant weight for any value of the parameter q [27]. These coefficients permit us to build-up the Mathieu–Gauss beams from the Bessel–Gaussian beams with good a accuracy. 4.1. Free-space propagation The propagation in free space is characterized by the following transfer matrix A B 1 z ¼ . C D 0 1 In Fig. 2, we present a comparison between the plots of the numerical calculations of the exact and the approximated axial propagation which correspond to Eqs. (8) and (15), respectively. From these plots we can see the coincidence of the curves in far-field and the disagreement in the near-field. The expression of Eq. (15) is advantageous in obtaining the field amplitude distribution because it is written as a summation of ten terms of Mathieu–Gauss beams. Thus, Eq. (15) can be used as an alternative expression in analyzing the propagation of the apertured Mathieu–Gauss beam. As it is pointed out in Refs. [9,10], the propagation properties of any nondiffracting beam modulated by the Gaussian envelope are generally determined by the following parameters: the width of Mathieu fringes D p/kt, the aperture radius q0 and the Gaussian width w0. In order to analyze in detail the dependence of the behavior of the Mathieu–Gauss beam on these parameters, we give in Fig. 3(a) plots of the axial intensity distribution of the propagated zeroth-order Mathieu– Gauss beams in free space for a = 16, q = 16, wavelength k = 632.8 nm and b = 0.01, 1, 100. In Fig. 3(b) the axial intensity of the unapertured beam is reported for q = 16, 200 and w0 = 3 mm (a and k are the same as in Fig. 3(a)). In order to
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Fig. 2. Comparison of the axial intensity distribution of zeroth-order Mathieu–Gauss beam propagating in free space for q = 25, a = 10, b = 0.5, k = 632.8 nm and q0 = 2 mm evaluated approximately using the circ function and exact evaluation of Eq. (8).
Fig. 3. Axial intensity distribution of zeroth-order Mathieu–Gauss beam propagating in free space for k = 632.8 nm and q0 = 2 mm: (a) apertured beam for a = 16, q = 16, the values of the truncation parameter are b = 0.01, 1, 100, and (b) unapertured beam for a 0 = 16, w0 = 3 mm and q = 16, 200.
analyze the behavior of the beam intensity in Fig. 3, we introduce the parameter c ¼ 12 k t w0 which depends on D, q0 and w0. This parameter plays an important role in determining the behavior of Mathieu–Gauss beam upon propagation [10]. For a given value of D, we can distinguish three cases: (i) c 1: the beam retains the nondiffracting propagation properties of
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ideal Mathieu beam until a ‘maximum’ propagation distance zmax, (ii) c 1: the beam have a Gaussian behavior and (iii) c = 1: this case corresponds to the transition between the two behaviors. In the case of Fig. 3, we have c 1 for both b 6 1 and the unapertured system, and c < 1 for b > 1. Thus, the beam retains the nondiffracting propagation properties of apertured ideal Mathieu beam in the first case, and the Gaussian behavior in the second case. On the other hand, for b = 0.01, the spot size of the Gaussian envelope (at z = 0) is 20 mm and the width of Mathieu fringes is 0.39 mm. This means that the wave front of the Gaussian components which construct the beam in the aperture plane is nearly flattened and therefore the beam extends far and far from the aperture. Following this picture, for b = 100, the diffractive Gaussian effect is imposed and zmax plane is near the aperture. In all cases, after the maximum propagation distance zmax the components of beam diverge and overlap lesser and lesser that’s why the axial irradiance decreases. In Fig. 4 we present the plots of the transverse intensity of apertured zeroth-order Mathieu–Gauss propagating in free space for different values of z. The curves
Fig. 4. Transverse intensity distribution of apertured zeroth-order Mathieu–Gauss beam propagating in free space at z = 0.5, 1, 1.8 m truncated by b = 0.01 (continuous lines), b = 1 (dotted lines) and b = 100 (dashed–dotted lines), the other parameters are q = 16, a = 16, q0 = 2 mm and k = 632.8 nm: (a)–(c) evolution along (x, z) plane, (d)–(f) evolution along (y, z) plane.
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allow us to deduce that the beam propagates in both (y–z) and (x–z) planes without significant divergence of central peaks. When we consider the evolution of the apertured beam along the (y–z) plane distortions appear in its sides in the far-field region, whereas the propagated beam feature is not changed in the (x–z) plane. On the other hand, if we compare the comportment of both zeroth-order Mathieu–Gauss and Bessel–Gauss beams (plots are reported in Refs. [13,24]) we can deduce the similarity between the two beam propagation characteristics along (y–z) plane. In Fig. 5, we present plots of transverse intensity of the unapertured zeroth-order Mathieu–Gauss beam at z = 0.5, 1, 1.8, 3 m for a 0 = 16, w0 = 3 mm, q = 16 and k = 632.8 nm. This case corresponds to c = a 0 /2 = 8 and the plots show that the propagation properties of the beam are identical to those of the ideal Mathieu beam until zmax.
Fig. 5. Transverse intensity distribution of unapertured zeroth-order Mathieu–Gauss beam propagating in free space at z = 0.5, 1, 1.8, 3 m for q = 16, a 0 = 16, w0 = 3 mm and k = 632.8 nm: (a) evolution along (x, z) plane, and (b) evolution along (y, z) plane.
Fig. 6. Axial intensity of the focalized zeroth-order Mathieu–Gauss beam by a thin lens of focal length f = 0.5 m for q = 16, q0 = 2 mm, and k = 632.8 nm: (a) apertured beam with w0 = 3 mm and b = 0.01, 1, 10, 100, (b) unapertured beam with w0 = 0.5 mm and c = 0.01, 1, 10, and (c) unapertured beam with w0 = 4 mm and c = 0.01, 1, 10.
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4.2. Propagation through a thin lens The propagation through a thin lens is characterized by the following matrix transfer elements: A = 1 z/f, B = z, C = 1/f and D = 0, where f is the focal length of the lens. In Fig. 6(a), we present the axial intensity of the focused Mathieu–Gauss beam for f = 0.5 m, for a = 16, w0 = 3 mm and the truncation parameter b = 0.01, 1, 100. From these plots one can see the focalization properties of the apertured nondiffracting beam [28,29] for b 6 1 (i.e., c 1), particularly the presence of an apparent focus before the focal plane and a second peak beyond it. However, in the case of b > 1 (i.e., c < 1) only the first peak subsists and we have then a focused Gaussian behavior. In Figs. 6(b) and (c) we present the axial intensity of the focused unapertured Mathieu–Gauss beam for w0 = 0.5 mm and w0 = 4 mm, respectively. From the curves on can note the extinction of the axial oscillations due to the hard aperture, and the existence of two peaks before and after the focal plane for c 1. These peaks are clearly shown in Fig. 6(c). In Fig. 6(b) the second peak is less intense, i.e., the values of kt and w0 cannot lead to the reconstruction of the beam beyond the focal plane [29]. On the other hand, for c 1 (Fig. 6(b)) we retrieve the behaviors of the focalized Gaussian beam and the well known focal shift [30,31]. For c = 1, the focalization properties are similar to those of Gaussian beam in the case of Fig. 6(b) and to those of ideal Mathieu beam in Fig. 6(c). In the last case, the peaks of maximum intensity are reached near the geometrical focus, thus in this case there is approximately no focal shift for the unapertured Mathieu beam. 5. Conclusion By using the expansion of Mathieu–Gauss beams in terms of Bessel–Gauss beams, we have derived the amplitude expression of Mathieu–Gauss beams propagating through an ABCD optical system in both apertured and unapertured cases. An exact and also an approximate propagation equation of zeroth-order Mathieu–Gauss beams propagating through an ABCD optical system have been derived. Moreover, the propagation properties of these beams in free space and through a thin lens have been illustrated with detailed numerical results. It is shown that, under particular conditions, Mathieu–Gauss beams have propagation characteristics similar to those of Bessel–Gauss beams, which are well known in the literature, especially if we consider the (y–z) plane as a plane of propagation. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]
J. Durnin, J. Opt. Soc. Am. A 4 (1987) 651. J. Durnin, J.J. Miceli Jr., J.H. Eberly, Phy. Rev. Lett. 58 (1987) 1499. S. Wang, Q. Lin, X. Lu, Optik 100 (1995) 8. J.C. Gutie´rrez-Vega, M.D. Iturbe-Castillo, S. Chavez-Cerda, Opt. Lett. 25 (2000) 1493. J.C. Gutie´rrez-Vega, M.D. Iturbe-Castillo, G.A. Ramirez, E. Tepichı´n, R.M. Rodriguez-Dagnino, S. Chavez-Cerda, G.H.C. New, Opt. Commun. 195 (2001) 35. A. Belafhal, Z. Hricha, Phys. Chem. News 16 (2004) 33. M.A. Bandres, J.C. Gutie´rrez-Vega, S. Chavez-Cerda, Opt. Lett. 29 (2004) 44. C. Lopez-Mariscal, M.A. Bandres, J.C. Gutie´rrez-Vega, S. Chavez-Cerda, Opt. Exp. 13 (2005) 2364. F. Gori, G. Guattari, C. Padovani, Opt. Commun. 64 (1987) 491. J.C. Gutie´rrez-Vega, M.A. Bandres, J. Opt. Soc. Am. A 22 (2005) 289. Y. Li, H. Lee, E. Wolf, J. Opt. Soc. Am. A 21 (2004) 640. A. Chafiq, Z. Hricha, A. Belafhal, Opt. Commun. 253 (2005) 223. D. Ding, X. Liu, J. Opt. Soc. Am. A 16 (1999) 1286. B. Lu¨, S. Luo, J. Mod. Opt. 48 (2001) 2169. D. Zhao, H. Mao, H. Liu, F. Jing, Q. Zhu, X. Wei, Optik 114 (2003) 504. D. Zhao, H. Mao, W. Zhang, S. Wang, Opt. Commun. 224 (2003) 5. B. Lu¨, X. Ji, J. Opt. A: Pure Appl. Opt. 6 (2004) 161. D. Zhao, H. Mao, M. Shen, H. Liu, F. Jing, Q. Zhu, X. Wie, J. Opt. A: Pure Appl. Opt. 6 (2004) 148. Z. Mei, D. Zhao, J. Opt. A: Pure Appl. Opt. 6 (2004) 1005. X. Ji, B. Lu¨, Optik 116 (2005) 219. G. Indebetouw, J. Opt. Soc. Am. A 6 (1989) 150. I.S. Gradshteyn, I.M. Ryzhik, Tables of Integrals, Series, and Products, fifth ed., Academic Press, New York, 1994. S.A. Collins, J. Opt. Soc. Am. 60 (1970) 1168. P.L. Overfelt, C.S. Kenney, J. Opt. Soc. Am. A 8 (1991) 732. A. Belafhal, L. Dalil-Essakali, Opt. Commun. 177 (2000) 181. J.J. Wen, M.A. Breazeale, J. Acoust. Soc. Am. 83 (1988) 1752. J.C. Gutie´rrez-Vega, Theory and Numerical analysis of the Mathieu functions, Monterrey, NL, Me´xico, 2003. Available at: http:// homepages.mty.itesm.mx/jgutierr/. S. Chavez-Cerda, G.H.C. New, Opt. Commun. 181 (2000) 369. J.C. Gutie´rrez-Vega, R. Rodriguez-Masegosa, S. Chavez-Cerda, J. Opt. A: Pure Appl. Opt. 5 (2003) 276. Y. Li, Opt. Commun. 95 (1993) 13. B. Lu, W. Huang, Opt. Commun. 109 (1994) 43.
A detailed study of Mathieu–Gauss beams propagation through an apertured ABCD optical system A. Chafiq, Z. Hricha, A. Belafhal
*
Laboratoire de Physique Mole´culaire, De´partement de Physique, Universite´ Chouaı¨b Doukkali, Faculte´ des Sciences, B. P. 20, 24000 El Jadida, Morocco Received 27 November 2005; received in revised form 19 March 2006; accepted 21 March 2006
Abstract Using Collins formula and the expansion of Mathieu beams in terms of Bessel beams we derive the exact propagation equations of Mathieu–Gauss beams through an apertured paraxial ABCD optical system. A comparison between the exact propagation equations and the approximated ones, which are derived by expanding the circ function into a finite sum of Gaussian functions, is presented. The propagation characteristics of zeroth-order Mathieu–Gauss beams in (y–z) and (x–z) planes are analyzed with detailed numerical calculations. It is found that the profile of the propagated Mathieu–Gauss beam is similar to that of Bessel–Gauss beam. Furthermore, the focalization of the Mathieu–Gauss beams through a thin lens is illustrated and analyzed with numerical results. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Collins formula; Mathieu beams; Mathieu–Gauss beams; Bessel–Gauss beams; Aperture function
1. Introduction In recent years, an increasing amount of attention has been paid to the nondiffracting beams in both optics and acoustics. The importance of these beams is due to the fact that they can propagate indefinitely without any change in their transverse shape. Their potential applications in wireless communications, optical interconnections, laser machining, medical imaging and surgery, among others, make them very relevant. Durnin et al. [1,2] have introduced the first kind of these beams called Bessel beams which are solutions of Helmholtz wave equation expressed in circular cylindrical coordinates. Three other nondiffracting solutions of Helmholtz equation namely, cosine [3], Mathieu [4–6] and parabolic [7,8] beams were reported in last few years. The ideal nondiffracting beams have an infinite extent and energy, so they are not physically realizable. For that reason, many authors have been devoted to study approximated versions of these ideal beams. Gori et al. [9] have proposed to modulate them by a Gaussian function and have introduced the most important family of the approximated nondiffracting beams named as Bessel–Gaussian beams. More recently, Gutie´rrez-Vega and Bandres [10] have presented a detailed study of the so-called Helmholtz–Gauss beams which are a general nondiffracting beams modulated by a Gaussian function. Bessel–Gauss and Mathieu–Gauss beams are then regarded as members of these beam families. In Ref. [10], the propagation of the Helmholtz–Gauss beams was performed in a general way by the resolution of Helmholtz wave equation in paraxial approximation. However, only the propagation in free space has been treated. On the other hand, Li et al. [11] have proposed new beams obtained by modulating Bessel beams by flat-topped Gaussian envelopes. The authors showed that these beams can be expressed as a sum of Gori Bessel–Gaussian beams. The main contribution of this work is to give a detailed *
Corresponding author. Tel.: +212 68 50 43 44; fax: +212 23 35 34 54. E-mail address: [email protected] (A. Belafhal).
0030-4018/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.03.048
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analytical study of the propagation of Mathieu–Gauss beams through an apertured ABCD optical system by using the Collins diffraction formula and by expanding these beams in terms of Bessel–Gauss beams. A rigorous and also an approximated calculation of the propagation equations of these beams through an apertured and unapertured ABCD optical systems are derived using the fact that the aperture function can be expanded into a finite sum of Gaussian functions. This last technique has been extensively used by many authors in the last few years in deriving approximated analytical propagation equations and factors of beams propagating through an apertured paraxial optical systems (see for example Refs. [13–20]). This paper is organized as follows: the analytical expression of Mathieu–Gauss field expressed in terms of Bessel–Gauss beams is given in Section 2. In Section 3, the paraxial propagation equations of Mathieu–Gauss beams through an apertured and unapertured ABCD optical systems are derived. Some numerical calculations and analysis are given in Section 4. Finally, we end this paper with a conclusion in Section 5. 2. Mathieu–Gauss beams The expansion of Mathieu beam in terms of Bessel functions basis is given by (details are found in Ref. [12]) 8 1 P ð2nÞ j > uc2n ðq; u; q; z ¼ 0Þ ¼ A2j ðqÞð1Þ cosð2juÞJ 2j ðk t qÞ; > > > j¼0 > > > > 1 > P ð2nþ1Þ 2jþ1 > > cos½ð2j þ 1ÞuJ 2jþ1 ðk t qÞ; > < uc2nþ1 ðq; u; q; z ¼ 0Þ ¼ j¼0 A2jþ1 ðqÞðiÞ 1 P > ð2nþ2Þ > > us2nþ2 ðq; u; q; z ¼ 0Þ ¼ B2jþ2 ðqÞð1Þjþ1 sin½ð2j þ 2ÞuJ 2jþ2 ðk t qÞ; > > > j¼0 > > > 1 > > > us2nþ1 ðq; u; q; z ¼ 0Þ ¼ P Bð2nþ1Þ ðqÞðiÞ2jþ1 sin½ð2j þ 1ÞuJ 2jþ1 ðk t qÞ; : 2jþ1
ð1Þ
j¼0
where ucm and usm are the even and odd modes of Mathieu beams. q, u, z are the cylindrical variables, and kt = k sin /0, kz = k cos /0 are the transverse and longitudinal components of the wave vector ~ k of plane waves that reconstruct the beam [21]. /0 is the semi-angle of the cone on which lie the wave’s vectors associated with the plane waves. q is the ellipticity ð2nÞ ð2nþ1Þ ð2nþ2Þ parameter given by q ¼ h2 k 2t =4, where 2h is the interfocal separation. The coefficients A2j ðqÞ; A2jþ1 ðqÞ; B2jþ2 ðqÞ, and ð2nþ1Þ B2jþ1 ðqÞ are expansion coefficients of Mathieu functions [22], that respect of each ellipticity parameter q a set of recurrence equations and J is Bessel function of the first kind. For the sake of simplicity a constant factor of 2p was omitted in Eq. (1). The insertion of a Gaussian term in the expressions of Eqs. (1) leads to the field distribution of Mathieu–Gauss beams at the plane z = 0 q2 e MGm ðq; u; z ¼ 0Þ ¼ ucm ðq; u; q; z ¼ 0Þ exp 2 ; ð2:aÞ w0 q2 ð2:bÞ MGmo ðq; u; z ¼ 0Þ ¼ usm ðq; u; q; z ¼ 0Þ exp 2 ; w0 where the superscripts e and o design even and odd modes of Mathieu–Gauss beams, and w0 is the waist of the Gaussian beam. The exponential term of Eqs. (2) can be inserted in the summation of ucm and usm because it is independent on the index summation. Subsequently, one can easily deduce that the Mathieu–Gauss beams can be built-up by a superposition of Bessel–Gauss beams. In Fig. 1, the two dimensional behaviors of both zeroth-order Mathieu–Gauss beam obtained from Eqs. (2) and the ideal Mathieu beam are presented. By comparing the reported profiles one can see the reduction of the intensity in the sides of the Mathieu–Gauss beam which is due the Gaussian envelope. 3. Propagation equations of Mathieu–Gauss through ABCD optical systems As it is well known, the propagation of beams through a paraxial optical system can be described by Collins formula [23]. Thus, if we consider an apertured paraxial ABCD optical system illuminated by one of the Mathieu–Gauss beams family of Eqs. (2), the relationship between the input and the output field distributions are given by Z q0 Z 2p ik k e, o e, o ðoutÞ ðinÞ 2 2 MGm ðq2 ; h; zÞ ¼ q1 MGm ðq1 ; u; z ¼ 0Þ exp ikz þ i ðAq1 2q1 q2 cosðu hÞ þ Dq2 Þ dq1 du; 2pB 0 2B 0 ð3Þ
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Fig. 1. Transverse intensity distribution of zeroth-order Mathieu–Gauss and Mathieu beams at plane z = 0 for the parameters kt = 15000 m1 and q = 25: (a) Mathieu–Gauss beam with a Gaussian waist w0 = 2 mm, and (b) ideal nondiffracting Mathieu beam.
oðoutÞ ðq ; h; zÞ is where MGme, oðinÞ ðq1 ; u; z ¼ 0Þ is the field distribution at the point (q1, u) in the input plane z = 0, and MGe, 2 m the field distribution at the point (q2, h) in the output plane located at z distance. A, B, C and D are the elements of the transfer matrix characterizing the optical system. k is the wave vector and q0 is the radius of the input aperture. For the sake of simplicity, we consider only the first family of the above beams i.e., the even modes MGe2n , the treatment of the other Mathieu–Gauss family is the same. Substituting from Eqs. (2) into Eq. (3) and using the following equality [22] 1 X in J n ðzÞein/ . ð4Þ eðiz cos /Þ ¼ 1
Eq. (3) becomes eðoutÞ
MG2n
1 X 2i 2 ðr; h; zÞ ¼ eðikzþiDr =gÞ A2n 2j ðqÞ cosð2jhÞ g j¼0
Z 0
1
2rq qJ 2j ðaqÞJ 2j expðQq2 Þdq; g
ð5Þ
where A Q¼bi ; g g ¼ B=ld ;
ð6:aÞ 2
ð6:bÞ
q0 w0
and where r = q2/q0, q = q1/q0 and a = ktq0 are the non-dimensional variables and b ¼ is the truncation parameter. In these expressions ld ¼ kq20 =2 is the diffraction length related to the aperture radius. From Eq. (5) it is clear that the propagation characteristics of Mathieu–Gauss beams can be easily deduced from those of Bessel–Gauss beams [9,13,24,25]. In the following, we will perform the integral expression of Eq. (5) for the apertured and unapertured optical systems. 3.1. Apertured case 3.1.1. Exact propagation equation of Mathieu–Gauss beams The finite integral of Eq. (5) can be performed in the same way as the method used in Ref. [24] by Overfelt and Kenney So, using the fact that 0 j 6¼ 0; J 2j ð0Þ ¼ ð7Þ 1 j¼0 the axial field distribution of the Mathieu–Gauss beam can be written as 8 1 P k > > j2Q=aj < 1; expðQÞ ð2Q=aÞ J k ðaÞ < i ikz ð2nÞ k¼1 eðoutÞ e A0 ðqÞ MG2n ð0; zÞ ¼ 1 P > Qg > : expða2 =4QÞ expðQÞ ða=2QÞk J k ðaÞ j2Q=aj > 1: k¼0
ð8Þ
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From Eq. (8), we can deduce that the axial intensity of the even family of Mathieu–Gauss beams is the product of the axial ð2nÞ intensity of the zeroth-order Bessel–Gauss beams and the coefficient A0 ðqÞ. We note that if we repeat the above calculations for the odd Mathieu–Gauss family we will find that their axial intensity is equal to zero. To perform the integral of Eq. (5) in the transverse plane one can use the product of Bessel functions formula [22] J m ðazÞJ l ðbzÞ ¼
1 2 2 ðaz=2Þm ðbz=2Þl X ð1Þk ðaz=2Þ2k 2 F 1 ðk; m k; l þ 1; b =a Þ ; k!Cðm þ k þ 1Þ Cðl þ 1Þ k¼0
ð9Þ
where C is the gamma function and 2F1 is the hypergeometric function. So, After some algebraic calculations Eq. (5) can be rewritten as 1 1 1 X X A2n i ð1Þm Qm X 2 2j ðqÞ eðoutÞ 2j ðar=2gÞ cosð2jhÞ MG2n ðr; h; zÞ ¼ eðikzþiDr =gÞ Gk ðj; m; a; g; rÞ; g ð2jÞ! m! j¼0 m¼0 k¼0
ð10Þ
where 8
2r 2 2 ð1Þk ða=2Þ2k > 2 F 1 k;2jk; 2jþ1; ð g Þ =a > < k!ð2jþkÞ!ð2jþmþkþ1Þ
Gk ðj; m; a; g; rÞ ¼ k 2k 2 2r 2 > > : ð1Þ ðr=gÞ2 F 1 k;2jk; 2jþ1; a =ð g Þ k!ð2jþkÞ!ð2jþmþkþ1Þ
2r g =a < 1; 2r g =a > 1:
ð11Þ
If the exponential term involved in the integral of Eq. (5) is not written in its series form, the summation over the index m in Eq. (10) can be substituted by the incomplete gamma function. It is to be noted that Eq. (8) can be obtained from Eq. (10) by taking r = 0. 3.1.2. Approximate propagation equation of Mathieu–Gauss beams To describe approximately the propagation of Mathieu–Gauss beams through an apertured ABCD optical system, we use the important technique in which the finite integral of Eq. (5) is transformed into an infinite one, by using the circ function 1 0 6 q 6 1; circðqÞ ¼ ð12Þ 0 q > 1; and afterwards expanding it into a finite sum of complex Gaussian functions circðqÞ ¼
kX ¼10
Ek expðLk q2 Þ;
ð13Þ
k¼1
where Ek and Lk are the expansion and Gaussian coefficients, respectively which are found by a computer optimization [26]. So, by inserting this aperture function in the integral of Eq. (5) and by using the following integral result [22] 2 Z 1 1 rd r þ d2 xJ l ðrxÞJ l ðdxÞ expðmx2 Þdx ¼ I l exp ð14Þ 2m 2m 4m 0 with jarg mj < p4, Re r > 0 and Re d > 0, where Il is the modified Bessel function of the first kind, and after some algebraic calculations, the output field distribution reads X X ð2nÞ 2 eðoutÞ 2 j MG2n ðr; h; zÞ ¼ ieðikzþiDr =gÞ Ek ð1=S k gÞ exp½ððagÞ þ 4r2 Þ=4g2 S k A2j ðqÞð1Þ cosð2jhÞJ 2j ðiar=S k gÞ; k ¼10
1
k¼1
j¼0
ð15Þ
where S k ¼ Lk þ Q.
ð16Þ
Eq. (15) shows that the propagated Mathieu–Gauss beams have a similar expansion form of the propagated Mathieu beams [12] but with an added exponential term characterizing the paraxial approximation and the modulation by the Gaussian function. Eq. (15) leads to the axial intensity expression IðzÞ ¼
kX ¼10 m ¼10 X k¼1
m¼1
fk ða; b; A; BÞfm ða; b; A; BÞ;
ð17Þ
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where fn ða; b; A; BÞ ¼ En ð1=S n gÞ expða2 =4S n Þ
ð18Þ
with the * representing the complex conjugation, n is an integer index. From Eq. (18) it is shown that the axial intensity depends essentially on the parameter of truncation b, the transfer matrix coefficients A and B implicit in Sk and kt implicit in a. On the other hand, it must be stressed out that the method used above in solving the finite integral of Eq. (5) can be applied to the other family of Helmholtz–Gauss beams. Particularly, Ding and Liu [13] and Zhao et al. [15,16] have used this technique to derive approximated propagation equations of Bessel–Gauss beams and cosine-Gaussian beams, respectively, through apertured ABCD optical systems. 3.2. Unapertured case Let us now consider a Mathieu–Gauss beam propagating through an ideal unapertured ABCD optical system, i.e., the aperture radius is equal to infinity. By using in a second time the equality of Eq. (14), the expression of the output field distribution (Eq. (5)) reduces to h i 1 eðoutÞ 02 0 0 2 02 0 0 MG2n ðr0 ; h0 ; zÞ ¼ expðikz þ iDr =g Þ exp ððag Þ þ 4r Þ=4g ðg iAÞ A þ ig0 1 X ð2nÞ j A2j ðqÞð1Þ cosð2jh0 ÞJ 2j ðia0 r0 =g0 iAÞ; ð19Þ j¼0
where r = q2/w0, a = ktw0 and g 0 = B/zR are the non-dimensional variables in this paragraph. With zR ¼ kw20 =2 is the Rayleigh length related to the Gaussian waist. From Eq. (19) one can deduce the axial intensity 0
IðzÞ ¼
0
1 2
A þ ðg0 Þ
2
2
2
exp½ðag0 Þ =2ððg0 Þ þ A2 Þ.
ð20Þ
It is to be noted that Eq. (20) is in accordance with results of Ref. [10] and it can also be regarded as a generalized form of Eq. (15) in that reference. Furthermore, Eq. (19) can be obtained from Eq. (10) when the aperture radius approaches infinity. 4. Numerical calculation results and discussion To study the propagation properties of Mathieu–Gauss beams through an ABCD optical system, the exact analytical form of Eqs. (8) and (10) can be used. Nevertheless Eq. (10) will lead to numerical summation of hypergeometric functions. To avoid this difficulty the approximated expression of Eq. (15) should be used. In this case we have to truncate the expansion series building-up Mathieu–Gauss beams, this operation can be done without any problem because there are only some coefficients that possess a significant weight for any value of the parameter q [27]. These coefficients permit us to build-up the Mathieu–Gauss beams from the Bessel–Gaussian beams with good a accuracy. 4.1. Free-space propagation The propagation in free space is characterized by the following transfer matrix A B 1 z ¼ . C D 0 1 In Fig. 2, we present a comparison between the plots of the numerical calculations of the exact and the approximated axial propagation which correspond to Eqs. (8) and (15), respectively. From these plots we can see the coincidence of the curves in far-field and the disagreement in the near-field. The expression of Eq. (15) is advantageous in obtaining the field amplitude distribution because it is written as a summation of ten terms of Mathieu–Gauss beams. Thus, Eq. (15) can be used as an alternative expression in analyzing the propagation of the apertured Mathieu–Gauss beam. As it is pointed out in Refs. [9,10], the propagation properties of any nondiffracting beam modulated by the Gaussian envelope are generally determined by the following parameters: the width of Mathieu fringes D p/kt, the aperture radius q0 and the Gaussian width w0. In order to analyze in detail the dependence of the behavior of the Mathieu–Gauss beam on these parameters, we give in Fig. 3(a) plots of the axial intensity distribution of the propagated zeroth-order Mathieu– Gauss beams in free space for a = 16, q = 16, wavelength k = 632.8 nm and b = 0.01, 1, 100. In Fig. 3(b) the axial intensity of the unapertured beam is reported for q = 16, 200 and w0 = 3 mm (a and k are the same as in Fig. 3(a)). In order to
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Fig. 2. Comparison of the axial intensity distribution of zeroth-order Mathieu–Gauss beam propagating in free space for q = 25, a = 10, b = 0.5, k = 632.8 nm and q0 = 2 mm evaluated approximately using the circ function and exact evaluation of Eq. (8).
Fig. 3. Axial intensity distribution of zeroth-order Mathieu–Gauss beam propagating in free space for k = 632.8 nm and q0 = 2 mm: (a) apertured beam for a = 16, q = 16, the values of the truncation parameter are b = 0.01, 1, 100, and (b) unapertured beam for a 0 = 16, w0 = 3 mm and q = 16, 200.
analyze the behavior of the beam intensity in Fig. 3, we introduce the parameter c ¼ 12 k t w0 which depends on D, q0 and w0. This parameter plays an important role in determining the behavior of Mathieu–Gauss beam upon propagation [10]. For a given value of D, we can distinguish three cases: (i) c 1: the beam retains the nondiffracting propagation properties of
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ideal Mathieu beam until a ‘maximum’ propagation distance zmax, (ii) c 1: the beam have a Gaussian behavior and (iii) c = 1: this case corresponds to the transition between the two behaviors. In the case of Fig. 3, we have c 1 for both b 6 1 and the unapertured system, and c < 1 for b > 1. Thus, the beam retains the nondiffracting propagation properties of apertured ideal Mathieu beam in the first case, and the Gaussian behavior in the second case. On the other hand, for b = 0.01, the spot size of the Gaussian envelope (at z = 0) is 20 mm and the width of Mathieu fringes is 0.39 mm. This means that the wave front of the Gaussian components which construct the beam in the aperture plane is nearly flattened and therefore the beam extends far and far from the aperture. Following this picture, for b = 100, the diffractive Gaussian effect is imposed and zmax plane is near the aperture. In all cases, after the maximum propagation distance zmax the components of beam diverge and overlap lesser and lesser that’s why the axial irradiance decreases. In Fig. 4 we present the plots of the transverse intensity of apertured zeroth-order Mathieu–Gauss propagating in free space for different values of z. The curves
Fig. 4. Transverse intensity distribution of apertured zeroth-order Mathieu–Gauss beam propagating in free space at z = 0.5, 1, 1.8 m truncated by b = 0.01 (continuous lines), b = 1 (dotted lines) and b = 100 (dashed–dotted lines), the other parameters are q = 16, a = 16, q0 = 2 mm and k = 632.8 nm: (a)–(c) evolution along (x, z) plane, (d)–(f) evolution along (y, z) plane.
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allow us to deduce that the beam propagates in both (y–z) and (x–z) planes without significant divergence of central peaks. When we consider the evolution of the apertured beam along the (y–z) plane distortions appear in its sides in the far-field region, whereas the propagated beam feature is not changed in the (x–z) plane. On the other hand, if we compare the comportment of both zeroth-order Mathieu–Gauss and Bessel–Gauss beams (plots are reported in Refs. [13,24]) we can deduce the similarity between the two beam propagation characteristics along (y–z) plane. In Fig. 5, we present plots of transverse intensity of the unapertured zeroth-order Mathieu–Gauss beam at z = 0.5, 1, 1.8, 3 m for a 0 = 16, w0 = 3 mm, q = 16 and k = 632.8 nm. This case corresponds to c = a 0 /2 = 8 and the plots show that the propagation properties of the beam are identical to those of the ideal Mathieu beam until zmax.
Fig. 5. Transverse intensity distribution of unapertured zeroth-order Mathieu–Gauss beam propagating in free space at z = 0.5, 1, 1.8, 3 m for q = 16, a 0 = 16, w0 = 3 mm and k = 632.8 nm: (a) evolution along (x, z) plane, and (b) evolution along (y, z) plane.
Fig. 6. Axial intensity of the focalized zeroth-order Mathieu–Gauss beam by a thin lens of focal length f = 0.5 m for q = 16, q0 = 2 mm, and k = 632.8 nm: (a) apertured beam with w0 = 3 mm and b = 0.01, 1, 10, 100, (b) unapertured beam with w0 = 0.5 mm and c = 0.01, 1, 10, and (c) unapertured beam with w0 = 4 mm and c = 0.01, 1, 10.
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4.2. Propagation through a thin lens The propagation through a thin lens is characterized by the following matrix transfer elements: A = 1 z/f, B = z, C = 1/f and D = 0, where f is the focal length of the lens. In Fig. 6(a), we present the axial intensity of the focused Mathieu–Gauss beam for f = 0.5 m, for a = 16, w0 = 3 mm and the truncation parameter b = 0.01, 1, 100. From these plots one can see the focalization properties of the apertured nondiffracting beam [28,29] for b 6 1 (i.e., c 1), particularly the presence of an apparent focus before the focal plane and a second peak beyond it. However, in the case of b > 1 (i.e., c < 1) only the first peak subsists and we have then a focused Gaussian behavior. In Figs. 6(b) and (c) we present the axial intensity of the focused unapertured Mathieu–Gauss beam for w0 = 0.5 mm and w0 = 4 mm, respectively. From the curves on can note the extinction of the axial oscillations due to the hard aperture, and the existence of two peaks before and after the focal plane for c 1. These peaks are clearly shown in Fig. 6(c). In Fig. 6(b) the second peak is less intense, i.e., the values of kt and w0 cannot lead to the reconstruction of the beam beyond the focal plane [29]. On the other hand, for c 1 (Fig. 6(b)) we retrieve the behaviors of the focalized Gaussian beam and the well known focal shift [30,31]. For c = 1, the focalization properties are similar to those of Gaussian beam in the case of Fig. 6(b) and to those of ideal Mathieu beam in Fig. 6(c). In the last case, the peaks of maximum intensity are reached near the geometrical focus, thus in this case there is approximately no focal shift for the unapertured Mathieu beam. 5. Conclusion By using the expansion of Mathieu–Gauss beams in terms of Bessel–Gauss beams, we have derived the amplitude expression of Mathieu–Gauss beams propagating through an ABCD optical system in both apertured and unapertured cases. An exact and also an approximate propagation equation of zeroth-order Mathieu–Gauss beams propagating through an ABCD optical system have been derived. Moreover, the propagation properties of these beams in free space and through a thin lens have been illustrated with detailed numerical results. It is shown that, under particular conditions, Mathieu–Gauss beams have propagation characteristics similar to those of Bessel–Gauss beams, which are well known in the literature, especially if we consider the (y–z) plane as a plane of propagation. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]
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