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Flat-topped Mathieu-Gauss beam and its transformation by paraxial optical systems
Optics Communications 278 (2007) 142–146 www.elsevier.com/locate/optcom
Flat-topped Mathieu-Gauss beam and its transformation by paraxial optical systems A. Chafiq, Z. Hricha, A. Belafhal
*
Laboratoire de Physique Mole´culaire, De´partement de Physique, Universite´ Chouaı¨b Doukkali, Faculte´ des Sciences, BP 20, 24000 El Jadida, Morocco Received 21 February 2007; received in revised form 15 May 2007; accepted 21 May 2007
Abstract In order to avoid the severe axial intensity oscillations which appear when an ideal Mathieu beam is truncated, we propose to modulate it by a flattened multi-Gaussian envelope. The obtained beam, which is referred as flattened Mathieu-Gauss (FTMG) beam, can be expanded into a finite series of Mathieu-Gauss beams with various waists. The propagation study of this beam reveals that the axial intensity is unchanged within a certain propagation distance as for super-Gaussian-Bessel beams or Gori’s flattened Bessel beams. By using Collins formula, we derive closed-form expressions of FTMG beam propagating through a paraxial axisymmetric ABCD optical system. Some numerical calculations and discussions are also given. Ó 2007 Published by Elsevier B.V. Keywords: Flattened Mathieu-Gauss (FTMG) beam; Mathieu beam; Collins formula; Bessel-Gauss beams
1. Introduction Mathieu beams constitute a complete and orthogonal family of nondiffracting optical beams, i.e. beams whose profile does not change with propagation distance. Their non-azimuthal symmetry and helical feature make them very relevant for theoretical and practical applications [1– 7]. However, ideal nondiffracting beams cannot really exist because they would carry an infinite energy. In practice these beams are apertured more or less. The hard-edged diffracted Mathieu beam has a near nondiffracting behavior within a maximal distance propagation zmax. Nevertheless, its axial intensity presents oscillations with the propagation distance. In order to avoid these oscillations, many authors have proposed to modulate the nondiffracting beams by a variety of aperture functions. Among them, the super-Gaussian [8] and Gori’s flattened apertures [9] are the most used. In the last decade, Jiang et al. [10,11] have demonstrated the existence of an optimal order of
*
Corresponding author. E-mail address: [email protected] (A. Belafhal).
0030-4018/$ - see front matter Ó 2007 Published by Elsevier B.V. doi:10.1016/j.optcom.2007.05.042
super-Gaussian envelope for which the Bessel beam axial intensity oscillations vanished. And they found that the relationships of the axial intensity versus propagation distance are similar to the radial distributions of the aperture. Particularly, the axial intensity does not oscillate when we use either airy or triangle apertures. Recently, Li et al. [12,13] have used the flat-topped aperture to derive a new expression of generalized Bessel-Gaussian beam. The authors showed that a generalized Bessel beam could be expressed as an arbitrary combination of Bessel-Gaussian beams. On the other hand, in a very recent works [14,15], we have demonstrated that Mathieu beam can be expanded into an infinite sum of higher-order Bessel beams with a well defined expansion coefficients and spatial frequencies. So, the theory of Li et al. [12,13] can be applied to Mathieu beams to produce FTMG beams. In the present work, we propose to study the propagation of FTMG beams through an ABCD optical system and particularly through free space and a simple thin lens. In Section 2, we give the analytical expressions of FTMG field expressed in terms of Mathieu-Gauss beams of various waists. The paraxial propagation equations of the beam through an unapertured ABCD optical system are
A. Chafiq et al. / Optics Communications 278 (2007) 142–146
derived in Section 3. Some numerical calculations and analysis are given in Section 4. We end this paper with a conclusion. 2. Field expressions of FTMG beams In the literature, many papers have used aperture functions which have a flattened profile to modulate the nondiffracting beams. However, most papers which deal with the propagation of the apertured beams have treated the problem numerically (see for example Refs. [10,11]). Indeed, as usual one could not find the closed-form equation, and tedious straightforward numerical integrals are unavoidable. Fortunately, if we consider the flattened aperture function introduced by Li et al. [12,13], one can resolve analytically the propagation problem of the apertured Mathieu beams. The expression of this aperture function is M
F M ðqÞ ¼ 1 ½1 expðq2 Þ ;
ð1Þ
where M is a non-negative number, which can be an integer, a fraction, a decimal or an irrational number. q is the radial cylindrical coordinate. It was demonstrated [12,13] that FM(q) can be re-scaled by introducing the beam width w0 of pure Gaussian profile, which correspond to M = 1. The new function reads M X q2 GM ðqÞ ¼ gm;M exp mb 2 ; ð2aÞ w0 m¼1
GM(q) versus the value of M. As it can be seen from the plots, the profile becomes more and more flattened when M is increased. The modulation of nth-order even and odd Mathieu beams by the above aperture function leads to the following expressions of FTMG beams at the plane z=0 M X Een;M ðn; g; k 1t ; z ¼ 0Þ ¼ gm;M W en ðn; g; k 1t ; z ¼ 0Þ m¼1
q2 exp mb 2 ; w0 M X gm;M W on ðn; g; k 1t ; z ¼ 0Þ Eon;M ðn; g; k 1t ; z ¼ 0Þ ¼
ð3aÞ
m¼1
q2 exp mb 2 ; w0
ð3bÞ
where W e;o n ð:Þ are the even (the superscripts e) and odd (the superscripts o) Mathieu modes. The integer index is n P 0 or n P 1 for even and odd modes, respectively. (n, g), q denote the transverse elliptic coordinate and the radial variable of cylindrical coordinate, respectively. And k1t is the wave vector transverse component related to the ellipticity parameter e by e ¼ h2 k 21t =4, where h is the semi-interfocal separation [16]. If we set pffiffiffiffiffiffiffi ð4Þ wm ¼ w0 = mb; FTMG beams can then be expanded in terms of MG beams of various waists as
with gm;M ¼ ð1Þ b¼
143
M X m¼1
1 ; m
mþ1
MðM 1Þ ðM m þ 1Þ ; m!
ð2bÞ ð2cÞ
where m! is the factorial of m. It is to be noted, that the summation of Eq. (2a) is extended to infinity if M is not an integer. In the following, we will be interested only in the integer values of M. In Fig. 1, we present the profile of the function
Ee;o n;M ðn; g; k 1t ; z ¼ 0Þ ¼
M X
gm;M MGe;o n;m ðn; g; k 1t ; z ¼ 0Þ;
ð5aÞ
m¼1
with e;o 2 2 MGe;o n;m ðn; g; k 1t ; z ¼ 0; Þ ¼ W n ðn; g; k 1t ; z ¼ 0Þ expðq =wm Þ:
ð5bÞ By adopting the series expansion [14], even and odd Mathieu modes can be written as a sum of Bessel functions of higher-order W en ðq; u; k 1t ; z ¼ 0Þ ¼
1 X
ðnÞ
j
Aj ðeÞðiÞ cosðjuÞJ j ðk 1t qÞ;
ð6aÞ
j¼0
W on ðq; u; k 1t ; z ¼ 0Þ ¼
1 X
ðnÞ
Bj ðeÞðiÞj sinðjuÞJ j ðk 1t qÞ;
ð6bÞ
j¼1 ðnÞ
Fig. 1. Plot of flat-topped profiles GM(q/w0) of Eq. (2a) for M = {1, 2, 5, 10, 20, 40}.
ðnÞ
where J is Bessel function of the first kind. Aj ð:Þ, Bj ð:Þ are expansion coefficients of Mathieu functions [16,17]. u denotes the angular variables of cylindrical coordinates. A constant factor in Eqs. (6) has been omitted. By inserting expressions of Eqs. (6) in Eqs. (5), one can easily deduce the expression of FTMG beam in polar coordinates. The obtained expression will be used in the next paragraph to derive a closed-form expression of the beam propagation through an unapertured ABCD optical system.
A. Chafiq et al. / Optics Communications 278 (2007) 142–146
3. Propagation equation of FTMG beam through a paraxial ABCD optical system Assume that the field distribution of FTMG beam at the input plane z = 0 of a paraxial ABCD optical system is given by Eqs. (5). The propagating field distribution of FTMG beam through this system is obtained straightforwardly using Collins diffraction integral formula [18] and the result of Ref. [15]. One obtains Ee;o n;M ðq; u; zÞ M X gm;M k 21t B kq2 i exp i ¼ A þ imbB=zR 2BðA þ imbB=zR Þ 2kðA þ imbB=zR Þ m¼1 q 2 e;o ; u; k 1t ; z ¼ 0 ð7Þ expðikz þ ikDq =2BÞW n A þ imbB=zR
where zR ¼ kw20 =2 is the Rayleigh length of the pure Gaussian profile. A, B, C and D are the elements of the transfer matrix characterizing the paraxial optical system. k is the wave vector. A constant phase term in Eq. (7) has been omitted which has non influence on the output intensity distribution. It is to be noted that the propagated FTMG beam can also be expressed in elliptic coordinates by using the very recent result on generalized Helmholtz-Gauss beams by Guizar-Sicairos and Gutie´rrez-Vega [19]. In this case, the field distribution reads M X k 1t k 2t e;o En;M ðn; g; zÞ ¼ gm;M exp i2k m¼1 GBm ðq; q2;m ÞW e;o ðn; g; k 2t ; z ¼ 0Þ;
ð8aÞ
where Aq1;m þ B ; Cq1;m þ D k 1t k 2t ¼ A þ B=q1;m
ð8bÞ
q2;m ¼
ð8cÞ
and
! expðikzÞ ikq2 GBm ðq; q2;m Þ ¼ exp : A þ B=q1;m 2q2;m
c ¼ 12k 1t w0 :
ð11Þ
The analysis of the propagation characteristics of FTMG beam allows us to distinguish three cases [16,21,22]: (i) c 1: the propagation properties are those of the flattopped envelope. (ii) c 1: the propagation is imposed by the nondiffracting beam and the nondiffracting behaviour will be significant until a maximum propagation distance, zmax zcR . (iii) c 1: we have a transition zone between the last cases. One can point out that a wide variety of beam shapes can be produced by choosing adequate values of the parameters k1t, w0, M and mode number n. 4. Numerical calculations and analysis In this section, we are interested in analyzing the propagation of FTMG beam for c 1. As a first example, we consider free space propagation for which the transfer matrix coefficients are: A = 1, B = z, C = 0 and D = 1. In order to confirm numerically the equivalence between Eqs. (7) and (8a), we present in Fig. 2 the axial intensity of ordinary even zeroth order FTMG beam calculated by these equations for k = 632.8 nm, e = 25, w0 = 50 mm, M = 6 and a = k1tw0 = 40. The plot shows that the considered equations give the same results. From numerical point of view, Mathieu beam field is calculated with two summations in elliptic coordinates (see for example the algorithms for calculating radial and angular Mathieu functions in Ref. [17]) whereas this field is calculated only with one
ð8dÞ
With q1,m, q2,m are the complex input and output parameters of the Gaussian beam transformed by the ABCD optical system [20], k2t is the output transverse wavenumber and the optical path length from the input to the output plane measured along the optical axis is taken equal to z. As it was pointed out in Ref. [19] the expression of Eq. (8) is a generalized formula of propagation of an arbitrary generalized Helmholtz-Gauss beam (gHzG) for which k1t and q1,m can be complex. For ordinary Mathieu-Gauss beam, k1t is real and the input parameter q1,m is purely imaginary, i.e., q1;m ¼ i12kw2m :
On the other hand, as it is shown in Fig. 1, the width of the flat-topped envelope is larger than the Gaussian waist w0 (which is equal to w1) for all M > 1. So, the effect of the modulation envelope on propagation properties will be examined by considering the parameter
ð9Þ
1.2
Eq. (8.a) Eq. (7)
1 Axial intensity
144
0.8 0.6 0.4 0.2 0 0
0.5
1
1.5
z/zmax
The irradiance distribution of FTMG field is given by I e;o n;M ðrÞ
2
¼ jEe;o n;M ðrÞj :
ð10Þ
Fig. 2. Normalized axial intensity of even zeroth order FTMG beam propagating in free space and calculated by using Eqs. (7) and (8a) for e = 25, w0 = 50 mm, a = 40 and M = 6.
A. Chafiq et al. / Optics Communications 278 (2007) 142–146
145
Fig. 3. Normalized axial intensity of even zeroth order FTMG beam propagating in free space for e = 25, w0 = 50 mm and various values of M = {1, 3, 6, 10, 20}. (a): a = 10; (b): a = 30.
summation in cylindrical coordinates. So, in our opinion the Eq. (7) is more convenient for numerical calculation than Eq. (8a). In Fig. 3, we present the normalized axial intensity of even zeroth order FTMG beam for the following parameters: e = 25, w0 = 50 mm, a = 2c = {10,30} and
k = 632.8 nm. From these curves, one can see that there is an optimal range of M values for which the beam is a best nondiffracting Mathieu beam. For higher values of M, the irradiance oscillations appear and their amplitude increase when M is increased. On other hand, because the radius of
Fig. 4. Transverse intensity of even zeroth order FTMG beam propagating in free space for e = 25, a = 40, w0 = 100 mm and M = 6 at z = 0.5zmax, z = zmax and z = 1.2zmax. (a): (x–z) plane propagation; (b): (y–z) plane propagation.
Fig. 5. Axial intensity of focused even zeroth order FTMG beam by a thin lens of focal length f = 0.5m for the following parameters e = 25 and M = {1, 6, 20, 40}. (a): a = 8, w0 = 1 mm; (b): a = 40, w0 = 5 mm.
146
A. Chafiq et al. / Optics Communications 278 (2007) 142–146
Mathieu fringes in the case of a = 10 is larger than the one corresponding to a = 30, the beam is hard-edged in the first case. Plots of the transverse intensity in Fig. 4 show that there is no significant divergence in central peaks in both cases of (x–z) and (y–z) planes propagation until a maximum propagation distance zmax. Beyond this distance, at z = 1.2zmax for example, one can see some distortions in transverse irradiance. In the second example, we present in Fig. 5, the axial irradiance of a focused even zeroth order FTMG beam. In this case, we have A = 1 z/f, B = z, C = 0 and D = 1, where f is the focal length of the thin lens. The parameter a is taken to be greater than 1, so that the propagation will be imposed by Mathieu beam. As it is shown, the beam exhibits the focusing profile of ideal nondiffracting beams. One can see the distortion at the first peak before the geometrical focus for M = 40 (Fig. 5a); this means that the Mathieu beam is hard-edged when a is small. 5. Conclusion In this paper, we have demonstrated that the behaviour of the FTMG beam is similar to the well known modulated nondiffracting beams like the super-Gaussian-Bessel beam and flat-topped Bessel-Gaussian beam. The closed-form expressions of the FTMG beam and its propagation equation through paraxial ABCD optical system have been derived. The analysis of the beam propagation in free space and through a thin lens reveals the existence of a well defined rectangular profile of the modulating envelope that avoids the axial irradiance oscillations. This adequate rectangular profile can be found by considering the competition between ideal nondiffracting Mathieu beam and
Gaussian envelope function as it is well known for Bessel-Gauss beams. References [1] J.C. Gutie´rrez-Vega, M.D. Iturbe-Castillo, S. Cha´vez-Cerda, Opt. Lett. 25 (2000) 1493. [2] J.C. Gutie´rrez-Vega, M.D. Iturbe-Castillo, G.A. Ramı´rez, E. Tepichı´n, R.M. Rodrı´guez-Dagnino, S. Cha´vez-Cerda, G.H.C. New, Opt. Commun. 195 (2001) 35. [3] A. Belafhal, Z. Hricha, Phy. Chem. News 16 (2004) 33. [4] Y.V. Kartashov, A.A. Egorov, V.A. Vysloukh, L. Torner, Opt. Lett. 31 (2006) 238. [5] S. Cha´vez-Cerda, M.J. Padgett, I. Allison, G.H.C. New, J.C. Gutie´rrez-Vega, A.T. O’Neil, I. MacVicar, J. Courtial, J. Opt. B: Quantum Semiclass. Opt. 4 (2002) S52. [6] C. Lo´pez-Mariscal, J.C. Gutie´rrez-Vega, G. Milne, K. Dholakia, Opt. Express 14 (2006) 4182. [7] C.A. Dartora, H.E. Herna´ndez-Figueroa, J. Opt. Soc. Am. A 21 (2004) 662. [8] S. De Silvestri, P. Laporta, V. Magni, O. Svelto, IEEE J. Quantum Electron QE-24 (1988) 1172. [9] F. Gori, Opt. Commun. 107 (1994) 335. [10] Z. Jiang, Q. Lu, Z. Liu, Appl. Opt. 34 (1995) 7183. [11] Z. Jiang, Opt. Commun. 125 (1996) 207. [12] Y. Li, Opt. Commun. 206 (2002) 225. [13] Y. Li, H. Lee, E. Wolf, J. Opt. Soc. Am. A 21 (2004) 640. [14] A. Chafiq, Z. Hricha, A. Belafhal, Opt. Commun. 253 (2005) 223. [15] A. Chafiq, Z. Hricha, A. Belafhal, Opt. Commun. 265 (2006) 594. [16] I.S. Gradshteyn, I.M. Ryzhik, Tables of Integrals, Series, and Products, fifth ed., Academic Press, New York, 1994. [17] J.C. Gutie´rrez-Vega, Theory and Numerical analysis of the Mathieu functions, Monterrey, NL, Me´xico, 2003. Available at:
Flat-topped Mathieu-Gauss beam and its transformation by paraxial optical systems A. Chafiq, Z. Hricha, A. Belafhal
*
Laboratoire de Physique Mole´culaire, De´partement de Physique, Universite´ Chouaı¨b Doukkali, Faculte´ des Sciences, BP 20, 24000 El Jadida, Morocco Received 21 February 2007; received in revised form 15 May 2007; accepted 21 May 2007
Abstract In order to avoid the severe axial intensity oscillations which appear when an ideal Mathieu beam is truncated, we propose to modulate it by a flattened multi-Gaussian envelope. The obtained beam, which is referred as flattened Mathieu-Gauss (FTMG) beam, can be expanded into a finite series of Mathieu-Gauss beams with various waists. The propagation study of this beam reveals that the axial intensity is unchanged within a certain propagation distance as for super-Gaussian-Bessel beams or Gori’s flattened Bessel beams. By using Collins formula, we derive closed-form expressions of FTMG beam propagating through a paraxial axisymmetric ABCD optical system. Some numerical calculations and discussions are also given. Ó 2007 Published by Elsevier B.V. Keywords: Flattened Mathieu-Gauss (FTMG) beam; Mathieu beam; Collins formula; Bessel-Gauss beams
1. Introduction Mathieu beams constitute a complete and orthogonal family of nondiffracting optical beams, i.e. beams whose profile does not change with propagation distance. Their non-azimuthal symmetry and helical feature make them very relevant for theoretical and practical applications [1– 7]. However, ideal nondiffracting beams cannot really exist because they would carry an infinite energy. In practice these beams are apertured more or less. The hard-edged diffracted Mathieu beam has a near nondiffracting behavior within a maximal distance propagation zmax. Nevertheless, its axial intensity presents oscillations with the propagation distance. In order to avoid these oscillations, many authors have proposed to modulate the nondiffracting beams by a variety of aperture functions. Among them, the super-Gaussian [8] and Gori’s flattened apertures [9] are the most used. In the last decade, Jiang et al. [10,11] have demonstrated the existence of an optimal order of
*
Corresponding author. E-mail address: [email protected] (A. Belafhal).
0030-4018/$ - see front matter Ó 2007 Published by Elsevier B.V. doi:10.1016/j.optcom.2007.05.042
super-Gaussian envelope for which the Bessel beam axial intensity oscillations vanished. And they found that the relationships of the axial intensity versus propagation distance are similar to the radial distributions of the aperture. Particularly, the axial intensity does not oscillate when we use either airy or triangle apertures. Recently, Li et al. [12,13] have used the flat-topped aperture to derive a new expression of generalized Bessel-Gaussian beam. The authors showed that a generalized Bessel beam could be expressed as an arbitrary combination of Bessel-Gaussian beams. On the other hand, in a very recent works [14,15], we have demonstrated that Mathieu beam can be expanded into an infinite sum of higher-order Bessel beams with a well defined expansion coefficients and spatial frequencies. So, the theory of Li et al. [12,13] can be applied to Mathieu beams to produce FTMG beams. In the present work, we propose to study the propagation of FTMG beams through an ABCD optical system and particularly through free space and a simple thin lens. In Section 2, we give the analytical expressions of FTMG field expressed in terms of Mathieu-Gauss beams of various waists. The paraxial propagation equations of the beam through an unapertured ABCD optical system are
A. Chafiq et al. / Optics Communications 278 (2007) 142–146
derived in Section 3. Some numerical calculations and analysis are given in Section 4. We end this paper with a conclusion. 2. Field expressions of FTMG beams In the literature, many papers have used aperture functions which have a flattened profile to modulate the nondiffracting beams. However, most papers which deal with the propagation of the apertured beams have treated the problem numerically (see for example Refs. [10,11]). Indeed, as usual one could not find the closed-form equation, and tedious straightforward numerical integrals are unavoidable. Fortunately, if we consider the flattened aperture function introduced by Li et al. [12,13], one can resolve analytically the propagation problem of the apertured Mathieu beams. The expression of this aperture function is M
F M ðqÞ ¼ 1 ½1 expðq2 Þ ;
ð1Þ
where M is a non-negative number, which can be an integer, a fraction, a decimal or an irrational number. q is the radial cylindrical coordinate. It was demonstrated [12,13] that FM(q) can be re-scaled by introducing the beam width w0 of pure Gaussian profile, which correspond to M = 1. The new function reads M X q2 GM ðqÞ ¼ gm;M exp mb 2 ; ð2aÞ w0 m¼1
GM(q) versus the value of M. As it can be seen from the plots, the profile becomes more and more flattened when M is increased. The modulation of nth-order even and odd Mathieu beams by the above aperture function leads to the following expressions of FTMG beams at the plane z=0 M X Een;M ðn; g; k 1t ; z ¼ 0Þ ¼ gm;M W en ðn; g; k 1t ; z ¼ 0Þ m¼1
q2 exp mb 2 ; w0 M X gm;M W on ðn; g; k 1t ; z ¼ 0Þ Eon;M ðn; g; k 1t ; z ¼ 0Þ ¼
ð3aÞ
m¼1
q2 exp mb 2 ; w0
ð3bÞ
where W e;o n ð:Þ are the even (the superscripts e) and odd (the superscripts o) Mathieu modes. The integer index is n P 0 or n P 1 for even and odd modes, respectively. (n, g), q denote the transverse elliptic coordinate and the radial variable of cylindrical coordinate, respectively. And k1t is the wave vector transverse component related to the ellipticity parameter e by e ¼ h2 k 21t =4, where h is the semi-interfocal separation [16]. If we set pffiffiffiffiffiffiffi ð4Þ wm ¼ w0 = mb; FTMG beams can then be expanded in terms of MG beams of various waists as
with gm;M ¼ ð1Þ b¼
143
M X m¼1
1 ; m
mþ1
MðM 1Þ ðM m þ 1Þ ; m!
ð2bÞ ð2cÞ
where m! is the factorial of m. It is to be noted, that the summation of Eq. (2a) is extended to infinity if M is not an integer. In the following, we will be interested only in the integer values of M. In Fig. 1, we present the profile of the function
Ee;o n;M ðn; g; k 1t ; z ¼ 0Þ ¼
M X
gm;M MGe;o n;m ðn; g; k 1t ; z ¼ 0Þ;
ð5aÞ
m¼1
with e;o 2 2 MGe;o n;m ðn; g; k 1t ; z ¼ 0; Þ ¼ W n ðn; g; k 1t ; z ¼ 0Þ expðq =wm Þ:
ð5bÞ By adopting the series expansion [14], even and odd Mathieu modes can be written as a sum of Bessel functions of higher-order W en ðq; u; k 1t ; z ¼ 0Þ ¼
1 X
ðnÞ
j
Aj ðeÞðiÞ cosðjuÞJ j ðk 1t qÞ;
ð6aÞ
j¼0
W on ðq; u; k 1t ; z ¼ 0Þ ¼
1 X
ðnÞ
Bj ðeÞðiÞj sinðjuÞJ j ðk 1t qÞ;
ð6bÞ
j¼1 ðnÞ
Fig. 1. Plot of flat-topped profiles GM(q/w0) of Eq. (2a) for M = {1, 2, 5, 10, 20, 40}.
ðnÞ
where J is Bessel function of the first kind. Aj ð:Þ, Bj ð:Þ are expansion coefficients of Mathieu functions [16,17]. u denotes the angular variables of cylindrical coordinates. A constant factor in Eqs. (6) has been omitted. By inserting expressions of Eqs. (6) in Eqs. (5), one can easily deduce the expression of FTMG beam in polar coordinates. The obtained expression will be used in the next paragraph to derive a closed-form expression of the beam propagation through an unapertured ABCD optical system.
A. Chafiq et al. / Optics Communications 278 (2007) 142–146
3. Propagation equation of FTMG beam through a paraxial ABCD optical system Assume that the field distribution of FTMG beam at the input plane z = 0 of a paraxial ABCD optical system is given by Eqs. (5). The propagating field distribution of FTMG beam through this system is obtained straightforwardly using Collins diffraction integral formula [18] and the result of Ref. [15]. One obtains Ee;o n;M ðq; u; zÞ M X gm;M k 21t B kq2 i exp i ¼ A þ imbB=zR 2BðA þ imbB=zR Þ 2kðA þ imbB=zR Þ m¼1 q 2 e;o ; u; k 1t ; z ¼ 0 ð7Þ expðikz þ ikDq =2BÞW n A þ imbB=zR
where zR ¼ kw20 =2 is the Rayleigh length of the pure Gaussian profile. A, B, C and D are the elements of the transfer matrix characterizing the paraxial optical system. k is the wave vector. A constant phase term in Eq. (7) has been omitted which has non influence on the output intensity distribution. It is to be noted that the propagated FTMG beam can also be expressed in elliptic coordinates by using the very recent result on generalized Helmholtz-Gauss beams by Guizar-Sicairos and Gutie´rrez-Vega [19]. In this case, the field distribution reads M X k 1t k 2t e;o En;M ðn; g; zÞ ¼ gm;M exp i2k m¼1 GBm ðq; q2;m ÞW e;o ðn; g; k 2t ; z ¼ 0Þ;
ð8aÞ
where Aq1;m þ B ; Cq1;m þ D k 1t k 2t ¼ A þ B=q1;m
ð8bÞ
q2;m ¼
ð8cÞ
and
! expðikzÞ ikq2 GBm ðq; q2;m Þ ¼ exp : A þ B=q1;m 2q2;m
c ¼ 12k 1t w0 :
ð11Þ
The analysis of the propagation characteristics of FTMG beam allows us to distinguish three cases [16,21,22]: (i) c 1: the propagation properties are those of the flattopped envelope. (ii) c 1: the propagation is imposed by the nondiffracting beam and the nondiffracting behaviour will be significant until a maximum propagation distance, zmax zcR . (iii) c 1: we have a transition zone between the last cases. One can point out that a wide variety of beam shapes can be produced by choosing adequate values of the parameters k1t, w0, M and mode number n. 4. Numerical calculations and analysis In this section, we are interested in analyzing the propagation of FTMG beam for c 1. As a first example, we consider free space propagation for which the transfer matrix coefficients are: A = 1, B = z, C = 0 and D = 1. In order to confirm numerically the equivalence between Eqs. (7) and (8a), we present in Fig. 2 the axial intensity of ordinary even zeroth order FTMG beam calculated by these equations for k = 632.8 nm, e = 25, w0 = 50 mm, M = 6 and a = k1tw0 = 40. The plot shows that the considered equations give the same results. From numerical point of view, Mathieu beam field is calculated with two summations in elliptic coordinates (see for example the algorithms for calculating radial and angular Mathieu functions in Ref. [17]) whereas this field is calculated only with one
ð8dÞ
With q1,m, q2,m are the complex input and output parameters of the Gaussian beam transformed by the ABCD optical system [20], k2t is the output transverse wavenumber and the optical path length from the input to the output plane measured along the optical axis is taken equal to z. As it was pointed out in Ref. [19] the expression of Eq. (8) is a generalized formula of propagation of an arbitrary generalized Helmholtz-Gauss beam (gHzG) for which k1t and q1,m can be complex. For ordinary Mathieu-Gauss beam, k1t is real and the input parameter q1,m is purely imaginary, i.e., q1;m ¼ i12kw2m :
On the other hand, as it is shown in Fig. 1, the width of the flat-topped envelope is larger than the Gaussian waist w0 (which is equal to w1) for all M > 1. So, the effect of the modulation envelope on propagation properties will be examined by considering the parameter
ð9Þ
1.2
Eq. (8.a) Eq. (7)
1 Axial intensity
144
0.8 0.6 0.4 0.2 0 0
0.5
1
1.5
z/zmax
The irradiance distribution of FTMG field is given by I e;o n;M ðrÞ
2
¼ jEe;o n;M ðrÞj :
ð10Þ
Fig. 2. Normalized axial intensity of even zeroth order FTMG beam propagating in free space and calculated by using Eqs. (7) and (8a) for e = 25, w0 = 50 mm, a = 40 and M = 6.
A. Chafiq et al. / Optics Communications 278 (2007) 142–146
145
Fig. 3. Normalized axial intensity of even zeroth order FTMG beam propagating in free space for e = 25, w0 = 50 mm and various values of M = {1, 3, 6, 10, 20}. (a): a = 10; (b): a = 30.
summation in cylindrical coordinates. So, in our opinion the Eq. (7) is more convenient for numerical calculation than Eq. (8a). In Fig. 3, we present the normalized axial intensity of even zeroth order FTMG beam for the following parameters: e = 25, w0 = 50 mm, a = 2c = {10,30} and
k = 632.8 nm. From these curves, one can see that there is an optimal range of M values for which the beam is a best nondiffracting Mathieu beam. For higher values of M, the irradiance oscillations appear and their amplitude increase when M is increased. On other hand, because the radius of
Fig. 4. Transverse intensity of even zeroth order FTMG beam propagating in free space for e = 25, a = 40, w0 = 100 mm and M = 6 at z = 0.5zmax, z = zmax and z = 1.2zmax. (a): (x–z) plane propagation; (b): (y–z) plane propagation.
Fig. 5. Axial intensity of focused even zeroth order FTMG beam by a thin lens of focal length f = 0.5m for the following parameters e = 25 and M = {1, 6, 20, 40}. (a): a = 8, w0 = 1 mm; (b): a = 40, w0 = 5 mm.
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A. Chafiq et al. / Optics Communications 278 (2007) 142–146
Mathieu fringes in the case of a = 10 is larger than the one corresponding to a = 30, the beam is hard-edged in the first case. Plots of the transverse intensity in Fig. 4 show that there is no significant divergence in central peaks in both cases of (x–z) and (y–z) planes propagation until a maximum propagation distance zmax. Beyond this distance, at z = 1.2zmax for example, one can see some distortions in transverse irradiance. In the second example, we present in Fig. 5, the axial irradiance of a focused even zeroth order FTMG beam. In this case, we have A = 1 z/f, B = z, C = 0 and D = 1, where f is the focal length of the thin lens. The parameter a is taken to be greater than 1, so that the propagation will be imposed by Mathieu beam. As it is shown, the beam exhibits the focusing profile of ideal nondiffracting beams. One can see the distortion at the first peak before the geometrical focus for M = 40 (Fig. 5a); this means that the Mathieu beam is hard-edged when a is small. 5. Conclusion In this paper, we have demonstrated that the behaviour of the FTMG beam is similar to the well known modulated nondiffracting beams like the super-Gaussian-Bessel beam and flat-topped Bessel-Gaussian beam. The closed-form expressions of the FTMG beam and its propagation equation through paraxial ABCD optical system have been derived. The analysis of the beam propagation in free space and through a thin lens reveals the existence of a well defined rectangular profile of the modulating envelope that avoids the axial irradiance oscillations. This adequate rectangular profile can be found by considering the competition between ideal nondiffracting Mathieu beam and
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