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Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams
ARTICLE IN PRESS 15 June 2002
Optics Communications 207 (2002) 29–34 www.elsevier.com/locate/optcom
Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams Kaicheng Zhu a,b,c,*, Huiqin Tang a,c, Xingming Sun b, Xuewen Wang c, Tienan Liu a b
a Department of Applied Physics, Central South University, Changsha 410083, Hunan, China Computer Science and Technology Department, Zhuzhou Institute of Technology, Zhuzhou 412008, Hunan, China c Department of Physics, Xiangtan Normal University, Xiangtan 411201, Hunan, China
Received 13 November 2002; received in revised form 27 February 2002; accepted 19 March 2002
Abstract Making use of combining 2ðN L þ 1Þ off-axial Gaussian beams, the beams with nearly flattened top profiles and an on-axial shadow are introduced. An analytical expression of the beams propagating through a first-order ABCD system is derived. The closed form of the beam propagation factor (M 2 -factor) is presented to characterize the quality of the beams. Additionally, it is showed that the beam is yet a special solution of the paraxial wave equation and the scheme to generate such a light beam is also proposed. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction In many applications of light beams, a field is required whose amplitude on a given plane is as uniform as possible within a certain area and practically vanishing outside. Therefore, there has been wide interest in the so-called flattened beams during the last years and many techniques have been developed to achieve the uniform laser beam experimentally. Of these, some typical examples include using specialized lenses to convert a Gaussian into a uniform beam profile [1], applying tapered reflectivity mirrors to produce a flat-top profile in a solid-state laser [2], and so on [3,4]. Theoretically, since 1994 Gori [5] introduced an
*
Corresponding author. E-mail address: [email protected] (K. Zhu).
analytical expression to describe the so-called flattened Gaussian beam (FGB), various forms of the expressions representing the flat-top beam distribution have been proposed and the beam characteristics of the flat-top beam have been extensively studied [6]. Very recently, Tovar [7] also suggested another analytical formula determining the flat-top field distribution through the combination of off-axial Gaussian beams and termed it as flat-topped multi-Gaussian laser beams. However, in the past, more attention was mainly paid to investigating relatively simple irradiance distributions such as those varying from Gaussian to flat-top distributions. In reality, in some high-average-power and high-beam quality laser sources unstable-resonator configurations were utilized so that resulted relatively complex output intensity distributions. One of the most typical examples is that copper-vapor lasers have
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 1 4 1 7 - 7
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K. Zhu et al. / Optics Communications 207 (2002) 29–34
output intensity profiles that approach a flat-top distribution with an axial shadow when on-axis unstable resonators and injection-seeded coppervapor laser oscillators were used [8]. And to model the beam with a doughnut-shaped profile in the near-field Saghafi et al. [9] proposed an analytical expression in the cylindrical coordinate system which is expressed as a superposition of two flattop Gaussian beams and obtained extremely in good agreement with the experimental result in a recent paper. In this paper, motivated by the interest to model the beam with top-hat profile with an axial hole and following Tovar’s research, we develop an analytic expression in rectangular coordinate system to describe the beam with nearly flattenedtop profile and an on-axial shadow, and for the sake of simplicity we will also term it as FMGB with a hole. The remaining parts of this paper are organized as follows: In Section 2 the definition of the beams is given and a simple scheme for generating such beams is proposed theoretically. And it is proved that the FMGBs defined in this paper is yet a possible solution of the paraxial equation. In Section 3 an analytical expression for the beams propagating through a first-order ABCD system is derived and the distribution characteristics of the field intensity during the propagation in free space is investigated numerically. Section 4 is focused on calculating the beam propagation factor because it is a useful beam parameter for characterizing the quality of laser beams. Section 5 represents some main results obtained in this paper.
2. Intensity distribution of FMGBs with a hole in transverse Cartesian coordinates Similar to the definition of flat-topped multiGaussian laser beam given by Tovar [7], in this paper, we introduce the analytical expression in rectangular coordinate system describing FMGBs with a hole as the superposition of 2ðN L þ 1Þ equally weighted off-axial Gaussian beams. Omitting an unimportant amplitude constant factor, the field distribution EN ðx; zÞ of the beam at the plane of z ¼ 0 is expressed as
EN ðx; 0Þ ¼
N X
" vn exp
jnj¼L
x nw0 w0
2 # ;
ð1Þ
where vn ¼ 1 12dn0 and dmn is the Kronecker symbol, while N and L ðL ¼ 0; 1; 2; . . . ; N P LÞ are two constant integers and w0 is the waist width associated with the Gaussian part. The three parameters will characterize the laser field distribution. The summation in Eq. (1) is carried out for n both from L to N and from L to N, so that the beam is a combination of 2ðN L þ 1Þ off-axial Gaussian components. When N ¼ L ¼ 0 the beam turns into the usual Gaussian beam, for N ¼ L 6¼ 0, the expression reduces to the coshGaussian beam, while for N 6¼ 0 and L ¼ 0 the expression stands for the flat-top multi-Gaussian beam defined by Tovar [7], and for N > L 6¼ 0 Eq. (1) represents a FMGB with a hole. Hence, expression (1) can uniformly describe the Gaussian beams, the cosh-Gaussian beams, the flat-topped multi-Gaussian beams, as well as the FMGBs with a hole. Introducing the scaled coordinate x0 ¼ x=w0 , equivalently, Eq. (1) yields EN ðx0 ; 0Þ ¼
N X
h i 2 exp ðx0 nÞ :
ð2Þ
jnj¼L
Therefore, in the scaled sense, this beam depends only on two parameters, namely, the two integers N and L. From Eq. (1), we can see that the FMGBs with a hole can be obtained through a coherent superposition involving 2ðN L þ 1Þ off-axial Gaussian beams with the same waist width and in phase with equal weighted, whose centers are, respectively, located at ðnw20 ; 0Þðn ¼ L; ðL þ 1Þ; . . . ; N Þ in the xz plane, and the 2ðN L þ 1Þ off-axial Gaussian beams are placed symmetrically about the original point. In fact, it has been one of the most effective means to obtain new beams with some useful features by making use of the beam superposition [10,11]. In Fig.R1 the normalized intensity distributions 1 of IN ðxÞ= 1 IN ðxÞ dx are shown as functions of 0 x ¼ x=w0 for the given value N ¼ 30 and several different L values at the z ¼ 0 plane, where IN ðxÞ ¼ jEN ðx; 0Þj2 is the field intensity distribu-
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31
easy to prove that a possible solution E of Eq. (3) can be expressed as w0 2 exp iUðzÞ iðx xn Þ En ðx; zÞ ¼ wðzÞ
1 ik ; ð4Þ þ w2 ðzÞ 2RðzÞ
R1 Fig. 1. Normalized intensity distributions of IN ðxÞ= 1 IN ðxÞ dx versus x at input plane (z ¼ 0) for N ¼ 30 and different L, where L are: (a) 5, (b) 10, and (c) 15, respectively.
tion. The numerical calculations show that for sufficiently great value of N (in fact, only N P 3Þ, the intensity distribution of the beam exhibits flattened profile with a central shadow only if L 6¼ 0. And for a given value of N, the central shadow becomes wider and wider with an increase of L. Hence, the FMGBs with a hole defined by Eq. (1) may afford a rather good tool for the study of flattened beams with a hole. Of course, it should be pointed out that the width of the central shadow also depends on w0 , that is, the waist width of the relative off-axial Gaussian components. Roughly speaking, the width of the central hole approximately equals to 2Lw0 . This is just the reason that we call it as FMGBs with a central hole. In the last part of this section, based on the superposition principle we can conclude that the FMGB with a hole should also be a possible solution of the paraxial wave equation because the equation is linearity. In practice, the paraxial wave equation in two-dimensional Cartesian coordinates for the field distribution U ðx; zÞ ¼ En ðx; zÞ expðikzÞ of an optical beam traveling in the z direction, assuming for the field amplitude En that o2 En =oz2 is negligible, is [12] 2
o En oEn ¼ 0; 2ik 2 ox oz
where xn and w0 are constant parameters independent of z, which represent the off-axial extent and the waist width, respectively, and " 2 # kz 2 2 w ðzÞ ¼ w0 1 þ ; pw20 " 2 2 # pw0 ð5Þ ; RðzÞ ¼ z 1 þ kz kz : U ¼ tan1 pw20 Secondly, setting xn ¼ nw0 and EN ðx; zÞ ¼ PN jnj¼L vn En ðx; zÞ we arrive at 2 o o EN ðx; zÞ 2ik oz ox2 N 2 o o X ¼ v En ðx; zÞ 2ik oz jnj¼L n ox2 2 N X o o ¼ vn ð6Þ En ðx; zÞ ¼ 0: 2ik oz ox2 jnj¼L Therefore, EN ðx; zÞ represents a special solution of the paraxial Helmholtz equation and would easily be realized experimentally. Especially, in the waist of z ¼ 0 the EN ðx; zÞ reduces to " 2 # N X x nw0 EN ðx; 0Þ ¼ vn exp ð7Þ w0 jnj¼L which is just the FMGBs with a hole defined by Eq. (1).
3. Propagation of FMGBs with a hole through an unapertured paraxial optical ABCD system
ð3Þ
where k ¼ 2p=k is the wavenumber while k is the wavelength of the monochromic field. Firstly, it is
The beam Eðx; 0Þ defined by Eq. (1) paraxially propagates through a general optical system parameterized by a transfer matrix
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K. Zhu et al. / Optics Communications 207 (2002) 29–34
A C
" rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X N q0 ik exp EN ðx; zÞ ¼ 2qðzÞ B þ Aq0 jnj¼L
B ; D
according to the Huygens–Fresnel diffraction theory, the distribution of the beam at the output plane of the system can be expressed as rffiffiffiffiffiffi Z b i Eðx; zÞ ¼ Eðx0 ; 0Þ kB b
ik ðAx20 2xx0 þ Dx2 Þ dx0 ; exp 2B ð8Þ where k has the same meaning as used in the previous section, b stands for the aperture width. In addition, here an unimportant phase factor has been neglected without loss of generality. Substituting the expression (1) of Eðx; 0Þ into Eq. (3) and setting b!1 we can write the field distribution Eðx; zÞ at z plane as
nw0 x D þ Cq0
2
# q0 n2 C ; D þ Cq0
ð9Þ
where ipw20 ; k B þ Aq0 qðzÞ ¼ D þ Cq0 q0 ¼
ð10Þ
and which turns out the closed-form propagation formula of FMGBs with a hole in the unapertured case. For the case of the propagation in free space A B 1 z ¼ : ð11Þ C D 0 1
Fig. 2. Same as Fig. 1 but in the Fresnel diffraction region (denoted by Nw ¼ 1:0Þ and in the Fraunhofer diffraction region (indicated by Nw ¼ 0:0025 and Nw ¼ 0:0002, respectively.)
ARTICLE IN PRESS K. Zhu et al. / Optics Communications 207 (2002) 29–34
So that Eq. (9) is reduced to sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " # 2 N ipNw X ipNw ð x nÞ Eðx; zÞ ¼ v exp ; 1 þ ipNw jnj¼L n ð1 þ ipNw Þw20 ð12Þ where Nw ¼ w20 =kz is the Fresnel number. Based on Eq. (6), we can also find that the field at z plane can still be expressed as the superposition form involving off-axis Gaussian beams except that the argument is complex. Fig. 2 displays the normalized intensity distributions of a FMGB with a hole where the parameters are the same as these in Fig. 1 but at different z plane. i.e., (A) in the Fresnel diffraction region denoted by Nw ¼ 1 and (B) in the Fraunhofer diffraction region indicated by Nw ¼ 0:0025 and Nw ¼ 0:0002, respectively. Obviously, the intensity distributions in the Fresnel diffraction region closely look like those at the input plane. But the distributions in the Fraunhofer diffraction region lose the distribution shapes at z ¼ 0 plane and expand into a much wider space. In particular, in the far field region the intensity distribution becomes a many like peak configuration and the envelope of the many-peak profile approximately behaves the Gaussian distribution.
4. Evaluations of beam propagation factor M 2 The design of laser systems requires appropriate evaluation of beam characteristics. Among these, the beam propagation factor M 2 has important advantages. As is well known, the beam propagation factor M 2 is a beam invariant for beam propagation through an ABCD-type optical system (and therefore along the beam axis). Following a standard mathematical manipulation, it is easily to prove the first-order moment for the FMGBs with a hole Z 1 Z 1 hxi ¼ xIN ðxÞ dx IN ðxÞ dx ¼ 0: ð13Þ 1
1
And according to the second-order moment definition of the variance r2x in the spatial domain, we can obtain
r2x ¼
Z
1
33
Z x2 IN ðxÞ dx
1
1
IN ðxÞ dx
1
"
N X 1 1 2 2 1þ ¼ vn vm exp ðn þ m Þ 4 2 n;m¼L 2 2 ðn þ m Þ coshðnmÞ þ 2nm sinhðnmÞ , #
N X 1 2 2 vn vm exp ðn þ m Þ coshðnmÞ 2 n;m¼L ð14Þ Similar calculations as above can be performed in the spatial-frequency domain. The field distribution in the spatial-frequency domain can be expressed as Z 1 E~N ðpÞ ¼ EN ðx; 0Þ expð2ippxÞ dx 1 N pffiffiffi X ¼ p vn expðp2 p2 Þ coshð2inppÞ;
ð15Þ
jnj¼L
where p is the spatial-frequency in the x direction. Hence, we can easily obtain the variance r2p in the spatial-frequency domain as Z 1 Z 1 r2p ¼ p2 IN ðpÞ dp IN ðpÞ dp 1 1 "
N X 1 1 2 2 vn vm exp ðn þ m Þ ¼ 2 1 4p 2 n;m¼L 2 ðn þ m2 Þ coshðnmÞ 2nm sinhðnmÞ # ,
N X 1 2 2 vn vm exp ðn þ m Þ coshðnmÞ : 2 n;m¼L ð16Þ 2
E~N ðpÞE~N ðpÞ
Here IN ðpÞ ¼ jE~N ðpÞj ¼ is the intensity distribution associated with the field distribution in the spatial-frequency domain. Therefore, the M 2 -factor of two-dimensional FMGBs with a hole can be computed based on the following expression: 1=2 M 2 ¼ 4p r2x r2p : ð17Þ In fact, if we take N ¼ L ¼ 0, these results immediately reduce to those of usual Gaussian beams and if we set N 6¼ 0 and L ¼ 0 they are just those associated with the flat-topped multi-Gaussian
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K. Zhu et al. / Optics Communications 207 (2002) 29–34
Fig. 3. Dependence of M 2 -factor of FMGBs with a hole on N for different L values, where M (N P L) are (a) 0, (b) 3, (c) 10, and (d) 17, respectively.
beams defined by Tovar [7] at once. Generally, some numerical results are pictured in Fig. 3. By way of comparison, the curve related to L ¼ 0 is plotted and represents the M 2 -factor of Tovar’s flattened-top multi-Gaussian beams. It indicates that M 2 increases monotonously from 1 with an increase of N, which represents that the more the component of Gaussian beam, the greater the value of M 2 . However, for a given value of L 6¼ 0, for sufficiently small value of N L, the M 2 -factor is greater, its variation with an increase of N is that at the small value of N L it quickly decreases and then goes up slowly for appropriately small L. In particular, the M 2 -parameter arrives at its greatest value at N L for a suitable large value of L, at which the beam only consists of two off-axial Gaussian beams centered at x ¼ L and is just a cosh-Gaussian beam.
5. Conclusions and discussions In summary, we have developed a new expression to describe the beam having a flattened-top profile with a hole in the middle. The beam may be generated through combining the off-axis Gaussian beams. The propagation features of such beams in free space are investigated. It is found that during free propagation, in the Fresnel diffraction region, FMGBs with a hole can still retain
their profiles unchangeable from numerical calculation. Then, the beam propagation factor (M 2 factor) is analytically derived. Hence, we expect that, in some sense, the FMGB with a hole introduced in this paper provides an alternative choice for describing the flattened light with an on-axis shadow as well as the evaluations presented here demonstrate some figures which could add information useful for the design of laser systems; in particular, for the design of optical resonators and systems of high power lasers for materials processing where beam characteristics are important for application performances. And, luckily, those relative parameters characterizing FMGBs with a hole can be treated analytically and have closed forms as those of Saghafi’s, and, more important, the beam is also a possible solution of the paraxial wave equation for propagation in complex optical systems.
Acknowledgements This work was supported in part by the Hunan Educational Commission and the Scientific Fund of Central South University.
References [1] D. Shafer, Opt. Laser Technol. 14 (1982) 159. [2] S. De Silvestri, P. Laporta, V. Magni, O. Svelto, IEEE J. Quantum Electron. 24 (1988) 1172. [3] M.R. Perrone, A. Piegari, S. Scaglione, IEEE J. Quantum Electron. 29 (1994) 1423. [4] R.V. Neste, C. Pare, R.L. Lachance, P.A. Blanger, IEEE J. Quantum Electron. 30 (1994) 2663. [5] F. Gori, Opt. Commun. 107 (1994) 335. [6] C.J.R. Sheppard, S. Saghafi, Opt. Commun. 132 (1996) 144. [7] A.A. Tovar, J. Opt. Soc. Am. A 18 (2001) 1897. [8] J.J. Chang, Appl. Opt. 33 (1994) 2255. [9] S. Saghafi, M.J. Withford, J.A. Piper, J. Opt. Soc. Am. A 18 (2001) 1634. [10] J.D. Strohschein, H.J.J. Sequin, C.E. Capjack, Appl. Opt. 37 (1998) 1045. [11] B.D. Lu, H. Ma, J. Opt. Soc. Am. A 17 (2000) 2005. [12] C.C. Davis, Lasers and Electro-Optics, Cambridge University Press, Cambridge, 1996.
Optics Communications 207 (2002) 29–34 www.elsevier.com/locate/optcom
Flattened multi-Gaussian light beams with an axial shadow generated through superposing Gaussian beams Kaicheng Zhu a,b,c,*, Huiqin Tang a,c, Xingming Sun b, Xuewen Wang c, Tienan Liu a b
a Department of Applied Physics, Central South University, Changsha 410083, Hunan, China Computer Science and Technology Department, Zhuzhou Institute of Technology, Zhuzhou 412008, Hunan, China c Department of Physics, Xiangtan Normal University, Xiangtan 411201, Hunan, China
Received 13 November 2002; received in revised form 27 February 2002; accepted 19 March 2002
Abstract Making use of combining 2ðN L þ 1Þ off-axial Gaussian beams, the beams with nearly flattened top profiles and an on-axial shadow are introduced. An analytical expression of the beams propagating through a first-order ABCD system is derived. The closed form of the beam propagation factor (M 2 -factor) is presented to characterize the quality of the beams. Additionally, it is showed that the beam is yet a special solution of the paraxial wave equation and the scheme to generate such a light beam is also proposed. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction In many applications of light beams, a field is required whose amplitude on a given plane is as uniform as possible within a certain area and practically vanishing outside. Therefore, there has been wide interest in the so-called flattened beams during the last years and many techniques have been developed to achieve the uniform laser beam experimentally. Of these, some typical examples include using specialized lenses to convert a Gaussian into a uniform beam profile [1], applying tapered reflectivity mirrors to produce a flat-top profile in a solid-state laser [2], and so on [3,4]. Theoretically, since 1994 Gori [5] introduced an
*
Corresponding author. E-mail address: [email protected] (K. Zhu).
analytical expression to describe the so-called flattened Gaussian beam (FGB), various forms of the expressions representing the flat-top beam distribution have been proposed and the beam characteristics of the flat-top beam have been extensively studied [6]. Very recently, Tovar [7] also suggested another analytical formula determining the flat-top field distribution through the combination of off-axial Gaussian beams and termed it as flat-topped multi-Gaussian laser beams. However, in the past, more attention was mainly paid to investigating relatively simple irradiance distributions such as those varying from Gaussian to flat-top distributions. In reality, in some high-average-power and high-beam quality laser sources unstable-resonator configurations were utilized so that resulted relatively complex output intensity distributions. One of the most typical examples is that copper-vapor lasers have
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 1 4 1 7 - 7
ARTICLE IN PRESS 30
K. Zhu et al. / Optics Communications 207 (2002) 29–34
output intensity profiles that approach a flat-top distribution with an axial shadow when on-axis unstable resonators and injection-seeded coppervapor laser oscillators were used [8]. And to model the beam with a doughnut-shaped profile in the near-field Saghafi et al. [9] proposed an analytical expression in the cylindrical coordinate system which is expressed as a superposition of two flattop Gaussian beams and obtained extremely in good agreement with the experimental result in a recent paper. In this paper, motivated by the interest to model the beam with top-hat profile with an axial hole and following Tovar’s research, we develop an analytic expression in rectangular coordinate system to describe the beam with nearly flattenedtop profile and an on-axial shadow, and for the sake of simplicity we will also term it as FMGB with a hole. The remaining parts of this paper are organized as follows: In Section 2 the definition of the beams is given and a simple scheme for generating such beams is proposed theoretically. And it is proved that the FMGBs defined in this paper is yet a possible solution of the paraxial equation. In Section 3 an analytical expression for the beams propagating through a first-order ABCD system is derived and the distribution characteristics of the field intensity during the propagation in free space is investigated numerically. Section 4 is focused on calculating the beam propagation factor because it is a useful beam parameter for characterizing the quality of laser beams. Section 5 represents some main results obtained in this paper.
2. Intensity distribution of FMGBs with a hole in transverse Cartesian coordinates Similar to the definition of flat-topped multiGaussian laser beam given by Tovar [7], in this paper, we introduce the analytical expression in rectangular coordinate system describing FMGBs with a hole as the superposition of 2ðN L þ 1Þ equally weighted off-axial Gaussian beams. Omitting an unimportant amplitude constant factor, the field distribution EN ðx; zÞ of the beam at the plane of z ¼ 0 is expressed as
EN ðx; 0Þ ¼
N X
" vn exp
jnj¼L
x nw0 w0
2 # ;
ð1Þ
where vn ¼ 1 12dn0 and dmn is the Kronecker symbol, while N and L ðL ¼ 0; 1; 2; . . . ; N P LÞ are two constant integers and w0 is the waist width associated with the Gaussian part. The three parameters will characterize the laser field distribution. The summation in Eq. (1) is carried out for n both from L to N and from L to N, so that the beam is a combination of 2ðN L þ 1Þ off-axial Gaussian components. When N ¼ L ¼ 0 the beam turns into the usual Gaussian beam, for N ¼ L 6¼ 0, the expression reduces to the coshGaussian beam, while for N 6¼ 0 and L ¼ 0 the expression stands for the flat-top multi-Gaussian beam defined by Tovar [7], and for N > L 6¼ 0 Eq. (1) represents a FMGB with a hole. Hence, expression (1) can uniformly describe the Gaussian beams, the cosh-Gaussian beams, the flat-topped multi-Gaussian beams, as well as the FMGBs with a hole. Introducing the scaled coordinate x0 ¼ x=w0 , equivalently, Eq. (1) yields EN ðx0 ; 0Þ ¼
N X
h i 2 exp ðx0 nÞ :
ð2Þ
jnj¼L
Therefore, in the scaled sense, this beam depends only on two parameters, namely, the two integers N and L. From Eq. (1), we can see that the FMGBs with a hole can be obtained through a coherent superposition involving 2ðN L þ 1Þ off-axial Gaussian beams with the same waist width and in phase with equal weighted, whose centers are, respectively, located at ðnw20 ; 0Þðn ¼ L; ðL þ 1Þ; . . . ; N Þ in the xz plane, and the 2ðN L þ 1Þ off-axial Gaussian beams are placed symmetrically about the original point. In fact, it has been one of the most effective means to obtain new beams with some useful features by making use of the beam superposition [10,11]. In Fig.R1 the normalized intensity distributions 1 of IN ðxÞ= 1 IN ðxÞ dx are shown as functions of 0 x ¼ x=w0 for the given value N ¼ 30 and several different L values at the z ¼ 0 plane, where IN ðxÞ ¼ jEN ðx; 0Þj2 is the field intensity distribu-
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31
easy to prove that a possible solution E of Eq. (3) can be expressed as w0 2 exp iUðzÞ iðx xn Þ En ðx; zÞ ¼ wðzÞ
1 ik ; ð4Þ þ w2 ðzÞ 2RðzÞ
R1 Fig. 1. Normalized intensity distributions of IN ðxÞ= 1 IN ðxÞ dx versus x at input plane (z ¼ 0) for N ¼ 30 and different L, where L are: (a) 5, (b) 10, and (c) 15, respectively.
tion. The numerical calculations show that for sufficiently great value of N (in fact, only N P 3Þ, the intensity distribution of the beam exhibits flattened profile with a central shadow only if L 6¼ 0. And for a given value of N, the central shadow becomes wider and wider with an increase of L. Hence, the FMGBs with a hole defined by Eq. (1) may afford a rather good tool for the study of flattened beams with a hole. Of course, it should be pointed out that the width of the central shadow also depends on w0 , that is, the waist width of the relative off-axial Gaussian components. Roughly speaking, the width of the central hole approximately equals to 2Lw0 . This is just the reason that we call it as FMGBs with a central hole. In the last part of this section, based on the superposition principle we can conclude that the FMGB with a hole should also be a possible solution of the paraxial wave equation because the equation is linearity. In practice, the paraxial wave equation in two-dimensional Cartesian coordinates for the field distribution U ðx; zÞ ¼ En ðx; zÞ expðikzÞ of an optical beam traveling in the z direction, assuming for the field amplitude En that o2 En =oz2 is negligible, is [12] 2
o En oEn ¼ 0; 2ik 2 ox oz
where xn and w0 are constant parameters independent of z, which represent the off-axial extent and the waist width, respectively, and " 2 # kz 2 2 w ðzÞ ¼ w0 1 þ ; pw20 " 2 2 # pw0 ð5Þ ; RðzÞ ¼ z 1 þ kz kz : U ¼ tan1 pw20 Secondly, setting xn ¼ nw0 and EN ðx; zÞ ¼ PN jnj¼L vn En ðx; zÞ we arrive at 2 o o EN ðx; zÞ 2ik oz ox2 N 2 o o X ¼ v En ðx; zÞ 2ik oz jnj¼L n ox2 2 N X o o ¼ vn ð6Þ En ðx; zÞ ¼ 0: 2ik oz ox2 jnj¼L Therefore, EN ðx; zÞ represents a special solution of the paraxial Helmholtz equation and would easily be realized experimentally. Especially, in the waist of z ¼ 0 the EN ðx; zÞ reduces to " 2 # N X x nw0 EN ðx; 0Þ ¼ vn exp ð7Þ w0 jnj¼L which is just the FMGBs with a hole defined by Eq. (1).
3. Propagation of FMGBs with a hole through an unapertured paraxial optical ABCD system
ð3Þ
where k ¼ 2p=k is the wavenumber while k is the wavelength of the monochromic field. Firstly, it is
The beam Eðx; 0Þ defined by Eq. (1) paraxially propagates through a general optical system parameterized by a transfer matrix
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K. Zhu et al. / Optics Communications 207 (2002) 29–34
A C
" rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X N q0 ik exp EN ðx; zÞ ¼ 2qðzÞ B þ Aq0 jnj¼L
B ; D
according to the Huygens–Fresnel diffraction theory, the distribution of the beam at the output plane of the system can be expressed as rffiffiffiffiffiffi Z b i Eðx; zÞ ¼ Eðx0 ; 0Þ kB b
ik ðAx20 2xx0 þ Dx2 Þ dx0 ; exp 2B ð8Þ where k has the same meaning as used in the previous section, b stands for the aperture width. In addition, here an unimportant phase factor has been neglected without loss of generality. Substituting the expression (1) of Eðx; 0Þ into Eq. (3) and setting b!1 we can write the field distribution Eðx; zÞ at z plane as
nw0 x D þ Cq0
2
# q0 n2 C ; D þ Cq0
ð9Þ
where ipw20 ; k B þ Aq0 qðzÞ ¼ D þ Cq0 q0 ¼
ð10Þ
and which turns out the closed-form propagation formula of FMGBs with a hole in the unapertured case. For the case of the propagation in free space A B 1 z ¼ : ð11Þ C D 0 1
Fig. 2. Same as Fig. 1 but in the Fresnel diffraction region (denoted by Nw ¼ 1:0Þ and in the Fraunhofer diffraction region (indicated by Nw ¼ 0:0025 and Nw ¼ 0:0002, respectively.)
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So that Eq. (9) is reduced to sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi " # 2 N ipNw X ipNw ð x nÞ Eðx; zÞ ¼ v exp ; 1 þ ipNw jnj¼L n ð1 þ ipNw Þw20 ð12Þ where Nw ¼ w20 =kz is the Fresnel number. Based on Eq. (6), we can also find that the field at z plane can still be expressed as the superposition form involving off-axis Gaussian beams except that the argument is complex. Fig. 2 displays the normalized intensity distributions of a FMGB with a hole where the parameters are the same as these in Fig. 1 but at different z plane. i.e., (A) in the Fresnel diffraction region denoted by Nw ¼ 1 and (B) in the Fraunhofer diffraction region indicated by Nw ¼ 0:0025 and Nw ¼ 0:0002, respectively. Obviously, the intensity distributions in the Fresnel diffraction region closely look like those at the input plane. But the distributions in the Fraunhofer diffraction region lose the distribution shapes at z ¼ 0 plane and expand into a much wider space. In particular, in the far field region the intensity distribution becomes a many like peak configuration and the envelope of the many-peak profile approximately behaves the Gaussian distribution.
4. Evaluations of beam propagation factor M 2 The design of laser systems requires appropriate evaluation of beam characteristics. Among these, the beam propagation factor M 2 has important advantages. As is well known, the beam propagation factor M 2 is a beam invariant for beam propagation through an ABCD-type optical system (and therefore along the beam axis). Following a standard mathematical manipulation, it is easily to prove the first-order moment for the FMGBs with a hole Z 1 Z 1 hxi ¼ xIN ðxÞ dx IN ðxÞ dx ¼ 0: ð13Þ 1
1
And according to the second-order moment definition of the variance r2x in the spatial domain, we can obtain
r2x ¼
Z
1
33
Z x2 IN ðxÞ dx
1
1
IN ðxÞ dx
1
"
N X 1 1 2 2 1þ ¼ vn vm exp ðn þ m Þ 4 2 n;m¼L 2 2 ðn þ m Þ coshðnmÞ þ 2nm sinhðnmÞ , #
N X 1 2 2 vn vm exp ðn þ m Þ coshðnmÞ 2 n;m¼L ð14Þ Similar calculations as above can be performed in the spatial-frequency domain. The field distribution in the spatial-frequency domain can be expressed as Z 1 E~N ðpÞ ¼ EN ðx; 0Þ expð2ippxÞ dx 1 N pffiffiffi X ¼ p vn expðp2 p2 Þ coshð2inppÞ;
ð15Þ
jnj¼L
where p is the spatial-frequency in the x direction. Hence, we can easily obtain the variance r2p in the spatial-frequency domain as Z 1 Z 1 r2p ¼ p2 IN ðpÞ dp IN ðpÞ dp 1 1 "
N X 1 1 2 2 vn vm exp ðn þ m Þ ¼ 2 1 4p 2 n;m¼L 2 ðn þ m2 Þ coshðnmÞ 2nm sinhðnmÞ # ,
N X 1 2 2 vn vm exp ðn þ m Þ coshðnmÞ : 2 n;m¼L ð16Þ 2
E~N ðpÞE~N ðpÞ
Here IN ðpÞ ¼ jE~N ðpÞj ¼ is the intensity distribution associated with the field distribution in the spatial-frequency domain. Therefore, the M 2 -factor of two-dimensional FMGBs with a hole can be computed based on the following expression: 1=2 M 2 ¼ 4p r2x r2p : ð17Þ In fact, if we take N ¼ L ¼ 0, these results immediately reduce to those of usual Gaussian beams and if we set N 6¼ 0 and L ¼ 0 they are just those associated with the flat-topped multi-Gaussian
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K. Zhu et al. / Optics Communications 207 (2002) 29–34
Fig. 3. Dependence of M 2 -factor of FMGBs with a hole on N for different L values, where M (N P L) are (a) 0, (b) 3, (c) 10, and (d) 17, respectively.
beams defined by Tovar [7] at once. Generally, some numerical results are pictured in Fig. 3. By way of comparison, the curve related to L ¼ 0 is plotted and represents the M 2 -factor of Tovar’s flattened-top multi-Gaussian beams. It indicates that M 2 increases monotonously from 1 with an increase of N, which represents that the more the component of Gaussian beam, the greater the value of M 2 . However, for a given value of L 6¼ 0, for sufficiently small value of N L, the M 2 -factor is greater, its variation with an increase of N is that at the small value of N L it quickly decreases and then goes up slowly for appropriately small L. In particular, the M 2 -parameter arrives at its greatest value at N L for a suitable large value of L, at which the beam only consists of two off-axial Gaussian beams centered at x ¼ L and is just a cosh-Gaussian beam.
5. Conclusions and discussions In summary, we have developed a new expression to describe the beam having a flattened-top profile with a hole in the middle. The beam may be generated through combining the off-axis Gaussian beams. The propagation features of such beams in free space are investigated. It is found that during free propagation, in the Fresnel diffraction region, FMGBs with a hole can still retain
their profiles unchangeable from numerical calculation. Then, the beam propagation factor (M 2 factor) is analytically derived. Hence, we expect that, in some sense, the FMGB with a hole introduced in this paper provides an alternative choice for describing the flattened light with an on-axis shadow as well as the evaluations presented here demonstrate some figures which could add information useful for the design of laser systems; in particular, for the design of optical resonators and systems of high power lasers for materials processing where beam characteristics are important for application performances. And, luckily, those relative parameters characterizing FMGBs with a hole can be treated analytically and have closed forms as those of Saghafi’s, and, more important, the beam is also a possible solution of the paraxial wave equation for propagation in complex optical systems.
Acknowledgements This work was supported in part by the Hunan Educational Commission and the Scientific Fund of Central South University.
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