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Decentered elliptical flattened Gaussian beam
Optics Communications 240 (2004) 245–252 www.elsevier.com/locate/optcom
Decentered elliptical flattened Gaussian beam Meixiao Shen *, Shaomin Wang
**
Department of Physics, Zhejiang University, Hangzhou 310027, China Received 28 June 2003; received in revised form 24 February 2004; accepted 19 June 2004
Abstract A decentered flattened Gaussian beam (DEFGB), is defined by a tensor method. The propagation formula for a DEFGB passing through an axially nonsymmetrical paraxial optical system is derived through vector integration. The derived formula can be reduced to the formula for a generalized decentered elliptical flattened Gaussian beam under certain condition. As an example application of the derived formula, the propagation characteristics of a DEFGB in free space are calculated and discussed. As another example we have studied the properties of superposition with radial array consisted by DEFGB. Ó 2004 Elsevier B.V. All rights reserved. PACS: 41.85.Ew; 41.85.Ja; 41.85.Ct Keywords: Flattened Gaussian beam; Nonsymmetrical optical system; Laser beam array
1. Introduction As yet, a growing interest has developed in flattened Gaussian beams introduced by Gori in 1994 [1], due to their advantage in the simulation of beams with a flat-top spatial profile in comparison with super-Gaussian beams. Its optical properties have been discussed in many publications [2–6].
*
Corresponding author. Tel./fax: +8657188805960. Corresponding author. Tel./fax: +86-571-87985970. E-mail addresses: [email protected] (M. Shen), [email protected] (S. Wang). **
For instance, the propagation characteristics of axially symmetric flattened Gaussian beams and its factor and kurtosis parameter have been studied and discussed [2–4]. The focusing of flattened Gaussian beams has been given [5]. The propagations of flattened Gaussian beams passing through an optical system with and without aperture have also been analyzed [6]. Until now, however, most studies have concentrated on the symmetric flattened Gaussian beams or on beams with separable variables x and y. Previously tensor methods were used to treat propagation and transmission of generalized elliptical Gaussian beams [7,8]. Recently Cai and Lin [9]
0030-4018/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.06.041
246
M. Shen, S. Wang / Optics Communications 240 (2004) 245–252
also defined the decentered elliptical Gaussian beam by tensor method. In this paper, we introduce a more general three-dimensional decentered elliptical flattened Gaussian beam (DEFGB), whose variables cannot be separated. The formula for propagation of a DEFGB through an axially nonsymmetrical optical system is derived based on the generalized Collins integral. By use of the derived propagation formula, the propagation of a DEFGB in free space is calculated and discussed. The properties of coherent and incoherent beam superposition with radial array for DEFGB are also studied.
2. Definition of a decentered elliptical flattened Gaussian beam and its propagation through paraxial optical system The generalized elliptical flattened Gaussian beam with M order can be expressed in tensor form as follows [7]: X m M ik 1 ik T 1 Eðr1 Þ ¼ A0 exp rT1 Q1 r r Q r ; 1 1M 1 2 m! 2 1 1M m¼0 ð1Þ where A0 is a constant, which will be set to unit in the following. k = 2p/k is the wave number, k is the wave length, and r1 = (x1,y1) is a position vector. 1 Q1 1M ¼ ðM þ 1ÞQ1 is a complex curvature tensor for the elliptical flattened Gaussian beam and Q1 1 is that for the elliptical Gaussian beam given by [7] " 1 # q1xx q1 1xy 1 Q1 ¼ : ð2Þ q1 q1 1xy 1yy In simple astigmatic case, q1 1xy ¼ 0. But in general, the cross elements exist. If the beam has a common waist position (z = 0) in two directions, e.g., the output beam of semiconductor lasers and slab lasers, then using both waistÕs width and Rayleigh range the complex radius at z = 0 can be given as " 1 # " 2 # 2 1 q q W W ik 1xx 1xy 0x 0xy Q1 ¼ ; ð3Þ 1 ¼ p W 2 q1 q1 W 2 1xy 1yy 0xy 0y
where W0x and W0y represent the width of waist in x and y directions, respectively. In general, the cross element W0xy for the width of waist does not equal zero. The decentered elliptical flattened Gaussian beam at z = 0 can be defined by use of the tensor method in the following form: ik T Eðr1 Þ ¼ A0 exp ðr1 r0 Þ Q1 ðr r Þ 0 1M 1 2 m M X 1 ik T 1 ðr1 r0 Þ Q1M ðr1 r0 Þ ; m! 2 m¼0 ð4Þ where rT0 ¼ ðxd þ ixi ; y d þ iy i Þ is a complex vector called the decentered parameter. The propagation of a DEFGB through a nonsymmetrical optical system can be treated by the general Collins formula, which can be written in tensor form as follows [7,8]: Z Z in1 1 E2 ðr2 Þ ¼ ½detðBÞ 2 E1 ðr1 Þ exp½ikldr1 ; k ð5Þ where l is an eikonal given by T r1 1 r1 l ¼ l0 þ R ; 2 r2 r2
ð6Þ
where l0 is the eikonal along the propagation axis. r1 and r2 are position vectors in the input and the output planes. R is an eikonal matrix which can be read as " # n1 B1 A n1 B1 R¼ ; ð7Þ n2 ðC DB1 AÞ n2 DB1 where n1 and n2 are the refractive indices of the input and the output spaces, respectively. For simplicity we assume that n1 = n2 = 1 in the following. A, B, C and D are the submatrices of the optical system, defined by r2 r1 A B ¼ : ð8Þ r02 r01 C D Substituting Eq. (4) as E1(r1) into Eq. (5), after a tedious vector integral operation we obtained the following expression for the output DEFGB:
M. Shen, S. Wang / Optics Communications 240 (2004) 245–252
247
12 ik T 1 ik T 1 1 r r Eðr2 Þ ¼ A0 detðA þ BQ1 Þ exp Q r ðQ þ A BÞ r exp 2 0 1M 2 2 2M 2 0 1M "rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# ik T 1 H 2m ðr2 Ar0 Þ B1T ðBQ1 1M þ AÞ ðr2 Ar0 Þ 2 m M h iX 1 1 1 m exp ikrT0 ðAQ1M þ BÞ r2 ½detðI þ Q1M B1 AÞ ; m! 4 m¼0
where 1
1 1 Q1 2M ¼ ðC þ DQ1M ÞðA þ BQ1M Þ :
ð10Þ
In the derivation of Eq. (9) the following relations were used: T
ðB1 AÞ ¼ B1 A; 1 T
T
ðB1 Þ ¼ ðC DB1 AÞ;
1
ðDB Þ ¼ DB :
ð11Þ
Eq. (10) is the well-known tensor ABCD law for the propagation of the generalized elliptical flattened Gaussian beam. When M = 0, Eq. (9) is reduced to the following form: Eðr2 Þ ¼ A0 detðA þ BQ1 expðikl0 Þ 1 Þ ik T 1 exp r2 Q2 r2 2 ik T 1 exp r0 ðQ1 þ A1 BÞ r0 2 h i exp ikrT0 ðAQ1 þ BÞ1 r2 :
A¼ ; 0 1 0 0 C¼ ; 0 0 1 0
B¼ D¼
z
0 1 0
0
z 0 1
ð9Þ
; ð13Þ :
Substituting Eq. (13) into Eqs. (9) and (10), we can obtain the three-dimensional relative intensity distribution of a DEFGB at various propagation distances, as shown in Fig. 1. The parameters used in the calculation are the following values: rT0 ¼ ð2 þ i; 2 þ iÞ; M ¼ 5;
k ¼ 632:8 nm;
W 0x ¼ 1:0 mm;
W 0y ¼ 1:5 mm;
W 0xy ¼ 2:0 mm:
12
According to Eqs. (2) and (3), we can obtain 1:209i 0:302i 1 1 Q1 ¼ ðM þ 1ÞQ ¼ ðmÞ : 1M 1 0:302i 0:538i ð14Þ ð12Þ
We can find that Eq. (12) is the same as the propagation formula for the generalized decentered elliptical Gaussian beam through the nonsymmetrical paraxial optical system in [9]
3. Propagation of a decentered elliptical flattened Gaussian beam in free space In this section we study the propagation properties of a DEFGB in the free space, using the derived propagation formula for a DEFGB. The submatrices for free space of distance z read as
The propagation distances are normalized to the Rayleigh distance in the x-direction, zx ¼ pw20x =k. From Fig. 1 we can find the peak position of the intensity distribution at z = 0 is located at r0 = (xd,yd), and the shape of the intensity distribution of the DEFGB at the input plane is closely related to the imaginary part of the decentered parameter. The short and long axis of the elliptical spot interchanges with propagation distances. We also can find that the position of peak intensity of the DEFGB will move during propagation when the imaginary part of the decectered parameter is not zero. In addition, the intensity distribution is also related to the order of the elliptical flattened Gaussian beam.
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M. Shen, S. Wang / Optics Communications 240 (2004) 245–252
Fig. 1. Three-dimensional relative intensity distributions of DEFGBs on the planes of various propagation distances. The calculation parameters are M = 5, rT0 ¼ ð2 þ i; 2 þ iÞ, and (a) z = 0, (b) z = zx, (c) z = 2zx, (d) z = 7zx.
4. The properties of superposition of DEFGB The superposition of decentered beams is a topic of current interest, because it provides a useful technique for coupling laser sub-systems in series or parallel to achieve high-power output. Recently, a number of theoretical and numerical analysis for coherent and incoherent beam superposition with linear, rectangular and radial array geometries for decentered Gaussian beams, Hermite–Gaussian beams, elliptical Gaussian beams, etc. [9–12] have been performed. However, in practice, particularly in high-power laser systems, an elliptical flattened Gaussian beam is often encountered. To our knowledge, as yet the properties of superposition of DEFGB have not been treated in the literature. As another application example of the derived formula, the properties of coherent and incoherent beam superposition with radial DEFGB array, or the phase-locked and nonphase-locked beam arrays, have been discussed. Assume that the beam array consists of N equal decentered elliptical flattened Gaussian beams,
whichpare located symmetrically on a ring with raffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi dius x2d þ y 2d . Then the field distribution of the beam array for coherent beam superposition case at z = 0 is given by N 1 X En ðr1n ; 0Þ; ð15Þ Eðr1 ; 0Þ ¼ n¼0
where En(r1n) is the nth element of the laser beam array, expressed by ik T 1 Eðr1n Þ ¼ A0 exp ðr1n r0 Þ Q1M ðr1n r0 Þ 2 m M X 1 ik T ðr1n r0 Þ Q1 ðr r Þ ; 0 1M 1n m! 2 m¼0 ð16Þ with r1n ¼
cos a sin a x1 cos a þ y 1 sin a ; r1 ¼ ; y 1 cos a x1 sin a sin a cos a ð17Þ
where a = na0, n = 0,1,2,. . .,N1, and a0 = 2p/N. Applying Eq. (9), we can easily get the field distri-
M. Shen, S. Wang / Optics Communications 240 (2004) 245–252
249
Fig. 2. Three-dinensional relative intensity distributions and corresponding contour graphs of the radial DEFGB array for coherent beam superposition on planes of several propagation distances. The calculation parameters are M = 5, N = 5, rT0 ¼ ð2; 2Þ, and (a) z = 0, (b) z = zx, (c) z = 2zx, (d) z = 7zx.
250
M. Shen, S. Wang / Optics Communications 240 (2004) 245–252
Fig. 3. Three-dimensional relative intensity distributions and corresponding contour graphs of the radial DEFGB array for incoherent beam superposition on planes of several propagation distances. The calculation parameters are M = 5, N = 5, rT0 ¼ ð2; 2Þ, and (a) z = 0, (b) z = zx, (c) z = 2zx, (d) z = 7zx.
M. Shen, S. Wang / Optics Communications 240 (2004) 245–252
bution of En(r2n) after its propagation through a paraxial system, which is expressed as
means the far field, there is a considerable spatial overlap between the elements of the array and con-
ik T 1 ik T 1 1 Eðr2n Þ ¼ A0 detðA þ expðikl0 Þ exp r2n Q2M r2n exp r0 ðQ1M þ A BÞ r0 2 2 m M h iX m 1 1 1 exp ikrT0 ðAQ1M þ BÞ r2n detðI þ Q1M B1 AÞ m! 4 m¼0 "rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# ik T 1 ðr2n Ar0 Þ B1T ðBQ1 H 2m 1M þ AÞ ðr2n Ar0 Þ ; 2
12 BQ1 1 Þ
where cos a sin a x2 cos a þ y 2 sin a r2n ¼ ; r2 ¼ : sin a cos a y 2 cos a x2 sin a ð19Þ Thus, for the coherent beam superposition, the field distribution of the resulting beam is written as Eðr2 Þ ¼
N 1 X
En ðr2n Þ
ð20Þ
n¼0
and the intensity reads I ¼ E ðr2 ÞEðr2 Þ;
ð21Þ
where * denote the complex conjugate and En(r2n) is given in Eq. (18). For the incoherent beam superposition case we have I¼
N 1 X n¼0
I n ðr2n Þ ¼
N 1 X
251
En ðr2n ÞEn ðr2n Þ:
ð22Þ
n¼0
The free space is also taken as the typical example. The submatrices ABCD is given in Eq. (13). Numerical calculations were performed using Eqs. (15)–(22). Typical results of the relative intensity for both coherent and incoherent superposition cases are complied in Figs. 2 and 3, where the calculation parameters are rT0 ¼ ð2; 2Þ, N = 5, M = 5. Q1 1M is given in Eq. (14). It turns out from Figs. 2(a) and 3(a) that the relative intensity distributions for the coherent and incoherent beam superposition cases at the input plane of z = 0 are identical, because there is no physical beam overlap that might create any interference. However, for the large value of z, which
ð18Þ
sequently the intensity distributions at the output plane become different. In general, the radial DEFGB arrays for coherent and incoherent beam superposition result in different intensity distributions and the intensity distribution for coherent beam superposition case is more complicated than that for the incoherent superposition one. In comparison with the results of superposition of radial elliptical Gaussian beam array [9] and rectangular Hermit–Gaussian beam array [10], there are different intensity distributions and symmetric characteristics of the radial DEFGB array for coherent and incoherent superposition, so that, to meet some practical requirement.
5. Conclusions In this study the decentered elliptical flattened Gaussian beam (DEFGB) has been defined by tensor method. The general propagation formula for the decentered elliptical flattened Gaussian beam has been derived. Using the derived formula, the propagation properties of a DEFGB in free space are illustrated with numerical examples. The numerical results show that the intensity distribution of a decentered elliptical flattened Gaussian beam depends upon the decentered parameter and the order of elliptical flattened Gaussian beam. We also investigated the properties of superposition of DEFGB. In general, the formula derived in this paper can be used to treat the propagation of the DEFGB in the non-symmetrical optical systems and the beam array constructed by DEFGB for coherent and incoherent
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M. Shen, S. Wang / Optics Communications 240 (2004) 245–252
superposition within the framework of paraxial approximations. Acknowledgement This work was supported by the National Natural Science Foundation of China Grant 60276035.
References [1] F. Gori, Opt. Commun. 107 (1994) 335.
[2] V. Bagini, R. Borghi, F. Gori, J. Opt. Soc. Am. A 13 (1996) 1385. [3] S.A. Amarande, Opt. Commun. 129 (1996) 311. [4] B. Lu¨, B. Zhang, S. Luo, Appl. Opt. 38 (1999) 4581. [5] M. Santarsiero, J. Mod. Opt. 44 (1997) 633. [6] B. Lu¨, S. Luo, B. Zhang, Opt. Commun. 164 (1999) 1. [7] Q. Lin, S. Wang, J. Alda, E. Bernabeu, Optik 85 (1990) 67. [8] J. Alda, S. Wang, E. Bernabeu, Opt. Commun. 80 (1991) 350. [9] Y. Cai, Q. Lin, Appl. Opt. 41 (2002) 4336. [10] B. Lu¨, H. Ma, Opt. Commun. 171 (1999) 185. [11] H.J. Baker, D.R. Hall, A.M. Hornby, R.J. Morley, M.R. Taghizadeh, E.F. Hall, IEEE J. Quantum Electron. 32 (1996) 400. [12] J.D. Strohschein, H.J.J. Seguin, C.E. Capjack, Appl. Opt. 37 (1998) 1045.
Decentered elliptical flattened Gaussian beam Meixiao Shen *, Shaomin Wang
**
Department of Physics, Zhejiang University, Hangzhou 310027, China Received 28 June 2003; received in revised form 24 February 2004; accepted 19 June 2004
Abstract A decentered flattened Gaussian beam (DEFGB), is defined by a tensor method. The propagation formula for a DEFGB passing through an axially nonsymmetrical paraxial optical system is derived through vector integration. The derived formula can be reduced to the formula for a generalized decentered elliptical flattened Gaussian beam under certain condition. As an example application of the derived formula, the propagation characteristics of a DEFGB in free space are calculated and discussed. As another example we have studied the properties of superposition with radial array consisted by DEFGB. Ó 2004 Elsevier B.V. All rights reserved. PACS: 41.85.Ew; 41.85.Ja; 41.85.Ct Keywords: Flattened Gaussian beam; Nonsymmetrical optical system; Laser beam array
1. Introduction As yet, a growing interest has developed in flattened Gaussian beams introduced by Gori in 1994 [1], due to their advantage in the simulation of beams with a flat-top spatial profile in comparison with super-Gaussian beams. Its optical properties have been discussed in many publications [2–6].
*
Corresponding author. Tel./fax: +8657188805960. Corresponding author. Tel./fax: +86-571-87985970. E-mail addresses: [email protected] (M. Shen), [email protected] (S. Wang). **
For instance, the propagation characteristics of axially symmetric flattened Gaussian beams and its factor and kurtosis parameter have been studied and discussed [2–4]. The focusing of flattened Gaussian beams has been given [5]. The propagations of flattened Gaussian beams passing through an optical system with and without aperture have also been analyzed [6]. Until now, however, most studies have concentrated on the symmetric flattened Gaussian beams or on beams with separable variables x and y. Previously tensor methods were used to treat propagation and transmission of generalized elliptical Gaussian beams [7,8]. Recently Cai and Lin [9]
0030-4018/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.06.041
246
M. Shen, S. Wang / Optics Communications 240 (2004) 245–252
also defined the decentered elliptical Gaussian beam by tensor method. In this paper, we introduce a more general three-dimensional decentered elliptical flattened Gaussian beam (DEFGB), whose variables cannot be separated. The formula for propagation of a DEFGB through an axially nonsymmetrical optical system is derived based on the generalized Collins integral. By use of the derived propagation formula, the propagation of a DEFGB in free space is calculated and discussed. The properties of coherent and incoherent beam superposition with radial array for DEFGB are also studied.
2. Definition of a decentered elliptical flattened Gaussian beam and its propagation through paraxial optical system The generalized elliptical flattened Gaussian beam with M order can be expressed in tensor form as follows [7]: X m M ik 1 ik T 1 Eðr1 Þ ¼ A0 exp rT1 Q1 r r Q r ; 1 1M 1 2 m! 2 1 1M m¼0 ð1Þ where A0 is a constant, which will be set to unit in the following. k = 2p/k is the wave number, k is the wave length, and r1 = (x1,y1) is a position vector. 1 Q1 1M ¼ ðM þ 1ÞQ1 is a complex curvature tensor for the elliptical flattened Gaussian beam and Q1 1 is that for the elliptical Gaussian beam given by [7] " 1 # q1xx q1 1xy 1 Q1 ¼ : ð2Þ q1 q1 1xy 1yy In simple astigmatic case, q1 1xy ¼ 0. But in general, the cross elements exist. If the beam has a common waist position (z = 0) in two directions, e.g., the output beam of semiconductor lasers and slab lasers, then using both waistÕs width and Rayleigh range the complex radius at z = 0 can be given as " 1 # " 2 # 2 1 q q W W ik 1xx 1xy 0x 0xy Q1 ¼ ; ð3Þ 1 ¼ p W 2 q1 q1 W 2 1xy 1yy 0xy 0y
where W0x and W0y represent the width of waist in x and y directions, respectively. In general, the cross element W0xy for the width of waist does not equal zero. The decentered elliptical flattened Gaussian beam at z = 0 can be defined by use of the tensor method in the following form: ik T Eðr1 Þ ¼ A0 exp ðr1 r0 Þ Q1 ðr r Þ 0 1M 1 2 m M X 1 ik T 1 ðr1 r0 Þ Q1M ðr1 r0 Þ ; m! 2 m¼0 ð4Þ where rT0 ¼ ðxd þ ixi ; y d þ iy i Þ is a complex vector called the decentered parameter. The propagation of a DEFGB through a nonsymmetrical optical system can be treated by the general Collins formula, which can be written in tensor form as follows [7,8]: Z Z in1 1 E2 ðr2 Þ ¼ ½detðBÞ 2 E1 ðr1 Þ exp½ikldr1 ; k ð5Þ where l is an eikonal given by T r1 1 r1 l ¼ l0 þ R ; 2 r2 r2
ð6Þ
where l0 is the eikonal along the propagation axis. r1 and r2 are position vectors in the input and the output planes. R is an eikonal matrix which can be read as " # n1 B1 A n1 B1 R¼ ; ð7Þ n2 ðC DB1 AÞ n2 DB1 where n1 and n2 are the refractive indices of the input and the output spaces, respectively. For simplicity we assume that n1 = n2 = 1 in the following. A, B, C and D are the submatrices of the optical system, defined by r2 r1 A B ¼ : ð8Þ r02 r01 C D Substituting Eq. (4) as E1(r1) into Eq. (5), after a tedious vector integral operation we obtained the following expression for the output DEFGB:
M. Shen, S. Wang / Optics Communications 240 (2004) 245–252
247
12 ik T 1 ik T 1 1 r r Eðr2 Þ ¼ A0 detðA þ BQ1 Þ exp Q r ðQ þ A BÞ r exp 2 0 1M 2 2 2M 2 0 1M "rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# ik T 1 H 2m ðr2 Ar0 Þ B1T ðBQ1 1M þ AÞ ðr2 Ar0 Þ 2 m M h iX 1 1 1 m exp ikrT0 ðAQ1M þ BÞ r2 ½detðI þ Q1M B1 AÞ ; m! 4 m¼0
where 1
1 1 Q1 2M ¼ ðC þ DQ1M ÞðA þ BQ1M Þ :
ð10Þ
In the derivation of Eq. (9) the following relations were used: T
ðB1 AÞ ¼ B1 A; 1 T
T
ðB1 Þ ¼ ðC DB1 AÞ;
1
ðDB Þ ¼ DB :
ð11Þ
Eq. (10) is the well-known tensor ABCD law for the propagation of the generalized elliptical flattened Gaussian beam. When M = 0, Eq. (9) is reduced to the following form: Eðr2 Þ ¼ A0 detðA þ BQ1 expðikl0 Þ 1 Þ ik T 1 exp r2 Q2 r2 2 ik T 1 exp r0 ðQ1 þ A1 BÞ r0 2 h i exp ikrT0 ðAQ1 þ BÞ1 r2 :
A¼ ; 0 1 0 0 C¼ ; 0 0 1 0
B¼ D¼
z
0 1 0
0
z 0 1
ð9Þ
; ð13Þ :
Substituting Eq. (13) into Eqs. (9) and (10), we can obtain the three-dimensional relative intensity distribution of a DEFGB at various propagation distances, as shown in Fig. 1. The parameters used in the calculation are the following values: rT0 ¼ ð2 þ i; 2 þ iÞ; M ¼ 5;
k ¼ 632:8 nm;
W 0x ¼ 1:0 mm;
W 0y ¼ 1:5 mm;
W 0xy ¼ 2:0 mm:
12
According to Eqs. (2) and (3), we can obtain 1:209i 0:302i 1 1 Q1 ¼ ðM þ 1ÞQ ¼ ðmÞ : 1M 1 0:302i 0:538i ð14Þ ð12Þ
We can find that Eq. (12) is the same as the propagation formula for the generalized decentered elliptical Gaussian beam through the nonsymmetrical paraxial optical system in [9]
3. Propagation of a decentered elliptical flattened Gaussian beam in free space In this section we study the propagation properties of a DEFGB in the free space, using the derived propagation formula for a DEFGB. The submatrices for free space of distance z read as
The propagation distances are normalized to the Rayleigh distance in the x-direction, zx ¼ pw20x =k. From Fig. 1 we can find the peak position of the intensity distribution at z = 0 is located at r0 = (xd,yd), and the shape of the intensity distribution of the DEFGB at the input plane is closely related to the imaginary part of the decentered parameter. The short and long axis of the elliptical spot interchanges with propagation distances. We also can find that the position of peak intensity of the DEFGB will move during propagation when the imaginary part of the decectered parameter is not zero. In addition, the intensity distribution is also related to the order of the elliptical flattened Gaussian beam.
248
M. Shen, S. Wang / Optics Communications 240 (2004) 245–252
Fig. 1. Three-dimensional relative intensity distributions of DEFGBs on the planes of various propagation distances. The calculation parameters are M = 5, rT0 ¼ ð2 þ i; 2 þ iÞ, and (a) z = 0, (b) z = zx, (c) z = 2zx, (d) z = 7zx.
4. The properties of superposition of DEFGB The superposition of decentered beams is a topic of current interest, because it provides a useful technique for coupling laser sub-systems in series or parallel to achieve high-power output. Recently, a number of theoretical and numerical analysis for coherent and incoherent beam superposition with linear, rectangular and radial array geometries for decentered Gaussian beams, Hermite–Gaussian beams, elliptical Gaussian beams, etc. [9–12] have been performed. However, in practice, particularly in high-power laser systems, an elliptical flattened Gaussian beam is often encountered. To our knowledge, as yet the properties of superposition of DEFGB have not been treated in the literature. As another application example of the derived formula, the properties of coherent and incoherent beam superposition with radial DEFGB array, or the phase-locked and nonphase-locked beam arrays, have been discussed. Assume that the beam array consists of N equal decentered elliptical flattened Gaussian beams,
whichpare located symmetrically on a ring with raffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi dius x2d þ y 2d . Then the field distribution of the beam array for coherent beam superposition case at z = 0 is given by N 1 X En ðr1n ; 0Þ; ð15Þ Eðr1 ; 0Þ ¼ n¼0
where En(r1n) is the nth element of the laser beam array, expressed by ik T 1 Eðr1n Þ ¼ A0 exp ðr1n r0 Þ Q1M ðr1n r0 Þ 2 m M X 1 ik T ðr1n r0 Þ Q1 ðr r Þ ; 0 1M 1n m! 2 m¼0 ð16Þ with r1n ¼
cos a sin a x1 cos a þ y 1 sin a ; r1 ¼ ; y 1 cos a x1 sin a sin a cos a ð17Þ
where a = na0, n = 0,1,2,. . .,N1, and a0 = 2p/N. Applying Eq. (9), we can easily get the field distri-
M. Shen, S. Wang / Optics Communications 240 (2004) 245–252
249
Fig. 2. Three-dinensional relative intensity distributions and corresponding contour graphs of the radial DEFGB array for coherent beam superposition on planes of several propagation distances. The calculation parameters are M = 5, N = 5, rT0 ¼ ð2; 2Þ, and (a) z = 0, (b) z = zx, (c) z = 2zx, (d) z = 7zx.
250
M. Shen, S. Wang / Optics Communications 240 (2004) 245–252
Fig. 3. Three-dimensional relative intensity distributions and corresponding contour graphs of the radial DEFGB array for incoherent beam superposition on planes of several propagation distances. The calculation parameters are M = 5, N = 5, rT0 ¼ ð2; 2Þ, and (a) z = 0, (b) z = zx, (c) z = 2zx, (d) z = 7zx.
M. Shen, S. Wang / Optics Communications 240 (2004) 245–252
bution of En(r2n) after its propagation through a paraxial system, which is expressed as
means the far field, there is a considerable spatial overlap between the elements of the array and con-
ik T 1 ik T 1 1 Eðr2n Þ ¼ A0 detðA þ expðikl0 Þ exp r2n Q2M r2n exp r0 ðQ1M þ A BÞ r0 2 2 m M h iX m 1 1 1 exp ikrT0 ðAQ1M þ BÞ r2n detðI þ Q1M B1 AÞ m! 4 m¼0 "rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi# ik T 1 ðr2n Ar0 Þ B1T ðBQ1 H 2m 1M þ AÞ ðr2n Ar0 Þ ; 2
12 BQ1 1 Þ
where cos a sin a x2 cos a þ y 2 sin a r2n ¼ ; r2 ¼ : sin a cos a y 2 cos a x2 sin a ð19Þ Thus, for the coherent beam superposition, the field distribution of the resulting beam is written as Eðr2 Þ ¼
N 1 X
En ðr2n Þ
ð20Þ
n¼0
and the intensity reads I ¼ E ðr2 ÞEðr2 Þ;
ð21Þ
where * denote the complex conjugate and En(r2n) is given in Eq. (18). For the incoherent beam superposition case we have I¼
N 1 X n¼0
I n ðr2n Þ ¼
N 1 X
251
En ðr2n ÞEn ðr2n Þ:
ð22Þ
n¼0
The free space is also taken as the typical example. The submatrices ABCD is given in Eq. (13). Numerical calculations were performed using Eqs. (15)–(22). Typical results of the relative intensity for both coherent and incoherent superposition cases are complied in Figs. 2 and 3, where the calculation parameters are rT0 ¼ ð2; 2Þ, N = 5, M = 5. Q1 1M is given in Eq. (14). It turns out from Figs. 2(a) and 3(a) that the relative intensity distributions for the coherent and incoherent beam superposition cases at the input plane of z = 0 are identical, because there is no physical beam overlap that might create any interference. However, for the large value of z, which
ð18Þ
sequently the intensity distributions at the output plane become different. In general, the radial DEFGB arrays for coherent and incoherent beam superposition result in different intensity distributions and the intensity distribution for coherent beam superposition case is more complicated than that for the incoherent superposition one. In comparison with the results of superposition of radial elliptical Gaussian beam array [9] and rectangular Hermit–Gaussian beam array [10], there are different intensity distributions and symmetric characteristics of the radial DEFGB array for coherent and incoherent superposition, so that, to meet some practical requirement.
5. Conclusions In this study the decentered elliptical flattened Gaussian beam (DEFGB) has been defined by tensor method. The general propagation formula for the decentered elliptical flattened Gaussian beam has been derived. Using the derived formula, the propagation properties of a DEFGB in free space are illustrated with numerical examples. The numerical results show that the intensity distribution of a decentered elliptical flattened Gaussian beam depends upon the decentered parameter and the order of elliptical flattened Gaussian beam. We also investigated the properties of superposition of DEFGB. In general, the formula derived in this paper can be used to treat the propagation of the DEFGB in the non-symmetrical optical systems and the beam array constructed by DEFGB for coherent and incoherent
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M. Shen, S. Wang / Optics Communications 240 (2004) 245–252
superposition within the framework of paraxial approximations. Acknowledgement This work was supported by the National Natural Science Foundation of China Grant 60276035.
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