- Email: [email protected]
Elliptical cosh-Gaussian beams
Optics Communications 265 (2006) 418–424 www.elsevier.com/locate/optcom
Elliptical cosh-Gaussian beams Xinyue Du, Daomu Zhao
*
Department of Physics, Zhejiang University, Hangzhou 310027, China Received 6 March 2006; received in revised form 6 April 2006; accepted 6 April 2006
Abstract Using tensor method, a new kind of light beams named elliptical cosh-Gaussian beam (EChGB) is introduced in this paper. An analytical propagation expression for the EChGB passing through axially nonsymmetrical ABCD optical systems is derived by using vector integration. The derived formula is easily reduced to the propagation formula of a fundamental Gaussian beam and that of a coshGaussian beam passing through optical systems. Some numerical simulations are illustrated for the propagation properties of EChGBs through the nonsymmetrical optical transforming systems, and further extensions are pointed out. 2006 Elsevier B.V. All rights reserved. PACS: 41.85.Ew; 42.60.Jf Keywords: Elliptical cosh-Gaussian beams; Nonsymmetrical optical systems; Diffraction integral
1. Introduction The sinusoidal-Gaussian beams are the solutions of the paraxial wave equation for the propagation of electromagnetic waves in free space and in complex media as shown by Casperson and Tovar [1]. The more general solutions are the Hermite-sinusoidal-Gaussian beams, which are obtained as rectangular-symmetrical solutions of the paraxial wave equation [2,3]. The propagation properties of cosh-Gaussian beams [4], elegant Hermitecosh-Gaussian beams [5], off-axial Hermite-cosh-Gaussian beams [6], Hermite-cosine-Gaussian beams [7], and offaxial Hermite-cosine-Gaussian beams [8] in axially symmetrical optical systems have been extensively studied. Eyyuboglu and Baykal have also shown the propagation properties of Hermite-cosh-Gaussian beams, cos-Gaussian beams and cosh-Gaussian beams in the turbulent atmosphere [9,10]. The cosh-Gaussian beams, which are regarded as the special cases of sinusoidal-Gaussian beams or Hermitesinusoidal-Gaussian beams, are of practical interest because *
Corresponding author. Tel./fax: +86 571 888 63887. E-mail address: [email protected] (D. Zhao).
0030-4018/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.04.011
of their efficient in extracting energy from conventional laser amplifiers, and they have some important applications because their profiles can resemble closely the flat-top field distribution by choosing suitable beam parameters of cosh parts. The propagation and transformation of generalized elliptical Gaussian beams were studied by use of tensor method, and the analytical propagation expression for the generalized elliptical Gaussian beams through axially nonsymmetrical optical systems was obtained [11,12]. Furthermore, the propagation of decentered elliptical Gaussian beams in apertured and nonsymmetrical optical systems was also studied [13]. In this paper we will introduce a new kind of light beams called elliptical cosh-Gaussian beam (EChGB). This paper is organized as follows. In Section 2, we give a definition of the EChGB by using the tensor method. In Section 3, the analytical propagation formula of an EChGB passing through axially nonsymmetrical paraxial optical systems is derived through vector integral. In Section 4, some numerical simulations are given to illustrate for the propagation properties of EChGBs through a free space and a focusing system. A simple conclusion is given in Section 5.
X. Du, D. Zhao / Optics Communications 265 (2006) 418–424
2. Definition of EChGBs It is known that the optical field distribution of the twodimensional cosh-Gaussian beam on the input plane is characterized by [1–3]: ! x21 y 21 E1 ðx1 ; y 1 Þ ¼ exp 2 exp 2 coshðux x1 Þ coshðuy y 1 Þ; w0x w0y ð1Þ where w0x and w0y are the waist sizes of a Gaussian beam in the x and y directions, respectively; ux and uy are the displacement parameters associated with the cosh parts. For a more general case, the astigmatic Gaussian beam can be expressed as: ! x21 y 21 2x1 y 1 Eðx1 ; y 1 Þ ¼ exp 2 2 2 : ð2Þ w0x w0y w0xy Using tensor method and after some manipulation (see Appendix A), we can define the EChGB as follows: 1 ik E1 ðr1 Þ ¼ exp rT1 Q1 r cosh rT1 uþ þ cosh rT1 u ; 1 1 2 2 ð3Þ where k = 2p/k is the wave number, k is the wavelength, r1 denotes a position vector on a transverse plane given by rT1 ¼ ðx1 y 1 Þ, T means the matrix transpose operation, u+ and u are associated with the cosh parts given by: ux ux uþ ¼ ; u ¼ ; ð4Þ uy uy and Q1 1 is the 2 · 2 complex curvature tensor for the generalized elliptical Gaussian beam given by [12]: " 1 # " 2 # 2 qxx q1 ik w0x w0xy xy 1 Q1 ¼ ¼ : ð5Þ p w2 q1 q1 w2 xy yy 0xy 0y In the following calculation, we choose the beam parameters as w0x = 1.0 mm, w0y = 1.5 mm, w0xy = 2.0 mm. Fig. 1 shows that the normalized intensity distributions and their contour graphs of EChGBs versus the displacement parameters of cosh parts. When ux and uy are small, we can find that the intensity profile of the EChGB is similar to the Gaussian distribution as shown in Fig. 1(a). When the parameters (ux = 1 mm1, uy = 0.83 mm1) increase, the profile becomes flat-topped as shown in Fig. 1(b). Furthermore, with the increasing of ux and uy the central hollow appears and becomes deeper and larger as shown in Fig. 1(c) and (d). 3. Propagation of EChGBs in nonsymmetrical optical systems With the framework of the paraxial approximation, the propagation of the EChGB through an axially nonsymmetrical optical system can be treated by the general
419
Collins formula. The tensor form can be written as follows [11]: Z 1Z 1 i 1=2 E1 ðr1 Þ E2 ðr2 Þ ¼ expðikL0 Þ½detðBÞ k 1 1 ik T 1 1 T 1 T exp r1 B Ar1 2r1 B r2 þ r2 DB r2 dr1 ; 2 ð6Þ where L0 is optical path along the propagation axis, A, B, C, and D are the sub-matrices of the optical system taking the form: ax 0 bx 0 cx 0 A¼ ; B¼ ; C¼ ; 0 ay 0 by 0 cy dx 0 D¼ : ð7Þ 0 dy On substituting Eq. (3) as E1(r1) into Eq. (6) and after a straightforward vector integral operation, the following analytical propagation expression of the EChGB through a nonsymmetrical ABCD optical system is obtained: 1=2 1 ik T 1 E2 ðr2 Þ ¼ det BQ1 þ A expðikL Þexp Q r r 0 2 1 2 2 2 2 n h i 1 1 uþ nþ cosh rT2 B1T Q1 1 þB A h io 1 1 þ B A u þ n cosh rT2 B1T Q1 ; ð8Þ 1 where n+ and n are small constant phase factors given by 1 i T 1 1 nþ ¼ exp uþ Q1 þ B A uþ ; 2k ð9Þ 1 i T 1 1 n ¼ exp u Q1 þ B A u ; 2k 1 and Q1 2 is related with Q1 by the formula: 1 1 1 1 B1T Q1 B Q1 2 ¼ DB 1 þB A 1 1 1 ¼ C þ DQ1 A þ BQ1 ;
ð10Þ
it is the tensor ABCD law for nonsymmetrical optical systems. Eq. (8) can be reduced to the previous existent result in some special cases. When the parameters of cosh parts ux = 0 and uy = 0, Eq. (8) can be simplified to the general formula for a fundamental Gaussian beam passing through a nonsymmetrical optical system as follows [11]: 1 1=2 ik T 1 E2 ðr2 Þ ¼ det BQ1 þ A expðikL0 Þ exp r2 Q2 r2 : 2 ð11Þ In Eq. (8), if Q1 1 is assumed to be 1 0 q1 : Q1 1 ¼ 0 0
ð12Þ
Eq. (8) is then reduced to the one-dimensional case as
420
X. Du, D. Zhao / Optics Communications 265 (2006) 418–424
Fig. 1. Three-dimensional normalized intensity distributions and their contour graphs of the EChGBs on the input plane of z = 0. The displacement parameters of cosh parts: (a) ux = 0.5 mm1, uy = 0.5 mm1; (b) ux = 1 mm1, uy = 0.83 mm1; (c) ux = 1 mm1, uy = 1 mm1; (d) ux = 1 mm1, uy = 2 mm1.
ik 2 1 E2 ðx2 Þ ¼ ðb=q1 þ aÞ expðikL0 Þ exp x2 q2 2 i 2 b 1 exp ux cosh ux x2 ; 2k a þ b=q1 a þ b=q1 1=2
ð13Þ
where 1 c þ d=q1 ¼ : q2 a þ b=q1
ð14Þ
Eq. (13) is the same as the propagation formula of Ref. [4].
X. Du, D. Zhao / Optics Communications 265 (2006) 418–424
4. Numerical simulations and discussion Generally, it is very difficult or troublesome to calculate the output intensity distribution by using Eq. (6). In this section we will study the propagation properties of EChGBs passing through nonsymmetrical optical systems by using the derived propagation equation (8). First, we are going to illustrate the propagation of an EChGB through a free space for the sake of simplicity. The sub-matrices for a free space of distance z read as 1 0 z 0 0 0 A¼ ; B¼ ; C¼ ; 0 1 0 z 0 0 1 0 D¼ : ð15Þ 0 1 On substituting Eq. (15) into Eq. (8), we can obtain the three-dimensional intensity distributions on the output plane for an EChGB passing through a free space. The propagation distance is represented by the equivalent Fresnel number N x ¼ w20x =ðkzÞ in the following calculation. Fig. 2 shows that the normalized intensity distributions of EChGBs versus the parameters of the cosh parts at the near-field of Nx = 1 (in the Fresnel diffraction region). We can see the intensity distributions are similar to the initial EChGBs. When ux and uy are large, it becomes to be the superposition of two decentered Gaussian beams as shown in Fig. 2(c) and (d).
421
Fig. 3 shows that the normalized intensity distributions of EChGBs versus the parameters of the cosh parts at the far-field of Nx = 0.01 (in the Fraunhofer diffraction region). We can see only one lobe peak as shown in Fig. 3(a) and it is similar to be the elliptical Gaussian distribution. The smaller lobe peaks appear on the two sides of the main peak in Fig. 3(b), and the height and number increase with the increasing of the parameters of cosh parts as shown in Fig. 3(c) and (d). With the augment of the propagation distance, this change becomes more significant. Furthermore, we will study the transformation of EChGBs passing through a focusing system. The sub-matrices for a focusing system consisted of a focal lens with focal lengths fx and fy in the x and y directions, respectively, and a free space with a distance of z read as
1 z=fx
0 1=fx
A¼ C¼
0
z 0 ; B¼ ; 1 z=fy 0 z 0 1 0 ; D¼ ; 1=fy 0 1 0
ð16Þ
and it can be regarded as an axially nonsymmetrical ABCD optical system. Fig. 4 shows that the normalized intensity distributions of EChGBs on the two focal planes of the lens. In the calculation, we choose the focal lengths fx = 1580 mm and fy = 2000 mm. From Fig. 4 we can find that the intensity
Fig. 2. Three-dimensional normalized intensity distributions of the EChGBs passing through a free space on the plane of Nx = 1. The displacement parameters of cosh parts: (a) ux = 1 mm1, uy = 1 mm1; (b) ux = 1 mm1, uy = 1.5 mm1; (c) ux = 1 mm1, uy = 2 mm1; (d) ux = 2 mm1, uy = 4 mm1.
422
X. Du, D. Zhao / Optics Communications 265 (2006) 418–424
Fig. 3. As Fig. 2, but on the plane of Nx = 0.01.
Fig. 4. Three-dimensional normalized intensity distributions of the EChGBs passing through a focusing system consisted of a focal lens with focal lengths fx = 1580 mm and fy = 2000 mm. The same displacement parameters of cosh parts ux = 1 mm1 and uy = 2 mm1. The different propagation distances: (a) z = 1580 mm, (b) z = 2000 mm.
distributions are compressed in x and y directions respectively, and the feature of the EChGB is partly preserved. Fig. 5 shows that the normalized intensity distributions and their contours of EChGBs on the focal plane of the lens. For the sake of simplicity, we choose the propagation distance as z = fx = fy = 1580 mm. From Fig. 5 we can find that on the focal plane of the lens, the intensity distributions of EChGBs can be regarded as the cosine curves with certain oscillating frequency modulated by Gaussian envelopes and the oscillating frequency increases with the increasing of the parameters of cosh parts. By numerical calculation we found that the displacement parameters of cosh parts decide the space oscillating frequency, when the beam waist in Gaussian part decreases,
it will reduce the size of the intensity distribution in the far field and the number of the lobe peaks decreases. Otherwise, with the increasing of the beam waist it will increase the number of the peaks. Furthermore, the different focal lengths only influence the size of the intensity distribution but not its profile. We also found the reciprocity phenomena of coshGaussian to cos-Gaussian when they pass through a free space or a focusing system. That is to say, a cosh-Gaussian beam on the input plane is corresponding to a cos-Gaussian beam in the far field or on the focal plane, and the opposite transformation is also existent. These observations are consistent with the reciprocity property of cos-Gaussian and cosh-Gaussian beams in a turbulent atmosphere [9,10].
X. Du, D. Zhao / Optics Communications 265 (2006) 418–424
423
Fig. 5. Three-dimensional normalized intensity distributions and their contour graphs on the focal plane of z = fx = fy = 1580 mm for the EChGBs passing through a focusing system. The displacement parameters of cosh parts: (a) ux = 1 mm1, uy = 1 mm1; (b) ux = 1 mm1, uy = 2 mm1; (c) ux = 2 mm1, uy = 4 mm1; (d) ux = 4 mm1, uy = 8 mm1.
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X. Du, D. Zhao / Optics Communications 265 (2006) 418–424
5. Conclusion In conclusion, a new kind of light beams called the EChGB is introduced in this paper by using tensor method. The analytical propagation formula of an EChGB passing through an axially nonsymmetrical ABCD optical system is derived through vector integration. By virtue of numerical simulations, the propagation properties of the EChGB through nonsymmetrical optical transforming systems of a free space and a focusing system are respectively illustrated. It should be pointed out that the method introduced here can also be straightforwardly extended to the analyses of other kinds of light beams, such as decentered elliptical cosh-Gaussian beams, elliptical cosine-Gaussian beams, and so forth. Acknowledgements This work was supported by the National Natural Science Foundation of China under Grant 10276034. The authors are also grateful for the referee’s comments and suggestions for improving our paper. Appendix A. Derivation of the elliptical cosh-Gaussian beam (Eq. (3)) The sinusoidal-Gaussian beams have been obtained to be the solutions of the paraxial wave equation for the propagation of electromagnetic waves. As their special cases, the two-dimensional cosh-Gaussian beam on the input plane is characterized by [1–3]; ! x21 y 21 E1 ðx1 ; y 1 Þ ¼ exp 2 exp 2 coshðux x1 Þ coshðuy y 1 Þ: w0x w0y
x2 y2 ¼ exp 21 21 w0x w0y
!
1 1 ½expðux x1 þ uy y 1 Þ 2 2
1 þ expðux x1 uy y 1 Þ þ ½expðux x1 uy y 1 Þ 2 þ expðux x1 þ uy y 1 Þ ! x21 y 21 1 ¼ exp 2 2 ½coshðux x1 þ uy y 1 Þ 2 w0x w0y þ coshðux x1 uy y 1 Þ:
ðA2Þ
For a more general case, the astigmatic Gaussian beam can be expressed as; ! x21 y 21 2x1 y 1 E1 ðx1 ; y 1 Þ ¼ exp 2 2 2 ; ðA3Þ w0x w0y w0xy and by setting ux ux x1 r1 ¼ ; uþ ¼ ; u ¼ ; uy uy y1 " 1 # " # 2 2 qxx q1 ik w0x w0xy xy Q1 ¼ ; 1 ¼ p w2 q1 q1 w2 xy yy 0xy 0y
ðA4Þ
the elliptical cosh-Gaussian beam is defined using tensor method as follows: ! x21 y 21 2x1 y 1 E1 ðx1 ;y 1 Þ ¼ exp 2 2 2 w0x w0y w0xy 1 ½coshðux x1 þ uy y 1 Þ þ coshðux x1 uy y 1 Þ 2 1 ik T 1 ¼ exp r1 Q1 r1 cosh rT1 uþ þ cosh rT1 u : 2 2 ðA5Þ
ðA1Þ Beginning with the two-dimensional cosh-Gaussian beam of Eq. (A1), we have ! x21 y 21 E1 ðx1 ; y 1 Þ ¼ exp 2 exp 2 coshðux x1 Þ coshðuy y 1 Þ w0x w0y ! x2 y2 1 ¼ exp 21 21 ½expðux x1 Þ þ expðux x1 Þ 2 w0x w0y 1 ½expðuy y 1 Þ þ expðuy y 1 Þ 2 ! x21 y 21 1 ¼ exp 2 2 ½expðux x1 þ uy y 1 Þ 4 w0x w0y þ expðux x1 uy y 1 Þ þ expðux x1 þ uy y 1 Þ þ expðux x1 uy y 1 Þ
References [1] L.W. Casperson, D.G. Hall, A.A. Tovar, J. Opt. Soc. Am. A 14 (1997) 3341. [2] L.W. Casperson, A.A. Tovar, J. Opt. Soc. Am. A 15 (1998) 954. [3] A.A. Tovar, L.W. Casperson, J. Opt. Soc. Am. A 15 (1998) 2425. [4] B. Lu¨, H. Ma, B. Zhang, Opt. Commun. 164 (1999) 165. [5] S. Yu, H. Guo, X. Fu, W. Hu, Opt. Commun. 204 (2002) 59. [6] D. Zhao, H. Mao, H. Liu, J. Opt. A: Pure Appl. Opt. 6 (2004) 77. [7] N. Zhou, G. Zeng, Opt. Commun. 232 (2004) 49. [8] D. Zhao, H. Mao, W. Zhang, S. Wang, Opt. Commun. 224 (2003) 5. [9] H.T. Eyyuboglu, Opt. Commun. 245 (2005) 37. [10] H.T. Eyyuboglu, Y. Baykal, Opt. Exp. 12 (2004) 4659. [11] Q. Lin, S. Wang, J. Alda, E. Bernabeu, Optik 85 (1990) 67. [12] J. Alda, S. Wang, E. Bernabeu, Opt. Commun. 80 (1991) 350. [13] X. Du, D. Zhao, J. Opt. Soc. Am. A 23 (2006) 625.
Elliptical cosh-Gaussian beams Xinyue Du, Daomu Zhao
*
Department of Physics, Zhejiang University, Hangzhou 310027, China Received 6 March 2006; received in revised form 6 April 2006; accepted 6 April 2006
Abstract Using tensor method, a new kind of light beams named elliptical cosh-Gaussian beam (EChGB) is introduced in this paper. An analytical propagation expression for the EChGB passing through axially nonsymmetrical ABCD optical systems is derived by using vector integration. The derived formula is easily reduced to the propagation formula of a fundamental Gaussian beam and that of a coshGaussian beam passing through optical systems. Some numerical simulations are illustrated for the propagation properties of EChGBs through the nonsymmetrical optical transforming systems, and further extensions are pointed out. 2006 Elsevier B.V. All rights reserved. PACS: 41.85.Ew; 42.60.Jf Keywords: Elliptical cosh-Gaussian beams; Nonsymmetrical optical systems; Diffraction integral
1. Introduction The sinusoidal-Gaussian beams are the solutions of the paraxial wave equation for the propagation of electromagnetic waves in free space and in complex media as shown by Casperson and Tovar [1]. The more general solutions are the Hermite-sinusoidal-Gaussian beams, which are obtained as rectangular-symmetrical solutions of the paraxial wave equation [2,3]. The propagation properties of cosh-Gaussian beams [4], elegant Hermitecosh-Gaussian beams [5], off-axial Hermite-cosh-Gaussian beams [6], Hermite-cosine-Gaussian beams [7], and offaxial Hermite-cosine-Gaussian beams [8] in axially symmetrical optical systems have been extensively studied. Eyyuboglu and Baykal have also shown the propagation properties of Hermite-cosh-Gaussian beams, cos-Gaussian beams and cosh-Gaussian beams in the turbulent atmosphere [9,10]. The cosh-Gaussian beams, which are regarded as the special cases of sinusoidal-Gaussian beams or Hermitesinusoidal-Gaussian beams, are of practical interest because *
Corresponding author. Tel./fax: +86 571 888 63887. E-mail address: [email protected] (D. Zhao).
0030-4018/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.04.011
of their efficient in extracting energy from conventional laser amplifiers, and they have some important applications because their profiles can resemble closely the flat-top field distribution by choosing suitable beam parameters of cosh parts. The propagation and transformation of generalized elliptical Gaussian beams were studied by use of tensor method, and the analytical propagation expression for the generalized elliptical Gaussian beams through axially nonsymmetrical optical systems was obtained [11,12]. Furthermore, the propagation of decentered elliptical Gaussian beams in apertured and nonsymmetrical optical systems was also studied [13]. In this paper we will introduce a new kind of light beams called elliptical cosh-Gaussian beam (EChGB). This paper is organized as follows. In Section 2, we give a definition of the EChGB by using the tensor method. In Section 3, the analytical propagation formula of an EChGB passing through axially nonsymmetrical paraxial optical systems is derived through vector integral. In Section 4, some numerical simulations are given to illustrate for the propagation properties of EChGBs through a free space and a focusing system. A simple conclusion is given in Section 5.
X. Du, D. Zhao / Optics Communications 265 (2006) 418–424
2. Definition of EChGBs It is known that the optical field distribution of the twodimensional cosh-Gaussian beam on the input plane is characterized by [1–3]: ! x21 y 21 E1 ðx1 ; y 1 Þ ¼ exp 2 exp 2 coshðux x1 Þ coshðuy y 1 Þ; w0x w0y ð1Þ where w0x and w0y are the waist sizes of a Gaussian beam in the x and y directions, respectively; ux and uy are the displacement parameters associated with the cosh parts. For a more general case, the astigmatic Gaussian beam can be expressed as: ! x21 y 21 2x1 y 1 Eðx1 ; y 1 Þ ¼ exp 2 2 2 : ð2Þ w0x w0y w0xy Using tensor method and after some manipulation (see Appendix A), we can define the EChGB as follows: 1 ik E1 ðr1 Þ ¼ exp rT1 Q1 r cosh rT1 uþ þ cosh rT1 u ; 1 1 2 2 ð3Þ where k = 2p/k is the wave number, k is the wavelength, r1 denotes a position vector on a transverse plane given by rT1 ¼ ðx1 y 1 Þ, T means the matrix transpose operation, u+ and u are associated with the cosh parts given by: ux ux uþ ¼ ; u ¼ ; ð4Þ uy uy and Q1 1 is the 2 · 2 complex curvature tensor for the generalized elliptical Gaussian beam given by [12]: " 1 # " 2 # 2 qxx q1 ik w0x w0xy xy 1 Q1 ¼ ¼ : ð5Þ p w2 q1 q1 w2 xy yy 0xy 0y In the following calculation, we choose the beam parameters as w0x = 1.0 mm, w0y = 1.5 mm, w0xy = 2.0 mm. Fig. 1 shows that the normalized intensity distributions and their contour graphs of EChGBs versus the displacement parameters of cosh parts. When ux and uy are small, we can find that the intensity profile of the EChGB is similar to the Gaussian distribution as shown in Fig. 1(a). When the parameters (ux = 1 mm1, uy = 0.83 mm1) increase, the profile becomes flat-topped as shown in Fig. 1(b). Furthermore, with the increasing of ux and uy the central hollow appears and becomes deeper and larger as shown in Fig. 1(c) and (d). 3. Propagation of EChGBs in nonsymmetrical optical systems With the framework of the paraxial approximation, the propagation of the EChGB through an axially nonsymmetrical optical system can be treated by the general
419
Collins formula. The tensor form can be written as follows [11]: Z 1Z 1 i 1=2 E1 ðr1 Þ E2 ðr2 Þ ¼ expðikL0 Þ½detðBÞ k 1 1 ik T 1 1 T 1 T exp r1 B Ar1 2r1 B r2 þ r2 DB r2 dr1 ; 2 ð6Þ where L0 is optical path along the propagation axis, A, B, C, and D are the sub-matrices of the optical system taking the form: ax 0 bx 0 cx 0 A¼ ; B¼ ; C¼ ; 0 ay 0 by 0 cy dx 0 D¼ : ð7Þ 0 dy On substituting Eq. (3) as E1(r1) into Eq. (6) and after a straightforward vector integral operation, the following analytical propagation expression of the EChGB through a nonsymmetrical ABCD optical system is obtained: 1=2 1 ik T 1 E2 ðr2 Þ ¼ det BQ1 þ A expðikL Þexp Q r r 0 2 1 2 2 2 2 n h i 1 1 uþ nþ cosh rT2 B1T Q1 1 þB A h io 1 1 þ B A u þ n cosh rT2 B1T Q1 ; ð8Þ 1 where n+ and n are small constant phase factors given by 1 i T 1 1 nþ ¼ exp uþ Q1 þ B A uþ ; 2k ð9Þ 1 i T 1 1 n ¼ exp u Q1 þ B A u ; 2k 1 and Q1 2 is related with Q1 by the formula: 1 1 1 1 B1T Q1 B Q1 2 ¼ DB 1 þB A 1 1 1 ¼ C þ DQ1 A þ BQ1 ;
ð10Þ
it is the tensor ABCD law for nonsymmetrical optical systems. Eq. (8) can be reduced to the previous existent result in some special cases. When the parameters of cosh parts ux = 0 and uy = 0, Eq. (8) can be simplified to the general formula for a fundamental Gaussian beam passing through a nonsymmetrical optical system as follows [11]: 1 1=2 ik T 1 E2 ðr2 Þ ¼ det BQ1 þ A expðikL0 Þ exp r2 Q2 r2 : 2 ð11Þ In Eq. (8), if Q1 1 is assumed to be 1 0 q1 : Q1 1 ¼ 0 0
ð12Þ
Eq. (8) is then reduced to the one-dimensional case as
420
X. Du, D. Zhao / Optics Communications 265 (2006) 418–424
Fig. 1. Three-dimensional normalized intensity distributions and their contour graphs of the EChGBs on the input plane of z = 0. The displacement parameters of cosh parts: (a) ux = 0.5 mm1, uy = 0.5 mm1; (b) ux = 1 mm1, uy = 0.83 mm1; (c) ux = 1 mm1, uy = 1 mm1; (d) ux = 1 mm1, uy = 2 mm1.
ik 2 1 E2 ðx2 Þ ¼ ðb=q1 þ aÞ expðikL0 Þ exp x2 q2 2 i 2 b 1 exp ux cosh ux x2 ; 2k a þ b=q1 a þ b=q1 1=2
ð13Þ
where 1 c þ d=q1 ¼ : q2 a þ b=q1
ð14Þ
Eq. (13) is the same as the propagation formula of Ref. [4].
X. Du, D. Zhao / Optics Communications 265 (2006) 418–424
4. Numerical simulations and discussion Generally, it is very difficult or troublesome to calculate the output intensity distribution by using Eq. (6). In this section we will study the propagation properties of EChGBs passing through nonsymmetrical optical systems by using the derived propagation equation (8). First, we are going to illustrate the propagation of an EChGB through a free space for the sake of simplicity. The sub-matrices for a free space of distance z read as 1 0 z 0 0 0 A¼ ; B¼ ; C¼ ; 0 1 0 z 0 0 1 0 D¼ : ð15Þ 0 1 On substituting Eq. (15) into Eq. (8), we can obtain the three-dimensional intensity distributions on the output plane for an EChGB passing through a free space. The propagation distance is represented by the equivalent Fresnel number N x ¼ w20x =ðkzÞ in the following calculation. Fig. 2 shows that the normalized intensity distributions of EChGBs versus the parameters of the cosh parts at the near-field of Nx = 1 (in the Fresnel diffraction region). We can see the intensity distributions are similar to the initial EChGBs. When ux and uy are large, it becomes to be the superposition of two decentered Gaussian beams as shown in Fig. 2(c) and (d).
421
Fig. 3 shows that the normalized intensity distributions of EChGBs versus the parameters of the cosh parts at the far-field of Nx = 0.01 (in the Fraunhofer diffraction region). We can see only one lobe peak as shown in Fig. 3(a) and it is similar to be the elliptical Gaussian distribution. The smaller lobe peaks appear on the two sides of the main peak in Fig. 3(b), and the height and number increase with the increasing of the parameters of cosh parts as shown in Fig. 3(c) and (d). With the augment of the propagation distance, this change becomes more significant. Furthermore, we will study the transformation of EChGBs passing through a focusing system. The sub-matrices for a focusing system consisted of a focal lens with focal lengths fx and fy in the x and y directions, respectively, and a free space with a distance of z read as
1 z=fx
0 1=fx
A¼ C¼
0
z 0 ; B¼ ; 1 z=fy 0 z 0 1 0 ; D¼ ; 1=fy 0 1 0
ð16Þ
and it can be regarded as an axially nonsymmetrical ABCD optical system. Fig. 4 shows that the normalized intensity distributions of EChGBs on the two focal planes of the lens. In the calculation, we choose the focal lengths fx = 1580 mm and fy = 2000 mm. From Fig. 4 we can find that the intensity
Fig. 2. Three-dimensional normalized intensity distributions of the EChGBs passing through a free space on the plane of Nx = 1. The displacement parameters of cosh parts: (a) ux = 1 mm1, uy = 1 mm1; (b) ux = 1 mm1, uy = 1.5 mm1; (c) ux = 1 mm1, uy = 2 mm1; (d) ux = 2 mm1, uy = 4 mm1.
422
X. Du, D. Zhao / Optics Communications 265 (2006) 418–424
Fig. 3. As Fig. 2, but on the plane of Nx = 0.01.
Fig. 4. Three-dimensional normalized intensity distributions of the EChGBs passing through a focusing system consisted of a focal lens with focal lengths fx = 1580 mm and fy = 2000 mm. The same displacement parameters of cosh parts ux = 1 mm1 and uy = 2 mm1. The different propagation distances: (a) z = 1580 mm, (b) z = 2000 mm.
distributions are compressed in x and y directions respectively, and the feature of the EChGB is partly preserved. Fig. 5 shows that the normalized intensity distributions and their contours of EChGBs on the focal plane of the lens. For the sake of simplicity, we choose the propagation distance as z = fx = fy = 1580 mm. From Fig. 5 we can find that on the focal plane of the lens, the intensity distributions of EChGBs can be regarded as the cosine curves with certain oscillating frequency modulated by Gaussian envelopes and the oscillating frequency increases with the increasing of the parameters of cosh parts. By numerical calculation we found that the displacement parameters of cosh parts decide the space oscillating frequency, when the beam waist in Gaussian part decreases,
it will reduce the size of the intensity distribution in the far field and the number of the lobe peaks decreases. Otherwise, with the increasing of the beam waist it will increase the number of the peaks. Furthermore, the different focal lengths only influence the size of the intensity distribution but not its profile. We also found the reciprocity phenomena of coshGaussian to cos-Gaussian when they pass through a free space or a focusing system. That is to say, a cosh-Gaussian beam on the input plane is corresponding to a cos-Gaussian beam in the far field or on the focal plane, and the opposite transformation is also existent. These observations are consistent with the reciprocity property of cos-Gaussian and cosh-Gaussian beams in a turbulent atmosphere [9,10].
X. Du, D. Zhao / Optics Communications 265 (2006) 418–424
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Fig. 5. Three-dimensional normalized intensity distributions and their contour graphs on the focal plane of z = fx = fy = 1580 mm for the EChGBs passing through a focusing system. The displacement parameters of cosh parts: (a) ux = 1 mm1, uy = 1 mm1; (b) ux = 1 mm1, uy = 2 mm1; (c) ux = 2 mm1, uy = 4 mm1; (d) ux = 4 mm1, uy = 8 mm1.
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X. Du, D. Zhao / Optics Communications 265 (2006) 418–424
5. Conclusion In conclusion, a new kind of light beams called the EChGB is introduced in this paper by using tensor method. The analytical propagation formula of an EChGB passing through an axially nonsymmetrical ABCD optical system is derived through vector integration. By virtue of numerical simulations, the propagation properties of the EChGB through nonsymmetrical optical transforming systems of a free space and a focusing system are respectively illustrated. It should be pointed out that the method introduced here can also be straightforwardly extended to the analyses of other kinds of light beams, such as decentered elliptical cosh-Gaussian beams, elliptical cosine-Gaussian beams, and so forth. Acknowledgements This work was supported by the National Natural Science Foundation of China under Grant 10276034. The authors are also grateful for the referee’s comments and suggestions for improving our paper. Appendix A. Derivation of the elliptical cosh-Gaussian beam (Eq. (3)) The sinusoidal-Gaussian beams have been obtained to be the solutions of the paraxial wave equation for the propagation of electromagnetic waves. As their special cases, the two-dimensional cosh-Gaussian beam on the input plane is characterized by [1–3]; ! x21 y 21 E1 ðx1 ; y 1 Þ ¼ exp 2 exp 2 coshðux x1 Þ coshðuy y 1 Þ: w0x w0y
x2 y2 ¼ exp 21 21 w0x w0y
!
1 1 ½expðux x1 þ uy y 1 Þ 2 2
1 þ expðux x1 uy y 1 Þ þ ½expðux x1 uy y 1 Þ 2 þ expðux x1 þ uy y 1 Þ ! x21 y 21 1 ¼ exp 2 2 ½coshðux x1 þ uy y 1 Þ 2 w0x w0y þ coshðux x1 uy y 1 Þ:
ðA2Þ
For a more general case, the astigmatic Gaussian beam can be expressed as; ! x21 y 21 2x1 y 1 E1 ðx1 ; y 1 Þ ¼ exp 2 2 2 ; ðA3Þ w0x w0y w0xy and by setting ux ux x1 r1 ¼ ; uþ ¼ ; u ¼ ; uy uy y1 " 1 # " # 2 2 qxx q1 ik w0x w0xy xy Q1 ¼ ; 1 ¼ p w2 q1 q1 w2 xy yy 0xy 0y
ðA4Þ
the elliptical cosh-Gaussian beam is defined using tensor method as follows: ! x21 y 21 2x1 y 1 E1 ðx1 ;y 1 Þ ¼ exp 2 2 2 w0x w0y w0xy 1 ½coshðux x1 þ uy y 1 Þ þ coshðux x1 uy y 1 Þ 2 1 ik T 1 ¼ exp r1 Q1 r1 cosh rT1 uþ þ cosh rT1 u : 2 2 ðA5Þ
ðA1Þ Beginning with the two-dimensional cosh-Gaussian beam of Eq. (A1), we have ! x21 y 21 E1 ðx1 ; y 1 Þ ¼ exp 2 exp 2 coshðux x1 Þ coshðuy y 1 Þ w0x w0y ! x2 y2 1 ¼ exp 21 21 ½expðux x1 Þ þ expðux x1 Þ 2 w0x w0y 1 ½expðuy y 1 Þ þ expðuy y 1 Þ 2 ! x21 y 21 1 ¼ exp 2 2 ½expðux x1 þ uy y 1 Þ 4 w0x w0y þ expðux x1 uy y 1 Þ þ expðux x1 þ uy y 1 Þ þ expðux x1 uy y 1 Þ
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