GENERAL FLAT BEAM FORMULATION AND INTENSITY PROFILES OF THESE BEAMS ON SOURCE PLANE SEYRAN YILMAZ, FATİH GÜRAL Electronic and Communication Engineering Department, Cankaya University Literature cited [1] Arpalı Ç., Eyyuboğlu Halil T., Baykal Y. “Flat-Topped beams and their characteristics in turbulent media”, Optic Express, Vol.14, No.10, 2006 [2] Arpalı Ç., Arpalı Serap A., Yazıcıoğlu C., Eyyuboğlu Halil T., Baykal Y., “Yüksek dereceli halkasal Gauss ve düz tepeli ışık hüzmelerinin türbülanslı atmosferde yayılım özellikleri”, URSI,2006 [3]Li Yaung, “Light beams with Flat-topped profiles”, Optic Letters, Volume.27, No.12, 15 June [4] Shealy David L., Hoffnagle John A., “Laser beam shaping profiles and propagation”, Applied Optics, Volume.45, No.21, 20 July [5] Cai Yangjian, “Propagation of various flat-topped beams in a turbulent atmosphere”, Journal Of Optics A: Pure and Applied Optics 8, , 2006 [6] Bagini V., Borghi R., Gori F., Pacileo A.M., Santarsiero M., Ambrosini D., Spagnolo G.Schirripa, “Propagation of axially symmetric flattened Gaussian beams”, Optical Society of America, /96/071385, 1996 [7] Andrews, L.C. and Phillips, R.L., Laser Beam Propagation through Random Media, SPIE, Bellingham, Washington, Photo Put your photo here Abstract In this project, Flat-Topped, Super Gaussian, Super Lorentzian, Circular Flat-Topped, Elliptical Flat-Topped, and Rectangular Flat- Topped beams are studied and their source plane characteristics are investigated. The general flat beam formulation is found and using this general formula, the intensity graphics for each flat beam are sketched with using MATLAB program. Introduction Free space optical systems, become more important in corporate and civilian communication areas according to some properties, for example, high data rate, fast installation, portability and safety. Data can be sent narrower beam in free space optical systems so blocking or obtaining information by outside is more difficult. Flat beams are reduced the fundamental Gaussian beam by introducing an order of flatness parameter. These beams expose the less expansion during spreading therefore flat beams intensity profiles are affected smaller than other beams when propagating through the atmospheric turbulence [1]. Flat-Topped, Super Gaussian, Super Lorentzian, Circular Flat-Topped, Elliptical Flat-Topped and Rectangular Flat- Topped beams are investigated on source plane and the general flat beam formula is found out with using these beams. The intensity profiles of these flat beams are sketched and studied by using the general formula, and flat beams are compared according to the different flatness parameter N. Materials and methods The formulas on the source plane of the Flat-Topped, Super Gaussian, Super Lorentzian, Circular Flat-Topped, Elliptical Flat-Topped and Rectangular Flat-Topped beams, are used for determining the general beam formulation: Eq. 2 is obtained from multiplication of Eq. 1 and complex conjugate of Eq. 1 and it shows the intensity of the beams. In this formulation;, is obtained from the decomposition of the vector s into x and y components of the transverse source plane. and are the source sizes along and directions.,,,, and values are used for flatness parameter for different flat beam types formulation. A, B,,, and, each has a different value to determine the different flat beam formulation. Flat beams are obtained and compared with using this general formulation. N flatness parameter must be even number, and for N = 2 Gaussian beam is obtained. Following parameters and limitations can be used to represent Super Gaussian beam, from Eq. 1; (12) If the values Eq. 12 are substituted from Eq. 1, Eq. 11 can be obtained. 6. Flat-Topped Beam: This beam is derived from the Gaussian beam by introducing an order of flatness parameter, and formulation is; For N = 1, Gaussian beam is described. Following parameters and limitations can be used to represent Flat-Topped beam, from Eq. 1; (14) If the values Eq. 14 are substituted from Eq. 1, Eq. 13 can be obtained. 1. Circular Flat - Topped Beam: The circular symmetry of the Flat - Topped beam formula at z = 0 and the circular coordinates of the Gaussian beam is expressed Circular Flat-Topped beam formulation [5]: is a binomial parameter and N is a flatness parameter in Eq. 3. For N = 0 Gauss beam is obtained, and if N increases, the Circular - Flat Topped beam becomes flatter. Following parameters and limitations can be used to represent Circular Flat - Topped beam, from Eq. 1; (4) If the values of Eq. 4 are substituted from Eq. 1, Eq. 3 can be obtained. 2. Elliptical Flat - Topped Beam: The elliptical symmetry of the Flat - Topped beam formula at z = 0 is expressed Elliptical Flat - Topped beam formulation [5]: N is determined as a flatness parameter, and if N=1, Gaussian beam is obtained. Elliptical Flat - Topped beam is defined for different values of and, also if N parameter increases, the beam becomes flatter. Following parameters and limitations can be used to represent Elliptical Flat - Topped beam, from Eq. 1; (6) If the values Eq. 6 are substituted from Eq. 1, Eq. 5 can be obtained. 3. Rectangular Flat - Topped Beam: The rectangular symmetry of the Flat - Topped beam formula at z = 0, also the rectangular coordinates of the Elliptical Flat Topped beam is expressed Rectangular Flat - Topped beam formulation [5]: N and M are flatness parameters. For N = 1 and M = 1 Gaussian beam is defined. If the formula parameters are taken as M = N and, the Square Flat - Topped beam is obtained. Following parameters and limitations can be used to represent Rectangular Flat - Topped beam, from Eq. 1; (8) If the values Eq. 8 are substituted from Eq. 1, Eq. 7 can be obtained. 4. Super Lorentzian Beam: It is the simple flat beam and its formulation is: For N =2, Gaussian beam profile is defined, and also flat beams are obtained when N is greater than 2. Following parameters and limitations can be used to represent Super Lorentzian beam, from Eq. 1; (10) If the values Eq. 10 are substituted from Eq. 1, Eq. 9 can be obtained. 5. Super Gaussian Beam: This beam type is the most popular model for describing flat-topped beam profiles and expressed with this formulation; Results and Conclusions In this section, the general flat beam formulation is used for investigating intensity profiles of the Flat-Topped, Super Gaussian, Super Lorentzian, Circular Flat-Topped, Elliptical Flat-Topped and Rectangular Flat-Topped beams on the source plane. These intensity profiles are sketched for each beam and compared with same and different N parameters. While sketching the intensity profiles of the beams, the general beam formulation Eq. 1, is used with the variable parameters above by substituting Eq. 2 for each one and maximum intensity on source plane is normalized below [7] Hence, expression shows the maximum value of the. Figs. 1-6 show the Circular Flat-Topped, Elliptical Flat-Topped, Rectangular Flat-Topped, Super Gaussian, Super Lorentzian and Flat- Topped beam intensity profiles in 3D (three dimensional) scheme. Furthermore, N flatness parameter is defined N = 6 for all beams. Also these figures are controlled by Ref. 1,3,4,5. Fig.1. The intensity profile on Fig.2. The intensity profile on Fig.3. The intensity profile on the the source plane of Circular the source plane of Elliptical source plane of Rectangular Flat-Topped beam Flat-Topped beam Flat-Topped beam Fig.4. The intensity profile on Fig.5. The intensity profile on Fig.6. The intensity profile on the the source plane of Super the source plane of Super the source plane of Flat-Topped Gaussian beam Lorentzian beam beam In Fig. 7. (a,b,c) is defined for Flat-Topped, Circular Flat-Topped, Rectangular Flat-Topped, Super Gaussian and Super Lorentzian beams, also and values are defined for Elliptical Flat-Topped beam. Investigated beams are sketched for N = 2 in Fig. 7 (a). As can be seen, Super Gaussian and Super Lorentzian beams have a Gaussian profile, but, the other beams become flatter. All beams are compared with flatness parameter N = 8 in Fig. 7 (b). In this figure, Super Gaussian beam tail becomes smoother faster than other Flat Topped beams. Rectangular Flat-Topped beam has a smallest electrical field value, and approximate to zero. Super Gaussian, Flat-Topped and Super Lorentzian beams are sketched for N = 50 in Fig. 7 (c). The electrical field values of the Super Gaussian and Super Lorentzian beams are nearly same according to this figure. Moreover, Super Lorentzian beam tail becomes smoother lately again. Circular Flat-Topped, Elliptical Flat-Topped and Rectangular Flat-Topped beams are not sketched in this figure, because their electrical field values approximately zero. (a) (b) (c) Fig.7. Flat-Topped, Circular Flat-Topped, Elliptical Flat-Topped, Rectangular Flat- Topped, Super Gaussian and Super Lorentzian beams on source plane (a) N = 2, (b) N = 8, (c) N = 50