Eur.J.Phys.38(2017)025209 Q Ye et al p"= lim 1 Sid=z. 2lEof r cos g, S,dx ∑Eo2hcos xm→0 The power reflection and transmission coefficients for a beam are therefore governed also by ∑"R ∑"T R;= and T:- Zi cos et are, respectively, the power reflection and transmission coefficients for a single plane wave introduced in standard textbooks [1-11],and ri, t are Fresnel coefficients given in on(8), the weight wi is given Oil cos It is therefore concluded that the power reflection and transmission coefficients of a beam with finite transverse extent are the weighted sum of the corresponding coefficients of the constituent plane waves that make up the beam We have calculated Rb and Tb given by(15) as well as R and T given by (18) for 2D Gaussian beams with different beam widths. Typical results are shown in figure 4(a) for the case with Wo= 2A. The results for Rb and R (also for Tb and T)as a function of incident angle Binc are graphically indiscernible, as can be seen from the relative discrepancy ER and Er Rb-RI T The discrepancy between Rb and R (and also between Tb and T)originates from the numerical integration as well as the approximation of the Gaussian beam by a discrete set of plane waves, in which the residual fields away from the beam axis remain oscillating around zero rather than decay exponentially as a standard gaussian beam should do. The greater discrepancy at smaller incident angle Binc arises from the larger residual fields near x=0 since the incident and reflected waves come closer in this case, resulting in a somewhat discernible overlap effect. The more serious inconsistency at larger incident angle Binc comes similarly from the overlap effect between the incident and reflected waves due to larger oblique angle. Besides, it stems also from the fact that, in our calculation, we have removed some plane wave components that propagate upward from the plane wave spectrum of the incident beam. For a beam with greater beam width, the discrepancy can be decreased, since a wider beam has all its constituent plane waves more bent to the direction of beam propagation nd thus our simulation by omitting upward wave vectors generates more accurate results Figure 4(b)evidences this tendency, where the discrepancy decreases with the beam width is exhibited. And finally, in figure 4(c)we show a typical example for the power reflection and transmission coefficients of the 2d Gaussian beam as a function of beam width. It is manifestò ò å å q q = = =- = ¥ - ¥ - ∣ ∣ ∣ ∣ () p x S x Z E r p x S x Z E t lim 1 d 1 2 cos , lim 1 2 d 1 2 cos . 17 x x j j j j x x x j j j j re m 0 1 1 0 2 2 tr m 2 2 0 2 2 t m m m m m The power reflection and transmission coefficients for a beam are therefore governed also by å å = = = = ( ) R p p w R T p p w T , , 18 j j j j j j re in tr in where q q Rr T = = ( ) Z Z and t cos cos j j j 19 j j j 2 1 t 2 2 are, respectively, the power reflection and transmission coefficients for a single plane wave introduced in standard textbooks [1–11], and r , t j j are Fresnel coefficients given in equation (8), the weight wj is given by å q q = ∣ ∣ ∣ ∣ w ( ) E E cos cos j . 20 j j j j j 0 2 0 2 It is therefore concluded that the power reflection and transmission coefficients of a beam with finite transverse extent are the weighted sum of the corresponding coefficients of the constituent plane waves that make up the beam. We have calculated Rb and Tb given by (15) as well as R and T given by (18) for 2D Gaussian beams with different beam widths. Typical results are shown in figure 4(a) for the case with W0 = 2l. The results for Rb and R (also for Tb and T) as a function of incident angle qinc are graphically indiscernible, as can be seen from the relative discrepancy eR and eT defined by e = e - = ∣ ∣ ∣∣ - ( ) R R R T T T R , . 21 b T b The discrepancy between Rb and R (and also between Tb and T) originates from the numerical integration as well as the approximation of the Gaussian beam by a discrete set of plane waves, in which the residual fields away from the beam axis remain oscillating around zero rather than decay exponentially as a standard Gaussian beam should do. The greater discrepancy at smaller incident angle qinc arises from the larger residual fields near x = 0 since the incident and reflected waves come closer in this case, resulting in a somewhat discernible overlap effect. The more serious inconsistency at larger incident angle qinc comes similarly from the overlap effect between the incident and reflected waves due to larger oblique angle. Besides, it stems also from the fact that, in our calculation, we have removed some plane wave components that propagate upward from the plane wave spectrum of the incident beam. For a beam with greater beam width, the discrepancy can be decreased, since a wider beam has all its constituent plane waves more bent to the direction of beam propagation and thus our simulation by omitting upward wave vectors generates more accurate results. Figure 4(b) evidences this tendency, where the discrepancy decreases with the beam width is exhibited. And finally, in figure 4(c) we show a typical example for the power reflection and transmission coefficients of the 2D Gaussian beam as a function of beam width. It is manifest Eur. J. Phys. 38 (2017) 025209 Q Ye et al 10