Eur.J.Phys.38(2017)025209 Q Ye et al Figure 1 Schematic plot of a two-dimensional Gaussian beam with focal width 2Wo incident on a planar interface depicted by y =0 at an incident angle binc. The beam is ocused on the origin and propagates in direction y. Two magenta dashed arrows show the range of wave vectors used to describe the beam such that the plane wave components forming the beam with upwards wave vectors are all neglected. See tex for more details produces naturally energy flux transmitted through the interface, the ratio of which to the total incident energy flux characterises the power transmission coefficient of the beam. We demonstrate that the power reflection and transmission coefficients of a beam of finite transverse spatial extent turn out to be the weighted sum of the corresponding coefficients for the each constituent plane wave component that makes up the beam On the other hand, by increasing the waist width of the beam, it is found that the beam power reflection and transmission coefficients approach asymptotically to" the power reflection and transmission coefficients'of a single plane wave that are evaluated based on the procedure in the standard textbooks [l-ll], implying that the latter describes actually the energy reflection and trans- port ratios of a light beam in the limit of infinite beam width. As the beam widths Wo in usual experiments are typically greater than dozens of microns, while the operating wavelength is only of order of Wo/100, the reflection and transmission coefficients of a single plane wave serve as a good approximation to those for the real light beam 2. The power reflection and transmission coefficients of a light beam For greatest simplicity, let us consider a 2D transverse electric Gaussian beam with its electri field E normal to the plane of incidence. Generalisation to three dimensions as well as to the case with e parallel to the plane of incidence is straightforward. Let the beam of waist width 2Wo be focused at the origin of the coordinate system and propagate in direction y,as schematically shown in figure 1. The planar interface is located at y=0 and the incident angle finc depicts the angle between the beam propagation direction (y-axis) and the interface normal (y-axis). The E field polarised along z of such a 2D beam reads [14] Emc(x,y=如oWf+1 k昭d expliko(d'x+ B'y)]da,(1) here the wave number ko= 2/A with A being the wavelength in free space, 2Wo is the waist width. and 1-a. Here we have excluded the evanescent wave componentsproduces naturally energy flux transmitted through the interface, the ratio of which to the total incident energy flux characterises the power transmission coefficient of the beam. We demonstrate that the power reflection and transmission coefficients of a beam of finite transverse spatial extent turn out to be the weighted sum of the corresponding coefficients for the each constituent plane wave component that makes up the beam. On the other hand, by increasing the waist width of the beam, it is found that the beam power reflection and transmission coefficients approach asymptotically to ‘the power reflection and transmission coefficients’ of a single plane wave that are evaluated based on the procedure in the standard textbooks [1–11], implying that the latter describes actually the energy reflection and trans￾port ratios of a light beam in the limit of infinite beam width. As the beam widths W0 in usual experiments are typically greater than dozens of microns, while the operating wavelength is only of order of W0 100, the reflection and transmission coefficients of a single plane wave serve as a good approximation to those for the real light beam. 2. The power reflection and transmission coefficients of a light beam For greatest simplicity, let us consider a 2D transverse electric Gaussian beam with its electric field E normal to the plane of incidence. Generalisation to three dimensions as well as to the case with E parallel to the plane of incidence is straightforward. Let the beam of waist width 2W0 be focused at the origin of the coordinate system and propagate in direction y¢, as schematically shown in figure 1. The planar interface is located at y = 0 and the incident angle qinc depicts the angle between the beam propagation direction (y¢-axis) and the interface normal (y-axis). The E field polarised along z of such a 2D beam reads [14] ò p a ¢ ¢ = - aba ¢ ¢ ¢ + ¢¢ ¢ - + ⎛ ⎝ ⎜ ⎞ ⎠ E ( ) [ ( )] ( ) x y ⎟ k W k W , kx y 2 exp 4 inc exp i d , 1 0 0 1 1 0 2 0 2 2 0 where the wave number k0 = 2p l with λ being the wavelength in free space, 2W0 is the waist width, and b a ¢ = -1 ¢ 2 . Here we have excluded the evanescent wave components Figure 1. Schematic plot of a two-dimensional Gaussian beam with focal width 2W0 incident on a planar interface depicted by y = 0 at an incident angle qinc. The beam is focused on the origin and propagates in direction y¢. Two magenta dashed arrows show the range of wave vectors used to describe the beam such that the plane wave components forming the beam with upwards wave vectors are all neglected. See text for more details. Eur. J. Phys. 38 (2017) 025209 Q Ye et al 3