Eur.J.Phys.38(2017)025209 Q Ye et al with Io> l, which is well justified for loosely focused beam with Wo> 2A. Equation(1) ctually expresses a beam as a superposition of a series of homogeneous plane waves, which is known as the angular spectrum representation of optical field in optics [15, 16 ], except that we have excluded the evanescent wave components for simplicity. Based on the transformation between two coordinates (r, y)and (, y), rcos e the incident E field (1) can be written as 4 x exp[-iko(ar+ By)]-- cos(6+ binc) d cos Binc +B sin g B=B cos Binc -a sin Binc= cos(8+ bins) ∫a=sn, ∫a=sin6, The upper and lower bounds of integration are re-set to amax= l and amin =-cos(2nc)to guarantee that all the plane wave components constituting the incident propagate directions of their wave vectors lying between the regime bounded by the two magenta shown in Next we further approximate the integral (3) by a summation over discrete wave vectors This is done by simply casting the integral in equation(3)into summation Ein(,v=ko W A k2 w2d -)=m9=√1-可 6i=8i-Binc. So we have written a beam as a superposition of a finite number M y)=∑E,E each of which has the amplitude Eo given by 2T expl k2wo2cos 8 (7b)with ∣a¢ >∣ 1, which is well justified for loosely focused beam with W0  2l. Equation (1) actually expresses a beam as a superposition of a series of homogeneous plane waves, which is known as the angular spectrum representation of optical field in optics [15, 16], except that we have excluded the evanescent wave components for simplicity. Based on the transformation between two coordinates (x y ¢, ¢) and (x, y), q q q q ¢ = - ¢ =- - ⎧ ⎨ ⎩ ( ) xy x yx y sin cos , sin cos , 2 inc inc inc inc the incident E field (1) can be written as ò p a a b q q q a = - ¢ ´- + ¢ ¢ + a a ⎛ ⎝ ⎜ ⎞ ⎠ ( ) ⎟ [ ( )] ( ) ( ) E xy k W k W kx y , 2 exp 4 exp i cos cos d, 3 in 0 0 0 2 0 2 2 0 inc min max where aa q b q q q bb q a q q q = ¢ + ¢ = ¢ + = ¢ - ¢ = ¢ + ( ) ( ) () cos sin sin , cos sin cos , 4 inc inc inc inc inc inc with a q b q a q b q ¢ = ¢ ¢ = ¢ = = ⎧ ⎨ ⎩ ⎧ ⎨ ⎩ ( ) sin , cos , and sin , cos . 5 The upper and lower bounds of integration are re-set to amax = 1 and a q min inc = -cos 2( ) to guarantee that all the plane wave components constituting the incident beam propagate downward and distribute symmetrically with respect to the y¢-axis, namely, with the directions of their wave vectors lying between the regime bounded by the two magenta dashed arrows shown in figure 1. Next we further approximate the integral (3) by a summation over discrete wave vectors. This is done by simply casting the integral in equation (3) into summation å p a a b q q q d = - ¢ - + ´ ¢ ¢ + = ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ( ) [ ( )] ( ) ( ) E xy k W k W , kx y 2 exp 4 exp i cos cos , 6 j M j j j j j in 0 0 1 0 2 0 2 2 0 inc where d = ( ) a a - M max min , a a d qb a a q = +- = = - ¢ = ¢ j jj min ( ) j sin , 1 , sin jj j 1 2 2 , and q¢ j = - q q j inc. So we have written a beam as a superposition of a finite number M of plane waves = = å a b = - + ( ) () ( ) E xy E E E a , , e, 7 j M j j j in kxy 1 in in 0 i j 0 j each of which has the amplitude E0j given by p a q q q = - d ¢ ¢ ¢ + ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ( ) E ( ) k W k W b 2 exp 4 cos cos j . 7 j j j 0 0 0 0 2 0 2 2 inc Eur. J. Phys. 38 (2017) 025209 Q Ye et al 4