Eur.J.Phys.38(2017)025209 Q Ye et al Keywords: power reflection and transmission coefficients, a plane wave, a light beam, weighted sum, asymptote (Some figures may appear in colour only in the online journal) 1 Introduction A wave experiences partial transmission and partial reflection when the medium through which it travels suddenly changes. The power reflection coefficient is defined physically as he normal-to-interface component of energy flux of the reflected wave to that of the incident ave, while the power transmission coefficient describes the normal component of energy flux of the transmitted wave to that of the incident wave. In most textbooks [1-11, for simplicity, such concepts are introduced for the case where a plane wave strikes on a planar interface between two media. The energy flux is obtained by computing the normal-to- nterface component of the(period-averaged) Poynting vector. To be more specific, the flows of power incident on and reflected from the interface are evaluated by the Poynting vectors S=EX H of the incident and reflected plane waves, respectively, and the ratio of whose normal components gives the power(energy) reflection coefficient. However, this simplified choice of illustration is somewhat ambiguous among students for the following reasons. Since the plane wave is of infinite spatial extent, on the incident side of the interface, the total fields based on which the Poynting vector should be computed in order to carry the meaning of energy current density, are actually a superposition of the incident and reflected fields, namely, E=E+ Er and H =Hi+ Hr. It is indeed not obviously justified to take partial field, either incident electric and magnetic fields Ei and Hi or reflected fields Er and H, to compute the Poynting vector and assign the implication of energy flux to each individual part because the Poynting vector has a quadratic form in field quantities [12, 13]. In addition, plane wave is actually an ideal model that does not exist in the real world since it possesses infinite extent and energy. Then what do the reflection and transmission coefficients evaluated for a plane wave imply in real situation where real beams are limited in extent. Although maybe intuitively known, there is never an explicit numerical demonstration to answer this In this paper, we develop a clear physical understanding by studying the reflection and transmission at a planar interface of a more realistic but still easily tractable model, a fun- damental two-dimensional (2D)Gaussian beam. The finite extent of the light beam in space enables the spatial separation of the incident and reflected waves in the regime far enough away from the planar interface. One can therefore compute the Poynting vectors in terms of the total fields, E and H, on two sides of the interface normal, which reduce indeed to the fields of the incident and reflected waves, respectively. By doing so, implication of the Poynting vector as energy current density is truly justified. We then integrate the normal components of the Poynting vectors on both sides of the normal to obtain the total energy fluxes transporting towards and reflected from the interface, the ratio of which defines the power reflection coefficient of a light beam. On the transmitted side, since the refracted wave itself represents the total field, the integration of the normal component of the Poynting vector [12. 131. it is demonstrated that the normal component of the Poynting vector evalu is continuous across a planar interface between two isotropic lossless media, which, together with a proof that the nomral component of the mixed term Ei XH,+ Er X Hi vanishes, implies that the Poynting vector of the reflected fields from a planar interface can be understood as the energy flux of the reflected wave. We are obliged to one of the anonymous referees for pointing out this pointKeywords: power reflection and transmission coefficients, a plane wave, a light beam, weighted sum, asymptote (Some figures may appear in colour only in the online journal) 1. Introduction A wave experiences partial transmission and partial reflection when the medium through which it travels suddenly changes. The power reflection coefficient is defined physically as the normal-to-interface component of energy flux of the reflected wave to that of the incident wave, while the power transmission coefficient describes the normal component of energy flux of the transmitted wave to that of the incident wave. In most textbooks [1–11], for simplicity, such concepts are introduced for the case where a plane wave strikes on a planar interface between two media. The energy flux is obtained by computing the normal-to￾interface component of the (period-averaged) Poynting vector. To be more specific, the flows of power incident on and reflected from the interface are evaluated by the Poynting vectors SEH = ´ of the incident and reflected plane waves, respectively, and the ratio of whose normal components gives the power (energy) reflection coefficient. However, this simplified choice of illustration is somewhat ambiguous among students for the following reasons. Since the plane wave is of infinite spatial extent, on the incident side of the interface, the total fields, based on which the Poynting vector should be computed in order to carry the meaning of energy current density, are actually a superposition of the incident and reflected fields, namely, E = + E E i r and HH H = +i r. It is indeed not obviously justified to take partial field, either incident electric and magnetic fields Ei and Hi or reflected fields Er and Hr, to compute the Poynting vector and assign the implication of energy flux to each individual part, because the Poynting vector has a quadratic form in field quantities [12, 13] 3 . In addition, a plane wave is actually an ideal model that does not exist in the real world since it possesses infinite extent and energy. Then what do the reflection and transmission coefficients evaluated for a plane wave imply in real situation where real beams are limited in extent. Although maybe intuitively known, there is never an explicit numerical demonstration to answer this question. In this paper, we develop a clear physical understanding by studying the reflection and transmission at a planar interface of a more realistic but still easily tractable model, a fun￾damental two-dimensional (2D) Gaussian beam. The finite extent of the light beam in space enables the spatial separation of the incident and reflected waves in the regime far enough away from the planar interface. One can therefore compute the Poynting vectors in terms of the total fields, E and H, on two sides of the interface normal, which reduce indeed to the fields of the incident and reflected waves, respectively. By doing so, implication of the Poynting vector as energy current density is truly justified. We then integrate the normal components of the Poynting vectors on both sides of the normal to obtain the total energy fluxes transporting towards and reflected from the interface, the ratio of which defines the power reflection coefficient of a light beam. On the transmitted side, since the refracted wave itself represents the total field, the integration of the normal component of the Poynting vector 3 In [12, 13], it is demonstrated that the normal component of the Poynting vector evaluated based on the total fields is continuous across a planar interface between two isotropic lossless media, which, together with a proof that the nomral component of the mixed term EHEH i rri ´+´ vanishes, implies that the Poynting vector of the reflected fields from a planar interface can be understood as the energy flux of the reflected wave. We are obliged to one of the anonymous referees for pointing out this point. Eur. J. Phys. 38 (2017) 025209 Q Ye et al 2