loP Publishing Eur.J.Phys.38(2017)025209(12pp) ttps://do.org/10.1088/1361-6404/aa596 Understanding the power reflection and transmission coefficients of a plane wave at a planar interface Qian Ye, yikun Jiang and Haoze Lin Department of Physics, Fudan University, Shanghai 200433, People's Republic of High School Affiliated to Fudan University, Shanghai 200433, Peoples Republic of China E-mail:gyel4@fudan.edu.cn Received 2 November 2016. revised 23 December 2016 Accepted for publication 13 January 2017 Published 3 February 2017 Abstract In most textbooks, after discussing the partial transmission and reflection of a plane wave at a planar interface, the power(energy) reflection and transmis- sion coefficients are introduced by calculating the normal-to-interface com- ponents of the Poynting vectors for the incident, reflected and transmitted waves, separately. Ambiguity arises among students since, for the Poynting vector to be interpreted as the energy flux density, on the incident(reflected side, the electric and magnetic fields involved must be the total fields, namely, the sum of incident and reflected fields, instead of the partial fields which are just the incident (reflected)fields. The interpretation of the cross product of partial fields as energy flux has not been obviously justified in most textbooks Besides, the plane wave is actually an idealisation that is only ever found in textbooks then what do the reflection and transmission coefficients evaluated for a plane wave really mean for a real beam of limited extent? To provide a clearer physical picture, we exemplify a light beam of finite transverse extent by a fundamental Gaussian beam and simulate its reflection and transmission at a planar interface. Due to its finite transverse extent, we can then insert the incident fields or reflected fields as total fields into the expression of the Poynting vector to evaluate the energy flux and then power reflection and transmission coefficients. We demonstrate that the power reflection and sum of the corresponding coefficients for all constituent plane wave coas transmission coefficients of a beam of finite extent turn out to be the weight nents that form the beam. The power reflection and transmission coefficients of a single plane wave serve, in turn, as the asymptotes for the corresponding coefficients of a light beam as its width expands infinitely. 0143-0807/17/025209+12533.00 G 2017 European Physical Society Printed in the UK
Understanding the power reflection and transmission coefficients of a plane wave at a planar interface Qian Ye1 , Yikun Jiang1 and Haoze Lin2 1 Department of Physics, Fudan University, Shanghai 200433, Peopleʼs Republic of China 2 High School Affiliated to Fudan University, Shanghai 200433, Peopleʼs Republic of China E-mail: qye14@fudan.edu.cn Received 2 November 2016, revised 23 December 2016 Accepted for publication 13 January 2017 Published 3 February 2017 Abstract In most textbooks, after discussing the partial transmission and reflection of a plane wave at a planar interface, the power (energy) reflection and transmission coefficients are introduced by calculating the normal-to-interface components of the Poynting vectors for the incident, reflected and transmitted waves, separately. Ambiguity arises among students since, for the Poynting vector to be interpreted as the energy flux density, on the incident (reflected) side, the electric and magnetic fields involved must be the total fields, namely, the sum of incident and reflected fields, instead of the partial fields which are just the incident (reflected) fields. The interpretation of the cross product of partial fields as energy flux has not been obviously justified in most textbooks. Besides, the plane wave is actually an idealisation that is only ever found in textbooks, then what do the reflection and transmission coefficients evaluated for a plane wave really mean for a real beam of limited extent? To provide a clearer physical picture, we exemplify a light beam of finite transverse extent by a fundamental Gaussian beam and simulate its reflection and transmission at a planar interface. Due to its finite transverse extent, we can then insert the incident fields or reflected fields as total fields into the expression of the Poynting vector to evaluate the energy flux and then power reflection and transmission coefficients. We demonstrate that the power reflection and transmission coefficients of a beam of finite extent turn out to be the weighted sum of the corresponding coefficients for all constituent plane wave components that form the beam. The power reflection and transmission coefficients of a single plane wave serve, in turn, as the asymptotes for the corresponding coefficients of a light beam as its width expands infinitely. European Journal of Physics Eur. J. Phys. 38 (2017) 025209 (12pp) https://doi.org/10.1088/1361-6404/aa5960 0143-0807/17/025209+12$33.00 © 2017 European Physical Society Printed in the UK 1
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 point
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 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-tointerface 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 fundamental 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
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 components
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 [1–11], implying that the latter describes actually the energy reflection and transport 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
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
Eur.J.Phys.38(2017025209 Q Ye et al 20 20 y=10A Figure 2.(a)The of the standard gaussian beam depicted b (1).(b)The of a superposition of 100 lane waves used the beam intensity as a function of transverse coordinates x at different longitudinal pos The solid and dotted lines denote, respectively, the field intensity of the standard Gaussian beam given by quation (1)and that formed by a superposition of 100 plane waves given by(7a) and(7b) Figure 2 shows the results for mimicking a 2D Gaussian beam of waist width 2Wo= 4X propagating in free-space by M=100 plane waves given by equations(7a) and(7b),con- firming a quite satisfactory agreement, and, in particular, showing a field pattern confine within a finite transverse extent for our purpose of studying reflection and refraction at nterface Next we assume that the 2D Gaussian beam propagates along the direction at angle with respect to the y axis within the x-y plane, as shown in figure 1. It experiences partial reflection and transmission when it is incident on a planar interface at y=0 between, for 如rhm面amt Fresnel coefficients [1-11] sin O, cos e; -cos 8, sin 8; sin(e;-8j) sin e cos 0;+ cos ej sin e; sin(e;+ 0) 2 sin gt cos e 2 sin gt cos e sin e; cos B;+ cos ej sin e, sin(@;+ e)) for each single plane wave depicted by the electric field given in(7a) and (7b), where the incident angle 0,=sin-a and the refracted angle 0=sin-(o; /nm), the reflected and transmitted waves can be easily worked out to yield
Figure 2 shows the results for mimicking a 2D Gaussian beam of waist width 2W0 = 4l propagating in free-space by M = 100 plane waves given by equations (7a) and (7b), con- firming a quite satisfactory agreement, and, in particular, showing a field pattern confined within a finite transverse extent for our purpose of studying reflection and refraction at an interface. Next we assume that the 2D Gaussian beam propagates along the direction at angle qinc with respect to the y axis within the x–y plane, as shown in figure 1. It experiences partial reflection and transmission when it is incident on a planar interface at y = 0 between, for simplicity, a free space with both relative permittivity e1 and permeability m1 equal to 1, and a dielectric with e2 = 4 and m = 1 2 , so that the refractive index n = e2 . With the help of the Fresnel coefficients [1–11], q q qq q q qq q q q q q q q q qq q q q q = - + = - + = + = + ( ) ( ) ( ) ( ) r t sin cos cos sin sin cos cos sin sin sin , 2 sin cos sin cos cos sin 2 sin cos sin , 8 j j j j j j j j j j j j j j j j j j j j j j j j t t t t t t t t t t t for each single plane wave depicted by the electric field given in (7a) and (7b), where the incident angle q = a - j j sin 1 and the refracted angle q = a - sin ( n) j j t 1 , the reflected and transmitted waves can be easily worked out to yield Figure 2. (a) The contour plot of the electric field intensity of the standard Gaussian beam depicted by equation (1). (b) The field intensity profile of a superposition of 100 plane waves used to mimic the beam. (c) The electric field intensity as a function of transverse coordinates x at different longitudinal position y. The solid and dotted lines denote, respectively, the field intensity of the standard Gaussian beam given by equation (1) and that formed by a superposition of 100 plane waves given by (7a) and (7b). Eur. J. Phys. 38 (2017) 025209 Q Ye et al 5
Eur.J.Phys.38(2017025209 Q Ye et al ∑E,E=Ee-, E=∑明,E=1E0e-ko+ where Bi=n2-o. The r and y components of the corresponding incident, reflected and transmitted magnetic fields follow the Maxwell equation (10) Hir, Hjr H ,B=TE0e-(- H 后偿=、 e-iko(a-r+py) (11a) H E H=∑H,HB= 班=∑H,班 where Z Z2=v=50 are, respectively, the wave impedances in media I and 2, while Eo and Ho are the permittivity and permeability in free-space. The energy current densities in direction normal to the interface are then determined by ormal components of the Poynting vectors computed from the total fields St=-Re[(ein +ere)(Hin*+ Hre ) while the components parallel to the interface( components)are Re[(ein ere)(Hn*+ Hre f)] (12b) where the superscript represents the complex conjugate and the subscripts I and 2 denote, respectively, the incident and transmitted sides of the interface. For comparison and following he standard textbooks [1-10], we define two quantities evaluated in terms of partial fields by
å å = = = = a b a b = - - = - + ( ) ( ) ( ) E E E rE E E E tE , e, , e, 9 j M j j j j kxy j M j j j j kxy re 1 re re 0 i tr 1 tr tr 0 i j j j j 0 0 t where b a j j = - n t 2 2 . The x and y components of the corresponding incident, reflected and transmitted magnetic fields follow the Maxwell equation wm = ´ H ( ) E i , 10 reading å å å b b b = =- = = = =- a b a b a b = - + = - - = - + ( ) ( ) ( ) ( ) HHH Z E HHH Z r E HH H Z tE a , e, , e, , e , 11 x j M jx jx j j kxy x j M jx jx j j j kxy x j M jx jx j j j kxy in 1 in in 1 0 i re 1 re re 1 0 i tr 1 tr tr t 2 0 i j j j j j j 0 0 0 t and å å å a a a = =- = =- = =- a b a b a b = - + = - - = - + ( ) ( ) ( ) ( ) HHH Z E HHH Z r E HHH Z tE b , e, , e, , e , 11 y j M jy jy j j kxy y j M jy jy j j j kxy y j M jy jy j j j kxy in 1 in in 1 0 i re 1 re re 1 0 i tr 1 tr tr 2 0 i j j j j j j 0 0 0 t where = m m e e Z1 1 0 1 0 and = m m e e Z2 2 0 2 0 are, respectively, the wave impedances in media 1 and 2, while e0 and m0 are the permittivity and permeability in free-space. The energy current densities in direction normal to the interface are then determined by normal components of the Poynting vectors computed from the total fields * * * =++ = [( )( )] ( ) () S E EH H S EH a 1 2 Re , 1 2 Re , 12 x x x 1 in re in re 2 tr tr while the components parallel to the interface (x components) are * * * ¢ =- + + ¢ = - [( )( )] ( ) () S E EH H S EH b 1 2 Re , 1 2 Re , 12 y y y 1 in re in re 2 tr tr where the superscript ∗ represents the complex conjugate and the subscripts 1 and 2 denote, respectively, the incident and transmitted sides of the interface. For comparison and following the standard textbooks [1–10], we define two quantities evaluated in terms of partial fields by Eur. J. Phys. 38 (2017) 025209 Q Ye et al 6
Eur.J.Phys.38(2017025209 Q Ye et al 10 Reflected 10 Figure 3. (a) The contour plot of a simulated Gaussian beam incident on plane interface at incident angle Binc 40. The planar interface locates at y=0. The cyan line at 10 A helps to display spatial separation of incident and reflected waves.(b)and(c) The green line, red line, and blue line denote, respectively, the absolute values of the plitude of E field (b) and the y component of Poynting vector(c) of the total incident and reflected waves at y 10 A denoted by the horizontal cyan line in panel (a). The complete separation in space of incident and reflected waves shows SI= Sin (SI= S)for x>0(x<0), and confirms the validity to calculate the incident and reflected power in terms of the total fields Sm=Re(E吗H1) Re(oTHres) (13) It is remarked that the implication of Sand S as energy flux density is not justified by its definition imere igure 3 shows typically the results for a beam of waist width 2Wo=4A incident on the y=0 at angle Binc= 40. The incident field E In. reflected field ere and transmitted field E are exhibited in figure 3(a). Different from a plane wave incidence, the reflected
* * = = ( ) ( ) () S EH S EH 1 2 Re , 1 2 Re . 13 x x in in in re re re It is remarked that the implication of Sin and Sre as energy flux density is not justified by its definition. Figure 3 shows typically the results for a beam of waist width 2W0 = 4l incident on the interface y = 0 at angle qinc = 40. The incident field Ein, reflected field Ere and transmitted field Etr are exhibited in figure 3(a). Different from a plane wave incidence, the reflected Figure 3. (a) The contour plot of a simulated Gaussian beam incident on plane interface at incident angle qinc = 40. The planar interface locates at y = 0. The cyan line at y = 10 l helps to display spatial separation of incident and reflected waves. (b) and (c) The green line, red line, and blue line denote, respectively, the absolute values of the amplitude of E field (b) and the y component of Poynting vector (c) of the total, incident and reflected waves at y = 10 l denoted by the horizontal cyan line in panel (a). The complete separation in space of incident and reflected waves shows S1 = Sin (S1 = Sre) for x > 0 (x < 0), and confirms the validity to calculate the incident and reflected power in terms of the total fields. Eur. J. Phys. 38 (2017) 025209 Q Ye et al 7
Eur.J.Phys.38(2017)025209 beam and the incident Gaussian beam, although both appear in the incident side of the interface, are spatially well separated from each other at a distance away from the interface, as denoted by the horizontal cyan line. Figure 3(b)displays the absolute values of the electric field amplitude along x parallel to the interface at y= 10A, marked by the cyan line in figure 3(a). The green, red, and blue curves denote the total, incident and reflected fields, respectively. The spatial separation of the incident and reflected fields is manifest. On one side(x>0)of the interface normal, the total field reduces completely to incident field, while on the other side of the normal (x0 and x 0 while S1= sre for x 0 It can be easily understood that Pin characterises the energy flux incident perpendicularly downward to the interface, since its integrand S is evaluated in terms of the total fields and carries the implication of energy current density, and, in addition, as the total fields turn out to be identical to the incident field, the energy flux comes solely from the incident wave. Similarly, pre describes the energy flux reflected upward away from the interface, and Pr is the energy flux transmitted to the second medium. All the integrands in equations (14a(14c)are evaluated in terms of local total fields, so it is fully justified that so-evaluated y components of the Poynting vectors determine the energy current densities and, consequently, their integrations yield the powers being transported. It follows naturally that the power reflection and transmission coefficients for a light beam, denoted by Rb and Tb, respectively, can be On the other hand, a light beam can be decomposed into a set of plane waves, as we have done for a fundamental Gaussian beam in equations(7a)and(7b). As each plane wave is of patially infinite extent, in this case, one chooses to focus on the powers incident, reflected and transmitted normally on or through the unit length at the interface, which can be eval-
beam and the incident Gaussian beam, although both appear in the incident side of the interface, are spatially well separated from each other at a distance away from the interface, as denoted by the horizontal cyan line. Figure 3(b) displays the absolute values of the electric field amplitude along x parallel to the interface at y = 10l, marked by the cyan line in figure 3(a). The green, red, and blue curves denote the total, incident and reflected fields, respectively. The spatial separation of the incident and reflected fields is manifest. On one side (x > 0) of the interface normal, the total field reduces completely to incident field, while on the other side of the normal (x 0 and x 0 while S1 = Sre for x < 0. It can be easily understood that Pin characterises the energy flux incident perpendicularly downward to the interface, since its integrand S1 is evaluated in terms of the total fields and carries the implication of energy current density, and, in addition, as the total fields turn out to be identical to the incident field, the energy flux comes solely from the incident wave. Similarly, Pre describes the energy flux reflected upward away from the interface, and Ptr is the energy flux transmitted to the second medium. All the integrands in equations (14a)–(14c) are evaluated in terms of local total fields, so it is fully justified that so-evaluated y components of the Poynting vectors determine the energy current densities and, consequently, their integrations yield the powers being transported. It follows naturally that the power reflection and transmission coefficients for a light beam, denoted by Rb and Tb, respectively, can be defined as R = = ( ) P P T P P b , . 15 re in b tr in On the other hand, a light beam can be decomposed into a set of plane waves, as we have done for a fundamental Gaussian beam in equations (7a) and (7b). As each plane wave is of spatially infinite extent, in this case, one chooses to focus on the powers incident, reflected, and transmitted normally on or through the unit length at the interface, which can be evaluated at y far away from the interface by Eur. J. Phys. 38 (2017) 025209 Q Ye et al 8
Eur.J.Phys.38(2017025209 Q Ye et al Incident angle(degree) (b) 0.680 0.676 -- W./2 Figure 4.(a) The power reflection coefficient R(solid red line) and transmission coefficient T(solid blue line) of a 2D Gaussian beam with Wo= 2A. Also shown as dashed lines are the corresponding relative discrepancy between Rb and R (red) and Tb and T(blue) defined by equation (21).(b) The relative discrepancy ER and er for incident angle Binc 70 as a function of half waist width Wo of 2D Gaussian beam, showing that the discrepancy decays with beam width until a plateau appears due to the residual numerical errors.(c) The power reflection coefficient R(red) and transmission coefficient T(blue) of a 2D Gaussian beam versus the beam width at ]inc 60 uggesting that R and T of a beam approach asymptotically to those of a single plane wave as the beam width tends to infinity. The horizontal dashed green lines display R and T of a single plane wave at the same incident angle Binc evaluated based on the standard procedure in most textbooks, see, e.g., [1-51 Re(engin")dx ∫xR∑EEc 27, 2IEoPB=27, 2IEolP-cos B, (16) ere use has been made of the fact that the mixed terms with j=i' vanish in the limit of m- 00, since the integrals for such terms are finite while the denominator xm tends to
* * ò ò ò ò å å å b b q =- =- = - = = = aba b ¥ ¥ ¥ ¥ ¢ ¢ ¢ - +- -¢ ¢ ( ) [ ] ∣ ∣ ∣ ∣ () ( ) p x S x x S x x EH x x Z EE x Z E Z E lim 1 d lim 1 d lim 1 1 2 Re d lim 1 1 2 Re e d 1 2 1 2 cos , 16 x x x x x x x x x j j j j j kxy x y j j j j j j in m 0 1 m 0 in m 0 in in m 0 1 , 0 0 i 1 0 2 1 0 2 j j j j m m m m m m m m 0 where use has been made of the fact that the mixed terms with j j ¹ ¢ vanish in the limit of xm ¥, since the integrals for such terms are finite while the denominator xm tends to infinity. In a similarly way, one derives Figure 4. (a) The power reflection coefficient R (solid red line) and transmission coefficient T (solid blue line) of a 2D Gaussian beam with W0 = 2l. Also shown as dashed lines are the corresponding relative discrepancy between Rb and R (red) and Tb and T (blue) defined by equation (21). (b) The relative discrepancy eR and eT for incident angle qinc = 70 as a function of half waist width W0 of 2D Gaussian beam, showing that the discrepancy decays with beam width until a plateau appears due to the residual numerical errors. (c) The power reflection coefficient R (red) and transmission coefficient T (blue) of a 2D Gaussian beam versus the beam width at qinc = 60, suggesting that R and T of a beam approach asymptotically to those of a single plane wave as the beam width tends to infinity. The horizontal dashed green lines display R and T of a single plane wave at the same incident angle qinc evaluated based on the standard procedure in most textbooks, see, e.g., [1–5]. Eur. J. Phys. 38 (2017) 025209 Q Ye et al 9
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 manifest
ò 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
