电磁场与电磁波chap22垂直极化波斜入射解读

Field and Wave Electromagnetics

2008.3

ˆjzˆEaxExfe;x方向极化，z方向传播UPWlossless EEmejrˆxExmaˆyEymaˆzEzme＝ajrˆn＝aˆxxaˆyyaˆzzvector wave number：a(描述传输媒质的参数)ˆzˆEaxExfe;x方向极化，z方向传播UPW22loss Amplitude;phase propagationdirectionpolarizationEEme22ˆnrˆaˆxExmaˆyEymaˆzEzmeaˆr;ˆn＝aˆxxaˆyyaˆzz;ˆˆaˆxxaˆyyaˆzzc;raH1cˆnEaˆn波的传播方向a8.10 Oblique incidence at a plane dielectric boundary

均匀平面波以任意角度入射到无限大平面分界面时出现的反射与折射情况

(1）分界面的法单位矢量:

ˆnaˆnaˆiax

ˆraz＝0

(2)入射波的传播方向: (3)反射波的传播方向: (4)反射波的传播方向:

ˆiaˆraˆtay

ˆta入射线、反射线、折射线与分界面法向之间的交角称为入射角，反射角，透射角入射面、反射面、透射面是同一平面yoz面

入射线、反射线、折射线与分界面法向构成的平面称为入射面、反射面、透射面

3个分量分别讨论，场叠加原理

EEmeˆnrjajrˆˆˆaxExmayEymazEzme垂直极化波，平行极化波:相对于入射面

ˆnaˆraE∥ˆnˆiax z＝0 y ˆtaaiEirary 3个分量分别讨论，场叠加原理 EEmeˆnrjaˆxExmaˆyEymaˆzEzmeajrz a入射面：yoz 垂直极化波:E沿x方向，H在yoz面内（x，z分量） 平行极化波: E在yoz面内（y，z分量），H沿x方向 相对于入射面：电场矢量的方向垂直或平行于入射面 入射面、反射面、透射面是同一平面 8.10 Oblique incidence at a plane dielectric boundary

When an electromagnetic wave strikes a plane boundary with any arbitrary angle,we refer to it is as oblique incidence.In fact ,normal incidence is a special case of oblique incidence.We discuss oblique incidence because it leads to three well-known laws in optics:Snell’s law of reflection,Snell’s law of

refraction,and Brewster’s law directing polarization by reflection. Once again,we consider a wave that is linearly polarized and a boundary that constitutes a plane at z=0.

The plane that includes the unit normal to the boundary and the propagation constant of the incident wave is called the plane of incidence,as shown in figure 8.16.

In general,the E field of the incident wave can make any angle with the plane of incidence;however,we will limit our discussion to two special cases:

First case ,we will assume that E field is normal to the plane of incidence and refer to it as a perpendicularly polarized wave as shown in figure8.17.where the E field in in the ax direction and yoz is the plane of incidence. In this case the E field is parallel to the interface xoz plane

second case ,the E field lies in the plane of incidence,and is called a parallel polarized wave.In this case,the H field lies in the plane parallel to the interface.The superposition theorem allows us to obtain all the necessary information for an incident wave that makes an arbitrary angle with the plane of incidence

8.10.1 Perpendicular polarization Medium 1 x Medium 2 1,1,1Incidence wave 2,2,2z Transmitted wave EiˆniSiairEtrefraction ˆntStaHtHiErHrˆnrSraz=0 电场：全部沿＋x方向； Reflection wave 8.10.1 Perpendicular polarization (1)The normal form of wave Incident wave Reflected wave Transmitted wave refracted (2)using boundary condition incident Medium 2 EiHianHriraki2ˆ2cˆ2ctEtHtakrErreflected Medium 1 Transmitted refracted aktˆ1cˆzE1E2aˆzH1H2 a11cyzz00 7.11707.118电场：全部沿＋x方向；z0(3)讨论significance ˆjzˆEaxExfe;x方向极化，z方向传播UPWlossless EEmejrˆxExmaˆyEymaˆzEzme＝ajrˆn＝aˆxxaˆyyaˆzzvector wave number：a(描述传输媒质的参数)ˆzˆEaxExfe;x方向极化，z方向传播UPW22loss Amplitude;phase propagationdirectionpolarizationEEme22ˆnraˆxExmaˆyEymaˆzEzmeaˆr;ˆn＝aˆxxaˆyyaˆzz;ˆˆaˆxxaˆyyaˆzzc;raH1cˆnEaˆn波的传播方向a(1)The normal form of wave Incident wave EEmeˆaˆnrˆxExmaˆyEymaˆzEzmeaˆr;ˆn＝aˆxxaˆyyaˆzz;22c;raˆxxaˆyyaˆzzˆˆaˆirˆiaˆˆˆx:EiaˆxE0eˆxE0epolarization direction:aa;22ˆˆi;ˆi11c;raˆxxaˆyyaˆzz;iˆ1aˆirˆiaˆysiniaˆzcosi;ˆiaˆyˆ1siniaˆzˆ1cosiaˆyˆ1yaˆzˆ1zaˆirˆ1siniyˆ1cosizˆ1siniycosizˆˆxE0eEia8.124a;11ˆ1siniycosizˆˆiEiˆysiniaˆzcosiaˆxE0eHiaaˆ1cˆ1c1ˆˆ1siniycosizˆzsiniaˆycosi8.124bHiE0eaˆ1c(1)The normal form of wave Reflected wave ˆn反射波电场分量 z0入射波电场分量 z0Er  z0Ei z0ˆEr0ˆEi0z0EEmeˆnraˆ ˆˆˆˆEE8.98 E nn0r0z0 i0z0反射波振幅与入射波在分界面上的比值，包括初始相位在内，是一个复数 ˆrrˆrrˆraˆraˆˆˆnE0eˆx:EraˆxEr0eˆxpolarization direction:aa;22ˆˆˆˆ;;raˆxaˆyaˆz;ar1rr11cxyzˆraˆysinraˆzcosr;aˆrrsinrycosrzaˆ1sinrycosrzˆˆnE0eˆxEra8.125a;11ˆ1siniycosizˆˆnE0eˆnErˆysinraˆzcosraˆxHraaˆ1cˆ1c1ˆ1siniycosizˆˆnE0eˆzsinr－aˆycosrHra8.125bˆ1cText rong ˆra(1)The normal form of wave Transmitted wave or refracted EEmenˆn透射波电场分量 z0入射波电场分量 z0Et z0Ei z0ˆEt z0ˆˆˆˆˆ8.100 EEEniz0n0tz0ˆEi z0透射波振幅与入射波在分界面上的比值，包括初始相位在内，是一个复数 ˆtrˆrˆtaˆaˆˆˆx:EtaˆxEt0eˆxˆnE0epolarization direction:aa;222ˆˆ;ˆˆ;raˆxaˆyaˆz;ˆa2222cxyzˆaˆysinaˆzcos;aˆrsinycoszaˆ2sinycoszˆˆxˆnE0eEa8.126a;2text11ˆ2sinycoszˆˆnEˆysin2aˆzcos2aˆxˆnE0eHaaˆ2cˆ2c1ˆ2sin2ycos2zˆˆzsin2aˆycos28126bHˆnE0eaˆ2c(2)using boundary condition ˆzEiErEaz0ˆzE1E2az00 7.117ˆ1siniycosizˆˆxE0e0;Eia8.124a;ˆ1siniycosizˆ2sinycoszˆˆˆˆxnE0eˆxˆnE0eEra8.125a;Ea8.126a;ˆ1siniycosizˆ1siniycosizˆˆE0eˆnE0eˆzaˆx0;aˆ2sinycoszˆˆnE0ez0ˆayˆ1siniyˆ1siniyˆ2sinyˆˆˆˆnE0eˆnE0eE0ez00;Y为任何值时成立，则，y＝0也成立 ˆˆˆ1ˆˆnˆˆn8.128EEE0n0n0ˆ1siniyˆ1siniyˆ2sinyˆ1siniˆ1siniˆ2sin8.129ˆ1siniˆ1siniˆ2sin8.129snell's law of reflectionˆ1siniˆ1sinisinisinrir18.130The angle of incidence is equal to the angle of reflection. Snell’s law of refraction jˆsin11c无耗介质ˆˆt11sini2sin8.131siniˆ2j22c01r01rn111sint＝8.131sini2202r02rn2ˆˆˆ1ˆˆˆˆEEE0n0n0nn8.128(2)using boundary condition ˆzH1H2 a1ˆˆ1siniycosizˆzsiniaˆycosi8.124bHiE0eaˆ1c1ˆ1siniycosizˆˆnE0eˆzsinr－aˆycosrHra8.125bˆ1c1ˆ2sin2ycos2zˆˆzsin2aˆycos28126bHˆnE0eaˆ2cz007.118ˆzHiHrHaˆxHiyHryHya0;z0ˆzHiyaˆyHizaˆzHryaˆyHrzaˆzHyaˆyHzaˆza11ˆˆ1siniycosizˆ1siniycosizˆˆEecos－cosEe0irn0ˆˆ1c1cˆxacos1ˆEˆeˆ2sin2ycos2z2n0ˆ2cz01ˆˆ1siniy11ˆˆsinysiny1i22ˆeˆeˆnEˆxE0eˆacosi－cosrcosE002n0ˆ1cˆ1cˆ2cˆ1siniˆ1siniˆ2sin8.129ir1ˆcos－cosˆˆcosˆˆEEE0irn0n02ˆ1cˆ2cˆncos11－ˆncos28.132ˆ1cˆ2cˆnˆn8.1281ˆ2cos1ˆ1cos2Er0ˆn8.133ˆ2cos1ˆ1cos2Ei0ˆ2cos1Et02ˆn8.134ˆ2cos1ˆ1cos2Ei0Dielectric-dielectric interface 120, and nonmagnetic media 120,j11csintˆ18.131siniˆ2j22c11rˆ1j11closslessj11nonmagnetic media8.135ˆ2j22cj2222r1rsin28.137sin12r1r2cos21sin21sin18.1382r21ˆ1closslessˆ12ˆ2ˆ2c11nonmagnetic media22r8.136211r22rcos1sin211r2cos1n;n1n8.1392r2r2cos1sin1cos1sin211r1r1rsin28.137sin12rθ2 <θ1, 折射角<入射角 光疏→光密 Case 1: If medium 2 is denser than medium 1εr2 >εr1 光密→光疏 Case 2: If medium 1 is denser than medium 2 εr2 <εr1 θ2>θ1, 折射角>入射角 cos90时对应的入射角i,称为Critical angle，记为crt90Critical angle(临界) iEr02cos11cos2ˆn18.1332,1是实数Ei02cos11cos2Total reflection 全发射 1r1rn1sint，sini2r2rn2ic是否为全反射???11sint对于非磁性介质：sintsinisini22若12,即当平面波从光密媒质（折射率大）入射到光疏媒质时此时折射波将贴着分界面传播。对于这种情况称为全反射在t90，sint1，sinisinc（折射率小）时，ti，故存在一个入射角c，使得透射角t90，21发生全反射时的入射角称为临界角，用c表示Case3: If medium1 is denser than medium 2εr2 >εr1,andθi>θc, 11sintsintsinisini222sini;sint1;t90;full reflection,ic if c  11sintsin2sini1 8.141a222(full reflection)???1r2costcos21sintjsint1jsin118.1412r2'2costpure imaginaryCase3: If medium 2 is denser than medium 1εr2 >εr1,andθi>θc, 全发射的定义：反射系数绝对值为1 1r2costjsin118.1412rEr02cos11cos2ˆnEi02cos11cos212cos1j1sin2112cos1j1sin112＝1 2,1是实数，分子分母互为复共轭ic是全反射ic时发生全反射，此时介质2中的透射场为：1r1r22costjsin11＝j8.141sin1102r2rˆ2sinycoszˆˆxE0eEa8.126a;2textj2sinyjzˆˆEaxE0e8.126a;j2siny2zˆˆxE0eEae8.126a;?8.141取正，取负j2siny2zˆˆxE0eEae8.126a;8.142取负Nonuniform plane wave Propagation at y direction Attenuation at z direction 沿z方向衰减，y方向传播,电磁场只存在分界面附近,称为表面波 222Why 介质2中存在电磁场,为什么还叫全反射 1r2sin118.142b2rSurface wave Why HSav1介质2中存在电磁场,为什么还叫全反射 全反射是指能量全部反射回介质一中,而不是指场全部返回到介质1中,介质2中可以存在电磁场 j2siny2zˆˆxE0eEae8.142120, j2siny2zˆˆzsin2aˆycos28126bnE0eea21*ReEH2cost1sinijsini12211ˆj2siny2z2zj2sinyˆˆxnE0eˆzsin2aˆycos2eReaenE0ae22SavSav12ˆ222zˆysin2aˆzcos2ReE0ea2212ˆ212ˆ22z2zˆyaE0sin2eSavyE0sin2e；Savz022222222sincsinc8.140sint11110222rfor nonmagnetic media:120:sinc0111rIf the left-hand side of (8.140) is equal to the right-hand side when θ1>θc, ,then (a) θc is called the critical angle,(b)ρ=1(c) θ2=900(d)the transmitted wave will propagate entirely parallel to the interface.simply stated ,there will be no power propagating along the z direction in medium 2,and the reflected power density will be equal to the incident power density .This is called total reflection .For this reason the critical angle is also referred to as the angle of total reflection j2siny2zˆˆxE0eEae8.126a;8.142取负1r1r22costjsin118.142222sin118.142b2r2rWhere we have only retained the negative sign for the second exponential term because the wave cannot grow as a function of z(没有能量的输入)，This equation shows that the wave is propagating along the y direction and is attenuating in the z direction with an attenuation constant of（8.142b） These are the traits(特征) of a non-uniform plane wave.As it is decaying in the z direction and is propagating in the y direction, it is also called the surface wave.Once again,there is no real power flow in the z direction.Thus ,based upon experimental observations in optics,we expect total reflection for all angles of incidence greater than the critical angle Savy12ˆ22zE0sin2e；Savz0220,12Dielectric-perfect conductor interface ˆ2cos1ˆ1cos2ˆ2cos1Er0Et02ˆnˆn8.1338.134ˆ2cos1ˆ1cos2ˆ2cos1ˆ1cos2Ei0Ei0介质2 是理想导体 ˆ2ˆ1jωjωjω0ˆ11ˆ11111jω1111下角标1都去掉 ˆn1; ˆn0ˆ1siniycosizˆ1siniycosizˆˆˆnE0eˆxE0eˆxEia8.124a;Era8.125a;E1EiErˆxE0eaj1siniycosiznE0ej1siniycosiz8.125a;ˆxE0eE1aj1siniyej1cosizej1cosizj1siniyˆ axE02jsin1cosizeE12jE0sinzcosejsyinˆx8.143aaˆxejtE1ReE1ejtRe2jE0sinzcosejsyinaˆx2jE0sinzcosReacostysinjsintsinyˆx2E0sincoszsintsiny8.146aE1a1ˆˆ1siniycosizˆzsiniaˆycosi8.124bHiE0eaˆ1c1ˆ1siniycosizˆˆˆHrazsinr－aycosrE0e8.125bˆ1cHi1E0e1jsinycoszˆazˆycos8.124bsinazHrE0ejsinycoszˆaˆycos8.125bsin－a11jsinycoszjsinycoszˆyE0coseH1HiHr＝aE0cose11jsinycoszjsinycoszˆzE0e -asinE0esinE0E0jsinyjcoszjcoszjsinyjcoszjcoszˆyˆH1acosee-e-asinee+ezˆyH1aE0cosejsinyˆ2coscosz-aE0zsinejsiny2jsincoszH1y2E0E0cosesinejtjsinycoscosz8.143b2jsincosz8.143bE0ˆzH1z=ajsinyˆzH1z=ReH1ze＝ReaˆzH1z=ReaH1z＝-E0E0sinejsiny2jsincoszejtsin2jsincoszcostsinyjsintsinysinsincosz2sintsiny8.146cH1yReH1yReH1yE0E0E0cosejsiny2coscoszejt8.143bcos2coscoszcostsinyjsintsinycos2coscoszcostsiny8.146bE12jE0sinzcose2E0jsyinˆx8.143aaˆx2E0sincoszsintsiny8.146aE1aH1yE0cosesinejsinycoscosz8.143b2jsincosz8.143bˆzH1z=aH1yjsinyE0cos2coscoszcostsiny8.146bsinsincosz2sintsiny8.146cH1z＝-E0We realize that the fields propagate in the y direction(y方向的行波) and form a standing wave pattern along the z direction（z方向驻波）. ˆx2E0sincoszsintsiny8.146aH1yE1aH1z＝-E0sinsincosz2sintsiny8.146cE0cos2coscoszcostsiny8.146bthe nodal points for the electric field intensity and Z component of the magnetic field intensity are at: msincosz＝0；cosz＝mm0,12..8.147az8.147bcosΒcosθ：is the component of the phase constant in the z direction, thus we can define the wavelength in that direction as (8.148) In terms of the wavelength in the z direction,the nodal points are 2mz8.148znodez for m=1,2,3...8.149acos2Thus,the E field has nodes at z=0,and at multiples of half-wavelengths from the interface in the negative z direction.The y component of the H field has nodal points at odd multiples of quarter-wavelengths in the negative z direction coscosz＝0；cosz＝-m2m0,12..znode-m12z2m18.149bm0,1,2..cos4propagate in the y direction(y方向的行波) 等相位面传播速度 the phase velocity,upyˆx2E0sincoszsintsiny8.146aE1aequiphase:tsinyC,t，yt1C1y1(1) up=  8.150asin等相位面在Δt时间内，向z轴正向行进Δy upt2Cdyy2y1y2(2) upy＝＝8.150sindtt2t1sinsinSince sinθ<1,we expect that upy>up,if the medium is free space,we can expect the phase velocity in the y direction t o be greater than the speed of light.Does this mean that the energy propagation in the y direction can occur with a velocity greater than the speed of light?The answer,of course,is no Einstein 相对论决定 The energy velocity——The group velocity (a)The energy propagates with a velocity known as the group velocity.To determine the group velocity,we will calculate the average power flow using the Poynting vector and the energy jsyinˆx8.143aE2jEsinzcosea0density 1ˆyH1aE01*22ˆˆy8.151aSavReE1H2E0sincoszsina2cosejsinyˆz2coscosz-aE0sinejsiny2jsincoszAs the power density is independent of variations along the direction,we can calculate the total average power per unit length along the x direction and between any two nodal points in the z direction as mnz1,z28.1478.151b coscosTwo distinct node points for the E field such that mn ˆdsPSav12E0upsin222E0sincoszsindxdzmn8.151ccos0z2z1(b)Average power from energy density: wwe＋wm11*ReDEReBH*44u能速度＝ugˆSwtotalddFor simple medium: 1111121****wDEBHEEHHEH244444420E112122EEdieletric,H4Hc442w2Esinzcos8.1522ˆuSw＝ugw能速度ˆdsuwdsPSavgy10z2z12E0ug222E0sinzcosugydxdzmn8.153cos(8.151c)=(8.153) ugyupsin8.154up1up=  8.150aupy＝8.150sinsinugyupyu8.1552pthe phase velocity1 up=  8.150a upysin＝upsin8.150传播介质不同，相速度不同； 传播空间不同，相速度不同； Parallel-plate wave-guide(平行板波导,矩形波导，谐振腔) As explained earlier,the fields in the dielectric medium represent a pure standing wave in the z direction and a traveling wave in the y direction.We can place another conducting plate at any of the nodal points,without disturbing the field patterns.In this case, it appears as if the fields are being guided by the two perfectly conducting planes.These perfectly conducting planes are said to form a parallel-plate wave-guide, and the preceding equations are the solutions of Maxwell’s equations for this wave-guide znodeznodemz for m=1,2,3...8.149a21z2m1 m0,1,2.. 8.149b4导体板：1↑6 平行板波导,矩形波导，谐振腔 理想导体边界条件：Et＝0，Hn＝0 EEmeˆnrjaˆxxaˆyyaˆzz;22;raˆn＝aˆxxaˆyyaˆzzaˆxcoscosaˆycossinaˆzsinaEEmeˆnrjaˆxxaˆyyaˆzz;;ra22ˆn＝aˆxxaˆyyaˆzzaˆxcoscosaˆycossinaˆzsinaznodeznodemz for m=1,2,3...8.149a21z2m1 m0,1,2.. 8.149b4导体板：1↑6 平行板波导,矩形波导，谐振腔 理想导体边界条件：Et＝0，Hn＝0 垂直极化波 znodemz for m=1,2,3...8.149a2两板之间的最小距离d是半个波长，如果距离d再小，怎么样？ 截止频率，截止波长，截止波数 ˆx8.143aE12jE0sinzcosejsyinaˆx2E0sincoszsintsiny8.146aE1aH1y2E0E0cosejsinycoscosz8.143bsinejsinyH1yE0cos2coscoszcostsiny8.146bsinsincosz2sintsiny8.146cˆzH1z=a2jsincosz8.143bH1z＝-E0(1）在z方向为驻波（2）在y方向为行波 uP＝ksini；uP0＝k0sinic(3)等振幅面——z为常数的平面；等相位面——y为常数的面，是非均匀平面波（4）平行板波导的原理（5）电磁波沿y方向传播，Ex垂直于波的传播方向，为TE波（横磁波）TEM波、TE波及TM波 TEM波、TE波及TM波的电场方向及磁场方向与传播方向的关系如下图示。 E E E es H TEM波 H TE波 es H TM波 es Transverse electromagnetic wave Transverse electric wave (TE) Transverse magnetic wave (TM) 垂直极化波入射到理想导电体分界面上:TE波 作业：chap 8 P414

E:4 6 7 P:8;E 8 10 11 14 P 24 25(极化）

P:28 29 31 30 32垂直入射 斜入射：书上例题及课外例题 谢谢!