26.柱面波函数+贝塞尔函数(10-0)

V. CYLINDRICAL WAVE FUNCTIONS1. Wave FunctionsA. Scalar wave equation● As mentioned before, the wave equations in source-free region are220H k H ∇+=220E k E ∇+=220A k A ∇+=220k ∇Φ+Φ=where k is the intrinsic wave number,k ==● Let ψ represent the potentialΦor one of the rectangular components ofA ,E ,H, then this scalar function ψ in source-free region should satisfythe following scalar wave equation:22k ψψ∇+=● In cylindrical coordinate system, the scalar wave equation is222222110k zψψψρψρρρρϕ⎛⎫∂∂∂∂+++= ⎪∂∂∂∂⎝⎭B. Separation of variables● By using the method of separation of variables, the solution is assumed to bethe product of three functions of one coordinate each:()()()R Z z ψρϕ=Φ● Substitution of above product function into the cylindrical-coordinate scalarwave equation and division by ()()()R Z z ψρϕ=Φyields()()()()()()2222221110d R d d Z z d kR d d d Z z d zρϕρρρρρρϕϕΦ⎡⎤+++=⎢⎥Φ⎣⎦● Since the third term is independent of ρ and ϕ, and the equation is to sumto zero for all ρ, ϕ, z , then the third term should be independent of z ,()()221d Z z Z z d z=constant● For later convenience, set()()2221zd Z z k Z z d z=-● Substitution of this into the preceding equation and multiplication by 2ρgives()()()()()2222210zd R d d k k R d d d ρϕρρρρρρϕϕΦ⎡⎤++-=⎢⎥Φ⎣⎦● Similarly, the second term of above equation should be a constant setting tobe()()2221d nd ϕϕϕΦ=-Φ● The preceding equation becomes()()()22220z d R d n k k R d d ρρρρρρρ⎡⎤-+-=⎢⎥⎣⎦which is an equation in ρ only.● Let222zk k k ρ=-the equation is reduced to()()()220d R d k n R d d ρρρρρρρρ⎡⎤⎡⎤+-=⎢⎥⎢⎥⎣⎦⎣⎦● As a result, the cylindrical-coordinate scalar wave equation is separated intofollowing three equations:()()()220d R d k n R d d ρρρρρρρρ⎡⎤⎡⎤+-=⎢⎥⎢⎥⎣⎦⎣⎦()()2220d n d ϕϕϕΦ+Φ=()()222z d Z z k Z z d z+=● In these equations, the parameters satisfy the following separation equation:222zkk k ρ=+● The equation for()R ρis called the Bessel ’s equation of order n , and itssolution, denoting by()n B k ρρ, are termed the Bessel ’s function of order n .● The solutions to the second and third equations are harmonic functions,which are denoted by ()h n ϕ and ()z h k z , respectively.● The scalar wave function ψ to be determined in cylindrical coordinates is()()()(),,n z z B k h n h k z ρψρϕρϕ=● The general solution to the scalar wave equation22k ψψ∇+= is given by()()()(),,,zzn k n z nk z C B k h n h k z ρψρϕρϕ=∑∑where()()(),,n z B k h n h k z ρρϕhave the form as()()()()()()()12:,,,n n n n nB k J k N k H kH kρρρρρρρρρρ()()()()():s i n ,c o s ,e x p ,e x p h n n n j n j n ϕϕϕϕϕ-()()()()():s i n ,c o s ,e x p ,e x p z z z z z h k z k z k zj k zj k z-()n J k ρρis the Bessel function of first kind and order n ,()n N k ρρ is the Bessel function of second kind and order n ,()()1nH kρρ is the Hankel function of first kind and order n ,()()2nH kρρis the Hankel function of second kind and order n .C. Bessel functions● The Bessel functions will be introduced in absence of detailed derivation, theinterested readers are referred to some mathematical textbooks.● The Bessel equation of order v may be generally expressed as()220d d y xx x v y d x d x ⎛⎫+-= ⎪⎝⎭● The solution to this equation is the Bessel function of first kind as()()()()()()20201!!21!!2mm vv m mm vv m x J x m m v x J x m m v +∞=-∞-=-⎛⎫=⎪+⎝⎭-⎛⎫=⎪-⎝⎭∑∑● If v is not an integer,()v J xand()v J x -are linearly independent solutions,and the general solution to the Bessel equation becomes()()12v v y C J x C J x -=+● Ifv n= is an integer,()n J xand()n J x - are not independent, but rather,they are related by()()()1nn n J x J x -=-● To look for the second independent solution to the Bessel equation, a Besselfunction of second kind is defined by()()()co s sin v v v J x v J x N x v ππ--=● If v is not an integer,()v J xand()v N xare linearly independent solutions,and the general solution to the Bessel equation is()()12v v y C J x C N x =+● In the case of integer, v n =,()()()()()()()()()()()2110220lim co s limsin 1!212lo g 2!11!!2n v v nv v v nn mn n m mn mm N x N x J x v J x v n mx J x x m m m n x m m n ππγππφφπ→-→--=+∞==-=--⎛⎫=-⎪⎝⎭-++⎡⎤⎛⎫⎣⎦-⎪+⎝⎭∑∑where101.781,log 0.577γγ== ()111123m mφ=++++● It is proven that the limit function()n N xis existent, and is a solution to theBessel equation, as well as is linearly independent of()n J x .● Thus, no matter whether the subscript v is an integer or not, the generalsolution to the Bessel equation is always()()12v v y C J x C N x =+● For later use, the Hankel functions are defined as()()()()()()()()12v v v vv v H x J x jN x H x J x jN x =+=-in which the former is called the Hankel function of the first kind, and the later the Hankel function of second kind.● Small argument approximation()01,01,0!2vv x v J x x v v →=⎧⎪→⎨⎛⎫> ⎪⎪⎝⎭⎩()()1002log ,021!2,0v vx x v N x v v x γππ→⎧⎛⎫= ⎪⎪⎝⎭⎪→⎨-⎛⎫⎪-> ⎪⎪⎝⎭⎩Note that()v J xis finite atx =, whereas ()v N xis infinite atx =.● Large argument approximation()s 42v x v J x x ππ→∞⎛⎫→-- ⎪⎝⎭()42v x v N x x ππ→∞⎛⎫→-- ⎪⎝⎭()()()()12v jxvx vjxvx H x eH x e-+→∞+-→∞→→● It is important to note from the large argument approximation that()n J k ρρ is analogous to ()cos k ρρ, denoting a standing waves in e ρ direction ()n N k ρρis analogous to ()sin kρρ, denoting a standing waves ine ρdirection()()1n H k ρρ is analogous tojk e ρρ+, denoting a traveling wave in e ρ- direction ()()2nH kρρis analogous tojk eρρ-, denoting a traveling wave ine ρ+ direction● Let()v B x denote an arbitrary solution to the Bessel function, someimportant formulas are given as follows:()()()()()()()()()()()()11011221v v v v v v v v v v B x B x B x x v B x B x B x x B x B x v B x B x B x x-+--'=-'=-+'=--=-D. TM to z wave● As described previously, the EM field in source-free region can be given by11ˆE F Ayεμ=-∇⨯+∇⨯∇⨯and11ˆH A Fzμε=∇⨯+∇⨯∇⨯● As also discussed earlier, a TM to z wave can be constructed by potentialsz A e ψ=andF =so that the EM field becomes()()11ˆ111ˆˆ11ˆˆE F AyF zA Ay zA Ayεμεμμμμ=-∇⨯+∇⨯∇⨯=-∇⨯-+∇∇⋅=-+∇∇⋅and()11ˆ111ˆˆ1H A FzA yF Fz Aμεμεεμ=∇⨯+∇⨯∇⨯=∇⨯-+∇∇⋅=∇⨯● Since()11z A A A A zzρϕρψρρρϕ∂∂∂∂∇⋅=++=∂∂∂∂and()222211z zA z e e e z z z z e e e zzzρϕρϕψψψψρρϕψψψρρϕ∂⎛⎫∇∇⋅=∇ ⎪∂⎝⎭∂∂∂∂∂∂⎛⎫⎛⎫⎛⎫=++ ⎪ ⎪ ⎪∂∂∂∂∂∂⎝⎭⎝⎭⎝⎭∂∂∂=++∂∂∂∂∂and()1111zzz A A A A A A A e e e z z e e ϕϕρρρϕρϕρρϕρρρρϕψψρϕρ⎛⎫∂⎛⎫⎛⎫∂∂∂∂∂ ⎪∇⨯=-+-+-⎪⎪ ⎪ ⎪ ⎪∂∂∂∂∂∂⎝⎭⎝⎭⎝⎭∂∂=-∂∂then the EM field becomes()()222222222222211ˆˆ111ˆˆ11ˆˆˆ11ˆz z z z E zA Ayz e e e e y z z z e e e zy y z z z e e e k y z z z ρϕρϕρϕμμψψψψμμρρϕψψψμρρϕψψψμρρϕ=-+∇∇⋅⎛⎫∂∂∂=-+++ ⎪∂∂∂∂∂⎝⎭⎡⎤⎛⎫∂∂∂=++-⎢⎥ ⎪∂∂∂∂∂⎝⎭⎣⎦⎡⎤⎛⎫∂∂∂=+++⎢⎥ ⎪∂∂∂∂∂⎝⎭⎣⎦and111H Ae e ρϕμψψμρϕρ=∇⨯⎛⎫∂∂=- ⎪∂∂⎝⎭● The EM field components become222221ˆ1ˆ1ˆz E y z E yz E k y z ρϕψμρψμϕψμ∂=∂∂∂=∂∂⎛⎫∂=+ ⎪∂⎝⎭110zH H H ρϕψμρϕψμρ∂=∂∂=-∂=E. TE to z wave● As discussed earlier, a TE to z wave can be constructed by potentialsz F e ψ=andA =so that the EM field becomes11ˆ1E F AyFεμε=-∇⨯+∇⨯∇⨯=-∇⨯and()11ˆ11ˆˆH A FzyF Fz μεεε=∇⨯+∇⨯∇⨯=-+∇∇⋅By using the following aforementioned duality relationships:E lectricSources M agneticSources⇔E H ⇔ H E⇔- J M⇔ A F ⇔()()ˆˆy z ωω⇔ ()()ˆˆzyωω⇔k k⇔ 1ηη⇔ μεεμ⇔⇔the TE to z wave field components can be found as follows:TM to z wave TE to z wave222221ˆ1ˆ1ˆz E yz E yz E k y z ρϕψμρψμϕψμ∂=∂∂∂=∂∂⎛⎫∂=+ ⎪∂⎝⎭222221ˆ1ˆ1ˆzH zz H z z Hk z z ρϕψερψεϕψε∂=∂∂∂=∂∂⎛⎫∂=+ ⎪∂⎝⎭110zH H H ρϕψμρϕψμρ∂=∂∂=-∂=110zE E E ρϕψερϕψερ∂=-∂∂=∂=。