理想導體球?qū)ζ矫娌ǖ纳⑸?Mie級數(shù))

理想導體球?qū)ζ矫娌ǖ纳⑸淅硐雽w球?qū)ζ矫娌ǖ纳⑸?，基本配置如下圖所示。1理想導體球散射的幾何關(guān)系入射平面波可以表示為Ei?

Eixx

Eejkzx0

Eejkrcosx0

(1)Hi?Hi?0ejkz?

ejkr

(2)將平面波用球坐標表示

y xi?Ei?rr

y y0Ei?Ei

(3)Hi?r其中

Hi?r

Hi?Hi

(4)cos EiEisincosE

sincose

jkrcosE ejkr

(5)r x 0

0 jkrEiEicoscosEcoscosejkrcos x 0EiEisinEsinejkrcos

(6)(7) x 0E Esin HiHisinsin0sinsinejkrcos0 ejkr

(8)r x

jkrEHiHicossin0cossinejkrcos (9)E x EHiHicos0cosejkrcos (10)E x 用球面波函數(shù)展開平面波，得到EiE

cos

jn2n1

Pcos

(11)r 0

n0

n nEiE 0

coscosn0

jn2n1jn

krPcosn

(12)EiE 0

sinn0

jn2n1jn

Pcosn

(13)i Esin n

H0

j 2n1j kr

cos

(14)r

n0

n nHi0cossin n0

jn2n1jn

Pcosn

(15)E Hi E n0

jn2n1jn

Pcosn

(16)其中 Pn

表示 n 階 Legendre 多項式。由于jn

kr1kr

J?kr，nPcos1cosP1=，則 n

nEijE

cos

0jn2n?kr1cos

(17)r 0kr

n nn1EiE 0

coscoskrn0

jn2n?krP0cosn n

(18)EiE 0

sinkrn0

jn2n?krP0cosn n

(19)i Esin

n ?

1 Hj0

j 2n1J krP cos

(20)r kr

n nn1i Ecossin

n

?

0 H kr

jn0

2n1Jn

krPn

cos

(21)i Ecos n

?

0 H0

j 2n1J kr

cos

(22) kr

n nn0r方向的矢量位可以表示為iEcosr 0 n1

ann

kr1cosn

(23)i Esin

? 1 F0

aJ krP

(24)2n1其中anjnnn1。

r

n1

nn n散射場的矢量位可以表示為AsE

cos

b?2kr1cos

(25)r 0

n1

n n ns Esin

?2

1 F0 c

krP

cos

(26)r

n n nn1H2kr1?2kr?？偸噶课粸閚 A

nEcos

a

b2P1cos

(27)r 0

nnn1

n n nEsin00

?

?2

1 Frn1

aJ nn

cHn

kr Pn

cos

(28)根據(jù)矢量位可以得到總場的表達式1 2 E 2A

(29)r jr2 r1 12AEr jr

1 1 Frsin

(30)1 1 2AEr jrsin

11F r

(31)1 2 H2F

(32)r jr2 r1 1 A 1 12FHrsinrrrr

(33)11AHrr

1 1 2Frsinr

(34)

cEt0Et0，可以得到n nn? n?n? n?

ka于是，散射場為

n

nH2'kan

，cn

nH2kan

(35)EsjEcosb?2''kr?2kr1cos

(36)r 0 n n1

n nE

?

?

P1cosEs 0cos

jbH2

krsinP1'coscH2kr

(37)

n nn1

n n

sin E

?

P1cos

?

Es 0sin

jbH2kr

cH2krsinP1'cos

(38)

n nn1

sin n n n EsrEsjE 0

ekr

cos

jnbnn1

sinP1n

cos

cn

P1cosn sin

(39)ejkr

1cos

n n Esn n 0

sin

jnb nn1

sin csinP1'cos

(40)c**********************************************************************c ComputeScatteringfromPECSpherebyMieSeriesca INPUT,real(8)c Onentry,'a'specifiestheradiusofthespherec f INPUT,real(8)c Onentry,'f'specifiestheincidentfrequencyc r INPUT,real(8)c Onentry,'r'specifiesthedistancebetweentheobservationc pointandtheoriginofcoordinatesc th INPUT,real(8)c Onentry,'th'specifiestheobservationc EstOUTPUT,complex(8)c Onexit,'Est'specifiesthethcomponentoftheelectricc scatteringfieldc EspOUTPUT,complex(8)c Onexit,'Ephi'specifiesthephcomponentoftheelectricc scatteringfieldcc ProgrammedbyPandaBrewmasterc**********************************************************************subroutinedSca_PEC_Sph_Mie(a,f,r,th,Est,Esp)c**********************************************************************implicitnonec -InputParametersreal(8)a,r,f,thcomplex(8)Est,c -ConstantNumbersreal(8),parameter::pi=3.141592653589793d0real(8),parameter::eps0=8.854187817620389d-12real(8),parameter::mu0=pi*4.d-7c -TemporaryVariablesintegeri,j,k,nmaxreal(8)wavek,ka,kr,coe,RCSt,real(8),allocatable,dimension(:)::+Jn,Yn,kapa,taocomplex(8),allocatable,dimension(:)::+Hn,an,bnwavek=2.d0*pi*f*dsqrt(mu0*ka=wavek*akr=wavek*rnmax=ka+10.d0*ka**(1.d0/3.d0)+1if(nmax<5)nmax=5allocate(Jn(0:nmax),Yn(0:nmax),Hn(0:calldSBJ(nmax+1,ka,Jn(0:nmax))calldSBY(nmax+1,ka,Yn(0:doi=0,nmaxHn(i)=dcmplx(Jn(i),-enddoallocate(kapa(0:nmax),tao(0:nmax))calldLeg_Poly_dp(nmax,1.d0*dcosd(th),tao(0)=0.d0doi=1,nmaxtao(i)=i*dcosd(th)*kapa(i)-(i+1)*kapa(i-1)enddoallocate(an(nmax),bn(nmax))doi=1,nmaxan(i)=Jn(i)/Hn(i)bn(i)=(ka*Jn(i-1)-i*Jn(i))/(ka*Hn(i-1)-i*Hn(i))enddoEst=0.d0;Esp=doi=1,nmaxEst=Est+(2*i+1.0)/i/(i+1)*(an(i)*kapa(i)+bn(i)*tao(i))Esp=Esp+(2*i+1.0)/i/(i+1)*(an(i)*tao(i)+bn(i)*kapa(i))e