REFLECTION AND REFRACTION

Transcription

1 REFLECTION AND REFRACTION REFLECTION AND TRANSMISSION FOR NORMAL INCIDENCE ON A DIELECTRIC MEDIUM Assumptios: No-magetic media which meas that B H. No dampig, purely dielectric media. No free surface charges. No free surface curret charges. Flat iterfaces with ulimited extet. Boudary coditios: E: Tagetial compoet cotiuous. D: Normal compoet cotiuous B: Normal compoet cotiuous. H: Tagetial compoet cotiuous. Subscript 0, 1 ad represet icidet, reflected ad trasmitted wave, respectively. x E 0 Medium 1: ε ε 1 µ 1 σ 0 1 E Medium : ε ε µ 1 σ 0 > 1 e z -e z e z z E H 1 0 H H 1 y

2 Bo E. Serelius 1: E0 e 0 0 ikz 1 ωt xe e E1 e 1 0 i k1z ωt xe e E e 0 ikz ( ωt) xe e The scalar amplitudes E i 0 are time-idepedet ad may be complex valued, allowig for a phase differece betwee the waves. ω k1 1 ε ω 1 c c ω k ε ω c c Sice we have assumed o-magetic materials we have H B ˆ k E ad H0 e ikz 1 ωt ye e H1 e i k 1z ωt ye e H e 0 ikz ωt y E e The boudary coditios for tagetial compoets give E0 0 E1 0 E 0 ad H0 0 + H1 0 H 0 or 1( E0 0 + E1 0 ) E 0

3 Bo E. Serelius 1:3 This gives E1 0 1 E E 0 1 E ; H E E H H E E H ; We see that i the case we have here there is a phase chage of π for the electric compoet of the reflected wave but ot for the magetic compoet ad ot for the trasmitted wave. If istead medium is optically thier tha medium 1 there is a phase chage of π for the magetic compoet of the reflected wave but ot for the electric compoet. Thus oe has to specify if the phase of the wave is represeted by the electric or magetic compoet. The average eergy flux i the icidet wave is c S π Re * E H ad the power reflectio coefficiet which is defied as the relative amout of eergy that is reflected at the boudary: R S1 ez S0 ez * E1 H1 * E0 H0 E1 0 E0 0 R 1 + 1

4 Bo E. Serelius 1:4 The power trasmissio coefficiet is defied by T S S0 ez ez * E H * E0 H0 1 E 0 E0 0 so that T ( + 1) Eergy coservatio meas that R + T 1

5 Bo E. Serelius 1:5 OBLIQUE INCIDENCE-THE FRESNEL EQUATIONS Medium 1 x Medium k 0 θ 0 θ 1 θ z k 1 k k r E0 E0 0 i 0 ωt e H0 1 k0 E0 k0 k r E1 E1 0 i ( 1 ωt e ) H 1 1 k E 1 1 k 1 k r E E 0 i ωt e H k E k Note that we use the H fields istead of the fudametal B fields because the boudary coditios for these fields are the same as for the E fields.

6 Bo E. Serelius 1:6 Now, the tagetial compoets of the fields have to be cotiuous across the boudary. For this to be possible the periodicities of the field vectors have to be equal at the boudary: k0 ex k1 ex k ex For symmetry reasos all three propagatio vectors are coplaar, i.e. are i the same plae, the plae of icidece (The surface ormal is also i this plae). This meas that k0siθ0 k1siθ1 ksiθ This ca also be viewed as coservatio of the mometum parallel to iterface. This coservatio holds if the iterface is smooth. A rough iterface or a itetioally created periodic structure at the iterface ca relax this coditio. Mometum parallel to the iterface ca the be absorbed from, or provided to, the reflected ad refracted waves. Now, k0 k1 θ0 θ1 ad 1siθ1 siθ Sell's law Now we have determied the propagatio directio of the reflected ad refracted waves. The amplitudes of the field compoets are obtaied from usig the boudary coditios. It is eough to use the boudary coditios for the tagetial compoets of the E ad H fields. The coditios for the ormal compoets of the B ad D fields give o more iformatio. The boudary coditios are:

7 Bo E. Serelius 1:7 ( E0 + E1) E ; ( H0 + H1) H The last ca be rewritte i terms of the electric field vectors ad we have ( E0 + E1) E ; ( k0 E0 + k1 E1) ( k E) Ay plae wave impigig o the iterface ca be writte as a liear combiatio of two waves, oe with the electric vector polarized parallel to the plae of icidece (p-polarized), ad oe with the electric vector perpedicular to the plae of icidece (s-polarized). We may treat these compoets separately.

8 Bo E. Serelius 1:8 E PERPENDICULAR TO THE PLANE OF INCIDENCE. S-polarized waves. Medium 1 x Medium H 0 E 0 k 0 θ 0 θ 1 θ E 1 E H z k 1 k H 1 All E vectors are i the mius y directio. This meas that the first boudary coditio makes: E0 0 + E1 0 E 0 The secod we expad usig the triple curl product ( k0 E0)+ ( k1 E1) ( k E) k0( E ) E ( k ) k0cosθ0 + k1( E1) E1 ( k1) k( E) E( k) k cosθ 0 k cosθ or 1 1

9 Bo E. Serelius 1:9 E0k0cosθ0 E1k1cosθ1 Ekcosθ or sice k0 k1 ; θ1 θ0 ; k k1 1 we have ( E0 0 E1 0 ) cosθ0 E 0 cosθ 1 Combiig the two relatios gives E1 0 cosθ 0 1 cosθ cosθ0 +( 1) cosθ si θ θ0 E0 0 si( θ + θ0) E 0 cosθ 0 cosθ0 +( 1) cosθ cosθ 0siθ E0 ( 0 si θ + θ0) E0 0 E0 0

10 Bo E. Serelius 1:10 Reflectio ad trasmissio coefficiets R T c S ( ) E 1 0 kˆ 1 1 ( ) E π S0 c E 0 0 kˆ 0 0 E0 0 8π c S E 0 kˆ E 0 cosθ 8π S0 c E 0 0 0k ˆ 0 E0 0 1cosθ0 8π R E1 0 E0 0 si ( θ θ0) si ( θ + θ0) T E 0 cosθ E0 0 1cosθ0 cosθ 4 1cosθ0 cos θ 0 si θ si ( θ + θ0) siθ 0cosθ cos θ 0 si θ 4 siθ cosθ0 si ( θ + θ0) siθ 4 0 cosθ cosθ 0 siθ si ( θ + θ0) si θ0si θ si ( θ + θ0)

11 Bo E. Serelius 1:11 E PARALLEL TO THE PLANE OF INCIDENCE. P-polarized waves. Medium 1 x Medium E 0 H 0 k 0 θ 0 θ 1 θ H 1 H E z E 1 k 1 k Now all H vectors are poitig i the y directio, i.e., out of the plae of the figure. We ca make use of our results from the s-polarized case. We just chage the electric fields to the refractive idex times the electric fields i the coditio for the fields: E0 0 + E1 0 E 0 1( E0 0 + E1 0 ) E 0 ad i the coditio for the H fields we chage the refractive idex times the electric fields to the electric fields: E0 0 E1 0 ( ) cosθ0 E 0 cosθ ( E 0 0 E 1 0 ) cosθ0 E 0 cosθ 1

12 Bo E. Serelius 1:1 from which we obtai E1 0 cosθ 0 1 cosθ E0 0 cosθ0 +( 1 ) cosθ ta θ θ0 E0 0 ta( θ + θ0) E 0 cosθ 0 E0 0 cosθ +( 1) cosθ0 cosθ 0siθ E0 ( 0 si θ0 + θ) cos( θ0 θ)

13 Bo E. Serelius 1:13 Reflectio ad trasmissio coefficiets R// E1 0 E0 0 ta ( θ θ0) ta ( θ + θ0) T// E 0 cosθ E0 0 1cosθ0 cos θ 4 1cosθ 0 si cos θ 0 si θ θ0 + θ cos θ0 θ ( ) siθ cos cos si 0 θ θ 0 θ 4 siθcosθ 0 si ( θ 0 + θ) cos θ0 θ siθ 0cosθ0siθcosθ 4 si ( θ0 + θ) cos ( θ0 θ) si θ θ 0si si ( θ0 + θ) cos ( θ0 θ) We see that there is o reflectio if θ + θ1 θ + θ0 π This happes for the agle of icidece θ 0 1 θb ta ( ) 1 Brewster's Agle

14 Bo E. Serelius 1:14 1 air-glass iterface Reflectio coefficiet P (θ 0 ) Brewster's agle R S 0 R M R P θ 0