Today in Physics 218: stratified linear media II

Transcription

1 Today in Physics 28: stratified linear media II Characteristic matrix formulation of reflected and transmitted fields and intensity Examples: Single interface Plane-parallel dielectric in vacuum Multiple quarterwave stacks Electron micrograph of the cross section of a twenty-layer, alternating quarter-wave stack. February 2004 Physics 28, Spring 2004

2 Last time Here is the relationship between E and H on the surfaces of a plane parallel layer of linear material, illuminated from side : E, cosδ isin δ/ Y E,2 E,2 = M H iy, sinδ cosδ H,2 H,2 where, Y = Y, TE = cosθt µ = Y =, TM ε ε µ cosθ T TE : E incidence plane; TM : E incidence plane. February 2004 Physics 28, Spring

3 Last time (continued) Furthermore, the tangential components of E and H = B µ for the first and p+st surfaces of a stack of p layers are related by E, E, p+ = MM2 M p H, H, p+ It remains for us to show how to obtain the transmitted and reflected amplitudes from these fields.. February 2004 Physics 28, Spring

4 Characteristic matrix formulation of reflected and transmitted fields and intensity Recall that the tangential components of the fields at the first surface are related to the fields in the incident and reflected waves by E ( ), = E 0I + E 0R and H, = Y0 E 0I E 0R, while the tangential field components at the last surface are related to the fields of the final transmitted wave by E, p+ = E 0 T and H, p+ = Yp+ E 0T, so E 0I + E 0R E 0 Tp, + = M. Y ( ) 0 E 0I E 0R Yp+ E 0 T, p+ Multiply it out: February 2004 Physics 28, Spring

5 Characteristic matrix formulation of reflected and transmitted fields and intensity (continued) E + E = m E + m Y E 0I 0R 0 T, p+ 2 p+ 0 T, p+ ( ) Y E E m E m Y E 0 0I 0R = 2 0 T, p p+ 0 T, p+. In terms of the electric-field amplitude reflection and transmission coefficients, r E 0R/ E 0I and t E 0 T, p / E + 0I, this equations are + r = m + m Y t e j 2 p+ a f e j. Y r = m + m Y t p+ Multiply the first of these by and add, to produce t = Y 0 2Y m Y + m Y Y + m + m Y p p+. February 2004 Physics 28, Spring

6 Characteristic matrix formulation of reflected and transmitted fields and intensity (continued) and substitute back to get + r = ( + ) p+ 2Y m m Y 0 2 m Y + m Y Y + m + m Y p p+ r = = ( + ) ( ) p+ + p+ + + p+ 2Y m m Y m Y m Y Y m m Y m Y + m Y Y + m + m Y p p+ m Y + m Y Y m m Y p p+ m Y + m Y Y + m + m Y p p+. Usually the fraction of intensity or power reflected and transmitted is also of interest. The Poynting vector is: February 2004 Physics 28, Spring

7 Characteristic matrix formulation of reflected and transmitted fields and intensity (continued) c ˆ c ε 2 S= E H = k E, 4π 4π µ so the power per unit area flowing through any plane parallel to the surfaces is c ε ˆ 2 c 2 S = Sz = cos θ E = Y E. 8π µ 8π In terms of the amplitude coefficients, we can define the intensity reflection and transmission coefficients, as SR, S 2 Tp, +, Yp+ 2 ρ = = r and τ = = t. SI, SI, Y0 Of course τ + ρ =, as demanded by energy conservation, and as you will show in this week s homework. February 2004 Physics 28, Spring

8 Example: a single vacuum-linear medium interface Can our new formalism be used to re-derive the Fresnel equations? Yes, because in the expression E, cosδ isin δ/ Y E,2 E,2 = M H iy, sinδ cosδ H,2 H,2, we are free to choose zero thickness (no layer at all), for which the phase delay across it is also zero, and the characteristic matrix becomes 0 M = 0. February 2004 Physics 28, Spring

9 Example: a single vacuum-linear medium interface (continued) This leaves us with the two media, for which the admittances are ε Y0, TE = cosθi Y2, TE = cosθt µ so t TE Y ε = Y = θ µ cosθ 0, TM 2, TM cos I 2Y0 2Y0 = = m Y + m Y Y + m + m Y Y + Y = = =, Y + 2 cosθ T ε + αβ + Y cosθ µ I February 2004 Physics 28, Spring T

10 Example: a single vacuum-linear medium interface (continued) and r TE just as before. m Y + m Y Y m m Y Y Y = = m Y + m Y Y + m + m Y Y + Y ε cosθ I µ cosθt αβ = = ε cosθ I + αβ + µ cosθ T Similarly, 2 Y 2 TM and Y t Y α β = = = =, Y + Y α + β Y + Y α + β rtm February 2004 Physics 28, Spring

11 Example: plane parallel dielectric in vacuum Actually this isn t a worked example because you re doing it in this week s homework. For normal incidence on a planeparallel linear medium with thickness d and µ =, YTE = YTM = ε cos θt = n, 2πnd 2πnd δ = cos θt =, λ λ and the characteristic matrix is 2πnd i 2πnd cos sin cosδ isin δ/ Y λ n λ M = iysinδ cosδ = 2πnd 2πnd insin cos λ λ February 2004 Physics 28, Spring 2004

12 Example: plane parallel dielectric in vacuum (continued) In vacuum, the amplitude transmission coefficient turns out to be 2 2 t = = F m + m + m + m nd HG I, nd i n+ n K J 2 2 cos π sin π λ λ and the intensity transmission coefficient τ = tt* =, 2 2 n 2π nd + sin 2n λ just as advertised a couple of lectures back. This is plotted on the next page, for three values of the single-surface intensity 2 reflection coefficient, r = n n+. ( ) 2 February 2004 Physics 28, Spring

13 Example: plane parallel dielectric in vacuum (continued) Transmission, τ r = r = nkd 2 r = 0.95 February 2004 Physics 28, Spring

14 Example: quarter-wave stacks The case of normal incidence and dielectric (µ = ) layers with thickness equal to λ/4n at some design wavelength λ d is encountered frequently (c.f. Crawford 5.2 on this week s homework). Y = Y = Y = ε cos θ = n, TE TM T 2π n λd πλd δ = cos θt =, λ 4n 2λ πλd i πλd cos sin 2λ n 2λ M =. πλd πλd insin cos 2λ 2λ February 2004 Physics 28, Spring

15 Example: quarter-wave stacks (continued) In particular, alternating quarter-wave layers, arranged in pairs consisting of one high-index and one low-index layer, are the main building block in the interference-filter and mirror- and lens-coating industry. With H as the characteristic matrix of a high-index material (n = 3-5) and L that of a low-index dielectric (n =.3-2), series like HLHL... HL = ( HL) are useful in making the reflectivity or transmission change rapidly with wavelength, especially around the design wavelength. The more layer pairs, the sharper the change. February 2004 Physics 28, Spring m

16 Example: quarter-wave stacks (continued) For instance, consider a stack that s good at transmitting long wavelengths and reflecting short wavelengths (a low-pass filter): ( Air ) LHLHL... HL( Glass) = ( Air) L( HL) m ( Glass) Though it looks complicated, the whole characteristic matrix is a simple product of 2x2 matrices that don t actually take very long at all to calculate. See the results on the next page: February 2004 Physics 28, Spring

17 Example: quarter-wave stacks (continued) 0.8 Transmission Wavelength (microns) One layer pair Two layer pairs Four layer pairs Eight layer pairs February 2004 Physics 28, Spring