Today in Physics 218: stratified linear media I

Size: px

Start display at page:

Download "Today in Physics 218: stratified linear media I"

Angela Russell
6 years ago
Views:

1 Today in Physics 28: stratified linear media I Interference in layers of linear media Transmission and reflection in stratified linear media, viewed as a boundary-value problem Matrix formulation of the fields at the interfaces Incident light Low-n layers Substrate High-n layers 9 February 2004 Physics 28, Spring 2004

2 Interference in layers of linear media As a preamble to the general question of transmission and reflection by stratified media, we will ask a simpler one: what is the condition for completely constructive interference in a single layer of linear material? Consider two plane-parallel, partially reflecting surfaces separated by a linear medium with refractive index n = µε and thickness d (next slide). It doesn t matter what the index of refraction outside the reflectors is, but we will assume here that it is unity (vacuum) on both sides. If the transmitted or reflected rays are focussed then the waves interfere. By calculating the path-length differences, we can find out how they interfere. 9 February 2004 Physics 28, Spring

3 Interference in layers of linear media (continued) Vacuum n = µε Vacuum C D θ i θ i θ i θ t A sinθi = sin n B d 9 February 2004 Physics 28, Spring

4 Interference in layers of linear media (continued) The path length difference between any two successive transmitted waves is the same. For the first set, that s the length between AB and ACD: 2 sin θt AB = 2d tanθt sinθi = 2 dn, cosθt nsinθt = sinθi 2d ACD =. AD = 2d tanθt cosθt The wavelength is λ in vacuum and λ/n in the medium between the reflectors, so AB 4π dn 2 δ ( AB) = 2π = sin θt, λ λcosθt nacd 4π dn δ ( ACD) = 2 π =. λ λcosθ 9 February 2004 Physics 28, Spring t

5 Interference in layers of linear media (continued) If the phase difference is an integer multiple of 2π, then the interference between the two wavefronts corresponding to these paths is completely constructive: 4π dn ( ) ( ) ( 2 ) 4π dn cosθt δ = δ ACD δ AB = sin θt =, λcosθt λ = 2π m m= 0,,2,. ( ) Thus there are maxima in the spectrum of the transmission of the dielectric slab, at wavelengths given by 2dn cosθt λ m = ( m = 0,,2, ). m This, BTW, is the principle of the Fabry-Perot interferometer. 9 February 2004 Physics 28, Spring

6 Transmission and reflection in stratified linear media, viewed as a boundary-value problem Now we will set up the general solution to the problem of the transmission and reflection by a plane parallel layer, and find thereby a method for dealing with as many layers as we want. Consider light propagating in one medium, incident obliquely on a layer of a second medium, and emerging into a third (next slide). What are the amplitudes of the transmitted and reflected waves? As before, this can be broken into two parts, one with light polarized perpendicular to the plane of incidence (TE), and one with E parallel to the plane of incidence (TM). We ll do TE first, and fill out the boundary conditions at the surfaces. 9 February 2004 Physics 28, Spring

7 Transmission and reflection in stratified linear media as a boundary-value problem (continued) ε 0, µ 0 ε, µ 2 ε 2, µ 2 θ θ I T θ T 2 E I E T E T 2 k T k I B I B T B T 2 k T 2 B R B R2 θ I θ T k R E E R2 R k R2 d TE waves 9 February 2004 Physics 28, Spring

8 Transmission and reflection in stratified linear media as a boundary-value problem (continued) The electric fields look generically like this: E E = = E E 0 0 e e ( kr ωt) in ( kr ω ) i n t for waves propagating toward + z, the other way. And of course B = µε kˆ E. At surface, the boundary conditions on E = E + E = E + E,, 0I 0R 0T 0R2 H = B B ( cos cos ) I θi R θi, 0 0 µ 0 = ( B ) 0TcosθT B 0R2 cos θt, µ and H are 9 February 2004 Physics 28, Spring E

9 Transmission and reflection in stratified linear media as a boundary-value problem (continued) or E + E = E + E 0I 0R 0T 0R2 0 θi I R T T R 0 Next the wave traverses the layer filled with medium #, as follows:, ε cos ( E ) ε ( ) 0 E 0 = cos θ E 0 E 0 2. µ µ P θ T Q θ T 2 R S 9 February 2004 Physics 28, Spring

10 Transmission and reflection in stratified linear media as a boundary-value problem (continued) As a wave crosses the slab it travels a distance PQ = d /cos θ T. Compared to the undisplaced wave that would have resulted if the slab were not there, it undergoes a phase change of 2π n 2π n δ 2 = k + k22 = PQ RS λ λ 2πnd 2πn = 2 d tanθtsinθt2 λcosθt λ 2πnd ( 2 ) 2πnd = sin θ T = cos θt. λcosθ λ T (half that of the two reflections in slide 5) 9 February 2004 Physics 28, Spring

11 Transmission and reflection in stratified linear media as a boundary-value problem (continued) Thus the E and H boundary conditions at surface 2 are E E e E e E e iδ iδ iδ,2 = 0T + 0R2 = 0T2 H ε E e E ε e E e ( ) iδ iδ 2 iδ,2 = θt 0T 0R2 = θt2 0T2 µ µ 2 cos cos. At this point we have four equations that we can solve for the four unknown amplitudes, E 0R, E 0R2, E 0T, and E 0T2, for the TE case. You can proceed directly in this manner, to solve a couple of the problems in this week s homework (e.g. Crawford 5.2, Griffiths!9.34). But it would be incredibly tedious to treat more than one layer like this. Fortunately there s a better way 9 February 2004 Physics 28, Spring 2004,

12 Matrix formulation of the fields at the interfaces The clever way to solve these problems starts by rearranging the boundary conditions to obtain relations between the fields at the two interfaces. E 0T and E 0R2 appear in both sets of boundary conditions, so solve the latest result for these two amplitudes: + ε E ε E e E e µ µ ( iδ ) iδ cosθt,2 = cosθt 0T + 0R2 H ε E e E e ( iδ ) iδ,2 = cosθt 0T 0R2 µ ε ε µ µ θt,2,2 θt 0T cos E + H = 2 cos E e, or 9 February 2004 Physics 28, Spring iδ

13 Matrix formulation of the fields at the interfaces (continued) iδ µ H,2 E 0T = e E, ε cosθ T Put this back in the surface-2 boundary conditions, and solve for E 0R2 : iδ µ H,2 i i E δ δ,2 = e E,2 + e + E 0R2e, or 2 ε cosθ T iδ µ H,2 E 0R2 = e E,2. 2 ε cosθ T Now put both of these into the surface- boundary conditions: 9 February 2004 Physics 28, Spring

14 Matrix formulation of the fields at the interfaces (continued) iδ µ H,2 iδ µ H,2, =,2 + +,2 2 ε cosθ T 2 ε cosθ T µ i sinδ = E,2 cos δ H,2, ε cosθ E e E e E H e E T ε µ H iδ µ H,2 e E,2 2 ε cosθ iδ,2, = cosθt,2 + µ 2 ε cosθt,2 θt δ,2 δ µ T ε = E cos isin + H cos. 9 February 2004 Physics 28, Spring

15 Matrix formulation of the fields at the interfaces (continued) Now define ε 4π Y, TE = cosθt = cos θt, µ c Z and the results look suggestive of matrix arithmetic: i sin E δ, = E,2 cosδ H,2 and Y, TE H, = E,2Y, TEisinδ + H,2 cos δ, or E, cosδ isin δ/ Y, TE E,2 E,2 = M H iy,, TE sinδ cosδ H,2 H,2. M is called the characteristic matrix of layer. 9 February 2004 Physics 28, Spring

16 Matrix formulation of the fields at the interfaces (continued) We could repeat this procedure for TM waves (see following slide), but it s so similar to what we just did that we ll just skip to the result: E H µ cos isin E cos,, =,2 θt δ +,2 δ ε i sin H ε δ, = H,2 cos δ E,2. µ cosθt Thus if we define ε Y, TM =, µ cosθt we get the same matrix equation as before, which we will write as: 9 February 2004 Physics 28, Spring

17 Matrix formulation of the fields at the interfaces (continued) ε 0, µ 0 ε, µ 2 ε 2, µ 2 θ θ I T θ T 2 I B T B T 2 k T k I E I E T E T 2 B k T 2 E R E R2 k R θ I R k R2 θ T H H R2 d TM waves 9 February 2004 Physics 28, Spring

18 Matrix formulation of the fields at the interfaces (continued) E, cosδ isin δ/ Y E,2 E,2 = M H iy, sinδ cosδ H,2 H,2 If there were yet a third surface to the right, the parallel components of the fields there could therefore be determined from E,2 E,3 = M2, H,2 H,3 which can be combined with our first result to yield E, E,3 = MM 2. H, H,3 9 February 2004 Physics 28, Spring

19 Matrix formulation of the fields at the interfaces (continued) And so on. Evidently, for a stack of p layers, the parallel components of the fields at the first and p+th surface are related by E, E, p+ = MM2M p, H, H, p+ and the whole stack can be said to have a characteristic matrix M given by L m m M = M M Mp = N M O 2 m m Q P February 2004 Physics 28, Spring

Today in Physics 218: stratified linear media II

Today in Physics 218: stratified linear media II 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