Films minces à base de Si nanostructuré pour des cellules ...
Films minces à base de Si nanostructuré pour des cellules ... Films minces à base de Si nanostructuré pour des cellules ...
Appendix I 2 x 2 Matrix formulation for Isotropic Layered Media [Yeh 88] tel-00916300, version 1 - 10 Dec 2013 Let us consider the case of a plane wave incident on an interface between two media with refractive indices n 1 and n 2 respectively. It is know that a part of the incident wave will be transmitted into medium 2 and a part reected back at medium 1. If E i exp [i (ωt − k i .r)], E r exp [i (ωt − k r .r)] and E t exp [i (ωt − k t .r)] are the eld amplitudes of the incident plane wave, reected and transmitted waves respectively, following the boundary conditions all the three propagation vectors k i ,k r and k t must lie in a plane. This plane where the propagation vectors of incident, reected and transmitted wave lie is known as the plane of incidence. In such a plane, the tangential components θ i ,θ r and θ t of the wave vectors with regard to the normal of the plane interface satisfy the following condition, n 1 sinθ i = n 1 sinθ r = n 2 sinθ t Eqn (i) These kinematic properties hold true for many types of propagation vectors such as light waves, sound waves, matter waves etc.,. But, the dynamical properties such as the intensity of the reected and transmitted waves, phase changes and the polarization eect depend on the specic nature of wave propagation and their boundary conditions. Extending the case above to TE polarization in a thin lm with three interfaces at air, thin lm and susbstrate, −→ E (x) for the s wave is a continuous function of x. However when decomposed into the right and left travelling components (Ref. Fig.5.1), they are no longer continuous at the interfaces. The amplitude of the eld at the left and right side of the interface at x = 0 can be represented as column vectors and linked by ( ( A ′ 2 B ′ 2 ( ) ( A 1 = D1 −1 D 2 B 1 )=P 2 ( ) ( A 2 = B 2 ) ( A 2 =D2 −1 D 3 B 2 A ′ 2 B ′ 2 A ′ 3 B ′ 3 )≡ D 12 ( e iφ 2 0 0 e −iφ 2 )≡D 23 ( A ′ 2 B ′ 2 )( A ′ 3 B ′ 3 ) ) A 2 B 2 ) Eqn (ii) Eqn (iii) Eqn (iii) where D 1 ,D 2 and D 3 are the dynamical matrix which relates the refractive index 188
and the tangential components of the wave vectors in medium 1,2 and 3 and is given by, D α = ( ) 1 1 n α cosθ α −n α cosθ α Eqn (iv) tel-00916300, version 1 - 10 Dec 2013 where α = 1, 2, 3 and θ α is the EM wave angle in each of the media which is related to β and k αx by, ω β= n α c sinθ ω α, k αx = n α c cosθ α Eqn (v) P 2 is the propagation matrix which acounts for the propagation of the EM wave through the thickness of the lm and φ 2 is given by k 2x d. The matrices D 12 and D 23 may be regarded as the transmission matrices which link the amplitude of the eld on the two sides of the interfaces as: ⎛ ( 1 1 + k ) ( 2x 1 1 − k ) ⎞ 2x D 12 = ⎜ 2 k 1x 2 k 1x ( ⎝ 1 1 − k ) ( 2x 1 1 + k ) ⎟ 2x ⎠ Eqn (vi) 2 k 1x 2 k 1x ⎛ ( 1 1 + k ) ( 3x 1 1 − k ) ⎞ 3x D 23 = ⎜ 2 k 2x 2 k 2x ( ⎝ 1 1 − k ) ( 3x 1 1 + k ) ⎟ 3x ⎠ 2 k 2x 2 k 2x Similarly in the case ⎛of TM ( polarized ) waves, ( ) ⎞ 1 1 + n2 2k 1x 1 1 − n2 2k 1x D 12 = ⎜ 2 n 2 1k 2x 2 n 2 1k 2x ( ) ( ) ⎟ ⎝ 1 1 − n2 2k 1x 1 1 + n2 2k 1x ⎠ 2 n 2 1k 2x 2 n 2 1k 2x Eqn (vii) Eqn (viii) ⎛ ( ) ( ) ⎞ 1 1 + n2 3k 1x 1 1 − n2 3k 1x D 23 = ⎜ 2 n 2 2k 2x 2 n 2 2k 2x ( ) ( ) ⎟ ⎝ 1 1 − n2 3k 1x 1 1 + n2 3k 1x ⎠ Eqn (ix) 2 n 2 2k 2x 2 n 2 2k 2x The expressions for D 12 and D 23 are similar, except the fact that they are represented by suitable subscripts referring to their corresponding media. D 12 can be formally written as, ( ) D 12 = 1 1 r 12 Eqn (x) t 12 r 12 1 where t 12 and r 12 are the Fresnel transmission and reection coecients given by, r 12 = k 1x − k 2x and r 12 = n2 1k 2x − n 2 2k 2x for s wave and p wave respectivelyEqn k 1x + k 2x n 2 1k 2x + n 2 2k 2x (xi) 189
- Page 155 and 156: Chapter 5 Photoluminescence emissio
- Page 157 and 158: As seen from gure 5.1, a part of th
- Page 159 and 160: function of wavelength consists of
- Page 161 and 162: tel-00916300, version 1 - 10 Dec 20
- Page 163 and 164: In the case of our multilayers, the
- Page 165 and 166: ⎛ ⎜ ⎝ A ′ 2 B ′ 2 1 ⎞
- Page 167 and 168: dN 3 dt = N 2 τ 23 − (σ em. φ
- Page 169 and 170: Figure 5.12: The shapes of k(λ) an
- Page 171 and 172: 5.4 Discussion on the choice of inp
- Page 173 and 174: tting operations show the presence
- Page 175 and 176: (a) 270nm. (b) 300nm. tel-00916300,
- Page 177 and 178: (a) σ emis.max. = 8.78 x 10 −18
- Page 179 and 180: (a) 50(3/1.5) (b) 50(3/3) tel-00916
- Page 181 and 182: (a) Integrated population of excite
- Page 183 and 184: two kinds of emitters in SRSO subla
- Page 185 and 186: Conclusion and future perspectives
- Page 187 and 188: 4. Investigating the origin of phot
- Page 189 and 190: Bibliography [Abeles 83] B. Abeles
- Page 191 and 192: [Carlson 76] D. E. Carlson & C. R.
- Page 193 and 194: [Di 10] D. Di, I. Perez-Wur, G. Con
- Page 195 and 196: [Gritsenko 99] V. A. Gritsenko, K.
- Page 197 and 198: [Kaiser 56] W. Kaiser, P. H. Kech &
- Page 199 and 200: [Mandelkorn 62] J. Mandelkorn, C. M
- Page 201 and 202: [Pavesi 00] L. Pavesi, L. D. Negro,
- Page 203 and 204: [Sopori 96] B. L. Sopori, X. Deng,
- Page 205: [Weng 93] Y. M. Weng, Zh. N. fan &
- Page 209 and 210: Appendix II Trials σ em.max (x10
- Page 211: Résumé tel-00916300, version 1 -
and the tangential components of the wave vectors in medium 1,2 and 3 and is given<br />
by,<br />
D α =<br />
(<br />
)<br />
1 1<br />
n α cosθ α −n α cosθ α<br />
Eqn (iv)<br />
tel-00916300, version 1 - 10 Dec 2013<br />
where α = 1, 2, 3 and θ α is the EM wave angle in each of the media which is<br />
related to β and k αx by,<br />
ω<br />
β= n α<br />
c sinθ ω<br />
α, k αx = n α<br />
c cosθ α<br />
Eqn (v)<br />
P 2 is the propagation matrix which acounts for the propagation of the EM wave<br />
through the thickness of the lm and φ 2 is given by k 2x d. The matrices D 12 and D 23<br />
may be regar<strong>de</strong>d as the transmission matrices which link the amplitu<strong>de</strong> of the eld<br />
on the two si<strong>de</strong>s of the interfaces as:<br />
⎛ (<br />
1<br />
1 + k ) (<br />
2x 1<br />
1 − k ) ⎞<br />
2x<br />
D 12 = ⎜ 2 k 1x 2 k 1x (<br />
⎝ 1<br />
1 − k ) (<br />
2x 1<br />
1 + k ) ⎟<br />
2x<br />
⎠<br />
Eqn (vi)<br />
2 k 1x 2 k 1x<br />
⎛ (<br />
1<br />
1 + k ) (<br />
3x 1<br />
1 − k ) ⎞<br />
3x<br />
D 23 = ⎜ 2 k 2x 2 k 2x (<br />
⎝ 1<br />
1 − k ) (<br />
3x 1<br />
1 + k ) ⎟<br />
3x<br />
⎠<br />
2 k 2x 2 k 2x<br />
<strong>Si</strong>milarly in the case ⎛of TM ( polarized ) waves, ( ) ⎞<br />
1<br />
1 + n2 2k 1x 1<br />
1 − n2 2k 1x<br />
D 12 = ⎜ 2 n 2 1k 2x 2 n 2 1k 2x ( ) ( ) ⎟<br />
⎝ 1<br />
1 − n2 2k 1x 1<br />
1 + n2 2k 1x<br />
⎠<br />
2 n 2 1k 2x 2 n 2 1k 2x<br />
Eqn (vii)<br />
Eqn (viii)<br />
⎛ ( ) ( ) ⎞<br />
1<br />
1 + n2 3k 1x 1<br />
1 − n2 3k 1x<br />
D 23 = ⎜ 2 n 2 2k 2x 2 n 2 2k 2x ( ) ( ) ⎟<br />
⎝ 1<br />
1 − n2 3k 1x 1<br />
1 + n2 3k 1x<br />
⎠<br />
Eqn (ix)<br />
2 n 2 2k 2x 2 n 2 2k 2x<br />
The expressions for D 12 and D 23 are similar, except the fact that they are represented<br />
by suitable subscripts referring to their corresponding media. D 12 can be<br />
formally written as,<br />
( )<br />
D 12 = 1 1 r 12<br />
Eqn (x)<br />
t 12 r 12 1<br />
where t 12 and r 12 are the Fresnel transmission and reection coecients given<br />
by,<br />
r 12 = k 1x − k 2x<br />
and r 12 = n2 1k 2x − n 2 2k 2x<br />
for s wave and p wave respectivelyEqn<br />
k 1x + k 2x n 2 1k 2x + n 2 2k 2x<br />
(xi)<br />
189
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