Layered structures: transfer matrix formalism Petr Kužel • Interfaces between LHI media • Transfer matrix formalism • Practically only one formula is to be known in order to calculate any structure • Applications: • Antireflective coatings • Dielectric mirrors, Chirped mirrors • Laser output couplers • Beam-splitters • Beam-splitting mirrors • Interference filters Transfer matrix formalism E TE polarization H + − E k k 0 0 α α 0 0 n0 H E = + Ei0 r0 E01 Et1 Es1 y E01 E η H = ()E − E γ (tangential) Et1 s1 0 01 t1 s1 n1 x
α z 1 E
H − + E = E + E k k 12 ()i1 r1 (tangential) E 1 1 i1 Er1 η = − γ 0H12 Ei1 Er1 E12
n2 …
− δ γ = cosα E = E e i n1 1 i1 t1 2π −iδ δ = d k = n d cosα E = E e 1 z1 λ 1 1 1 s1 r1 Introduction of transfer matrix
isin δ E01 cosδ E12 = γ η H η H 0 01 iγ sin δ cosδ 0 12 Transfer matrix connects tangential fields on both ends of a layer For j-th layer: isin δ m m cosδ j = 11 12 = j γ M j m21 m22 γ δ δ i j sin j cos j
For the whole structure:
E E + E + 01 = M M M N ,N 1 = M N ,N 1 η 1 2 ! N η tot η 0H01 0H N ,N +1 0H N ,N +1 Reflection and transmission coefficients
E E + E E E + 01 = i0 r0 = M t = M N ,N 1 η − γ tot γ tot η 0H01 (Ei0 Er0 ) 0 t Et 0H N ,N +1
γ m + γ γ m − m − γ m r = 0 11 0 t 12 21 t 22 γ + γ γ + + γ 0m11 0 t m12 m21 t m22 2γ t = 0 γ + γ γ + + γ 0m11 0 t m12 m21 t m22
γ = α δ = ω α polarisation TE: j n j cos j j n j d j cos j / c γ = α δ = ω α polarisation TM: j n j cos j j n j d j cos j / c γ = δ = ω normal incidence: j n j j n j d j / c Generalization
The formalism is also valid for − κ • absorbing layers; j-th layer absorbs: Nj = nj i j N 2 − n2 sin2 α α = j 0 0 cos j N j • layers where total reflection occurs; total reflection on j-th layer:
n sin 2 α − n2 α = − 0 0 j cos j i n j δ γ ∆ δ Γ γ j and j become imaginary; one introduces: j = i j and j = i j, ∆ Γ where j and j are real. The transfer matrix becomes:
sh ∆ ch ∆ i j = j Γ M j j − Γ ∆ ∆ i j sh j ch j Applications
Optical elements and coatings are designed for given incidence angle Specific layer thicknesses are frequently used in stacks: • quarter-wave (λ/4) layer π π δ = 2 α = 0 i γ n1d1 cos 1 M = λ %"$ "# 2 γ λ 4 i 0 •half-wave (λ/2) layer
2π −1 0 δ = n d cosα = π M = λ %1"$1 "#1 0 −1 λ 2 • quarter-wave (λ/4-λ/4) bilayer
0 i γ 0 i γ − γ γ 0 M = 1 2 = 2 1 γ γ − γ γ i 1 0 i 2 0 0 1 2 Antireflective single layer Let’s try λ/4-layer (then waves with λ/2 phase delay will interfere)
γ = 1 0 i 1 n0 M = γ λ/4 = i 1 0 MgF2 n1 1.38
= BK7 glass ns 1.51 γ γ − γ2 n − n2 r = 0 s 1 = s 1 γ γ + γ2 + 2 0 s 1 ns n1 6
ns = 1.51
n = n 4 glass (not treated) 1 s Antireflective coating (for 550 nm) R (%) λ/4 Broadband 2 Less efficient (no degree 0 of freedom for n1) 400 500 600 700 800 Wavelength (nm) Antireflective bilayer
• quarter-wave (λ/4-λ/4) bilayer − γ γ 2 1 0 λ /4 CeF3 n = 1.65 M = 1 − γ γ λ = 2.1 0 1 2 /4 ZrO2 n2
2 2 2 2 BK7 glass n = 1.51 γ γ − γ γ n − n n s r = 2 0 s 1 = 2 s 1 γ2γ + γ γ2 2 + 2 2 0 s 1 n2 nsn1 6 n2 n = 1.51 = n s n s 1 glass (not treated) Antireflective coating 4 (for 550 nm) V-like shape (narrow λ/4
R (%) λ/4-λ/4 frequency range) 2 More efficient (one degree of freedom for n1, n2) 0 400 500 600 700 800 Wavelength (nm) Broadband AR coating
• trilayer structure (λ/4-λ/4-λ/4) γ 0 − i 2 γ γ M = 1 3 γ γ λ/4 = 1 3 n 1.38 − i 0 1 γ λ = 2 /4 n2 2.0 λ = /4 n3 1.8 γ2γ γ − γ2γ2 n2n − n2n2 r = 2 0 s 1 3 = 2 s 1 3 γ2γ γ + γ2γ2 n2n + n2n2 2 0 s 1 3 2 s 1 3 = BK7 glass ns 1.51
n1n3 = ns n2 Broadband AR coating
• trilayer structure (λ/4-λ/2-λ/4): similar to a quarter-wave bilayer at the resonant wavelength • half-wave layer helps to extend the antireflective range
γ γ λ = 0 /4 1.38 n M = 3 1 1 γ γ 0 1 3 λ = /2 n2 2.2
2 2 2 2 − γ γ + γ γ n n − n λ/4 n = 1.7 r = 3 0 s 1 = s 1 3 3 γ2γ + γ γ2 2 + 2 3 0 s 1 n3 nsn1 BK7 glass n = 1.51 n3 = s ns n1 Antireflective coating: summary
10 Antireflective coating for 550 nm λ 8 /4 λ/4-λ/4 λ/4-λ/2 n = 1.51 6 λ/4-λ/4-λ/4 s λ/4-λ/2-λ/4
R (%) glass (not treated) 4
2
0 400 500 600 700 800 Wavelength (nm) Dielectric mirrors λ /4 n 1 bilayer (n << n ): L H H λ /4 n − n n 0 L M = L H − 0 n n n H L H
N bilayers: nL
()− n n N 0 M = L H N ()− 0 nH nL
n H
Reflectivity: nL
2 ()1 n ()n n 2N −1 R = s L H n ()()2N + s 1 ns nL nH 1 Dielectric mirrors: example
1 1 0.6 0.8 0.2 -0.2
Phase change -0.6 0.6 -1 0.9×104 1.1×104 1.3×104 1.5×104
R Wavenumber (cm-1) 0.4 Dielectric mirrors for 800 nm
Number of bilayers (ZnS/MgF2): 0.2 20 8 3 0 400 600 800 1000 1200 1400 1600 Wavelength (nm) Chirped dielectric mirrors The resonant wavelength is linearly tuned along the stack of bilayers • Different wavelengths are reflected at different depths → different optical paths • Adding or compensating of a chirp of the pulses
0.3 Mirrors for 800 nm (12500 cm-1) (20 bilayers) 0.2 Standard dielectric mirror Chirped mirror (10 nm step) 0.1
0
-0.1 Departure from the linear phase shift phase linear the from Departure -0.2 1.1×104 1.2×104 1.3×104 1.4×104 Wavenumber (cm-1) Laser output coupler
λ = 514.5 nm
→ R0 1 Rx < 1
Rx
Reflective (~ 90%) coating
AR coating Beamsplitter
n , d R R 20%
n , d 1 1 45° n s n , d 2 2
80% Beamsplitting mirror
• Separation of harmonic frequencies (e.g. Nd:YAG fundamental beam at 1064 nm and its second harmonic at 532 nm) • Long-pass or cut-off filters Example of solution: 1) Stack of λ/4 bilayers forming a dielectric mirror for 1064 nm These layers are λ/2 for the second harmonics so it passes through unchanged 2) We add below an AR coating for 532 nm 1064 nm component does not penetrate down to these layers Result: 1064 nm is reflected, 532 nm is transmitted • Sometimes a detuning of a resonant wavelength is used in several layers of the HR coating: it can smooth the unwanted interference maxima and minima. Interference band-pass filters
Contain: • Stacks of high-reflecting bilayers • Antireflective coatings • Fabry-Pérot cavities • Detuning of the resonant wavelength is also often used for smoothing of the interferences