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Prism ... diffraction limited by entrance aperture
Chromatic Resolving Power
�
��⇥ B · dn
d�
Resolving Power
chromatic resolving powerFinesseinstrumental widthinstrumental line profile ...
Spectroscopic Resolution – some alternatives
�/��
F = FSR/FWHM
�⇥̄ = ��/�2
apparent spectral energy distribution for monochromatic light entering the device
f
FWHM
transmission
�� or
Multi-Layer Coatings
metalreflectance
reflectance ofcommercial
multi-layer coatings
Multi-Layer Coatings
anti-reflection coating
Anti-Reflection Coating
• reflected waves out of step• equal amplitudes } destructive interference
Multi-Layer Coatings
single layer ➙ FPI
l
nT = nglassn1 = nlayern0 = nair
Multi-Layer Coatings
single layer ➙ circulating wave
l
nT = nglassn1 = nlayern0 = nair
incoming
reflected
transmitted
E�0, H
�0 E�
1, H�1
ET , HTE1, H1E0, H0
Multi-Layer Coatings
single boundaryn1 = nlayern0 = nair
E�0, H
�0 E�
1, H�1
E1, H1E0, H0
from Maxwell‘s equations
Z =Ex
Hy=
�µµ0
��0=
Z0
n
continuity of E and H
E0 + E�0 = E1 + E�
1
H0 + H �0 = H1 + H �
1
ExHy
kz
E’x
H’y
k’z
Anti-Reflection CoatingnT = nglassn1 = nlayern0 = nair
E�0, H
�0 E�
1, H�1
ET , HTE1, H1E0, H0
�/4
Reflection
�E0
E�0
⇥= M01 · M�/4 · M1T ·
�ET
0
⇥= � i
2n1ET
�nT + n2
1
nT � n21
⇥
n1 �⇥
nTforR =����E�
0
E0
����2
=����nT � n2
1
nT + n21
����2
�⇥ 0
Anti-Reflection CoatingnT = nglassn1 = nlayern0 = nair
E�0, H
�0 E�
1, H�1
ET , HTE1, H1E0, H0
�/4
n1 �⇥
nTfor
Compare to Fabry-Perot Interferometer
R =����E�
0
E0
����2
=����nT � n2
1
nT + n21
����2
�⇥ 0
Iout = Imax
⇤1 +
�2F⇤
sin�
2
⇥2⌅�1
with � = 2kl cos ⇥
for � = 0 =� Iout = Imax if p⇥ = 2l l = �/2
l = �/4
AR-layer
FPI
Anti-Reflection CoatingnT = nglassn1 = nlayern0 = nair
E�0, H
�0 E�
1, H�1
ET , HTE1, H1E0, H0
�/4
n1 �⇥
nTfor
Compare to Fabry-Perot Interferometer
R =����E�
0
E0
����2
=����nT � n2
1
nT + n21
����2
�⇥ 0
Iout = Imax
⇤1 +
�2F⇤
sin�
2
⇥2⌅�1
with � = 2kl cos ⇥
for � = 0 =� Iout = Imax if p⇥ = 2l l = �/2
l = �/4
AR-layer
FPIphase jumps by π when reflecting of th
e optically thicker medium
Anti-Reflection Coating
reflection vs. layer thickness
Multi-layer Stack�
E0
E⇥0
⇥=
⇤k⇧
l=1
Ml�1,l · M�/4
⌅· MnT ·
�ET
0
⇥
R =����(n1n3n5...nk�1)2nT � n0(n2n4...nk)2
(n1n3n5...nk�1)2nT + n0(n2n4...nk)2
����2
R =����(n1n3n5...nk)2 � n0nT (n2n4...nk�1)2
(n1n3n5...nk)2 + n0nT (n2n4...nk�1)2
����2
k odd
k even
Anti-Reflection Coating
Dielectric Mirror
reflection vs. number of layers
n1 = 2.3 and n2 = 1.35
reflectivities up to 99.9999%
Interference filter; composed of multi-layer stacks
Multi-Layer Coatings
Interference Filter
single line
band pass
notch
edge
Impedance ... a sleeker way
�⌃ · �B = 0 �⌃ · �D = ⇤
�⌃⇤ �E = � ��t
�B �⌃⇤ �H = �j + ��t
�D
�D = �r�0 �E �B = µrµ0�H
c = 1/⇧
µ0�0 n = ⇧µr�r ⌅⇧
�r
Maxwell‘sequations
flux densities
speed
⇥2 ⌦E =1v2
⇤2 ⌦E
⇤t2; ⇥2 ⌦H =
1v2
⇤2 ⌦H
⇤t2; v =
c
n=
1�
µrµ0�r�0
wave equations for E and H
⇥2 ⌦E =1v2
⇤2 ⌦E
⇤t2; ⇥2 ⌦H =
1v2
⇤2 ⌦H
⇤t2; v =
c
n=
1�
µrµ0�r�0
�E(�r, 1) = �E0 ei(⌃k·⌃r��t); |�k| = k =2⇥
�; v =
⇤
k=
c
n
plane wave (same for H)
from Maxwell‘s equations: � ⇧B = ⇧k � ⇧E
Ex(z, t) = E0 ei(kz��t) =�
Ex
Hy= Z � k
|k| with Z =�
µrµ0
�r�0⇥ Z0
n
Impedance ... a sleeker way
Ex
Hy= Z � k
|k| with Z =�
µrµ0
�r�0⇥ Z0
n
Impedance ... a sleeker way
impedance Z; vacuum impedance Z0 = 377 Ω
continuity
E2 + E�2 = E3 + E�
3
H2 + H �2 = H3 + H �
3
E2
E�2
=Z3 � Z2
Z3 + Z2
Ex = E2 ei(k2z��t)
Hy = E2/Z2 ei(k2z��t)
H ⇥y = �E⇥
2/Z2 ei(�k2z��t)
E⇥x = E⇥
2 ei(�k2z��t)
0-l z
Z3Z1 Z2
Impedance ... a sleeker way
E2
E�2
=Z3 � Z2
Z3 + Z2
Ex = E2 ei(k2z��t)
Hy = E2/Z2 ei(k2z��t)
H ⇥y = �E⇥
2/Z2 ei(�k2z��t)
E⇥x = E⇥
2 ei(�k2z��t)
0-l z
Z3Z1 Z2
ZL =�
Ex + E�x
Hy + H �y
⇥
0
=E2 + E�
2
E2 � E�2
Z2load impedance at z=0
input impedance at z=-l(for a λ/4 layer) Zin =
�Ex + E⇥
x
Hy + H ⇥y
⇥
�l
�⇥ Z22
ZL
at boundaries
ZL,m = Zin,m+1
Impedance ... a sleeker way
E2
E�2
=Z3 � Z2
Z3 + Z2
Ex = E2 ei(k2z��t)
Hy = E2/Z2 ei(k2z��t)
H ⇥y = �E⇥
2/Z2 ei(�k2z��t)
E⇥x = E⇥
2 ei(�k2z��t)
0-l z
Z3Z1 Z2
ZL =�
Ex + E�x
Hy + H �y
⇥
0
=E2 + E�
2
E2 � E�2
Z2load impedance at z=0
input impedance at z=-l(for a λ/4 layer) Zin =
�Ex + E⇥
x
Hy + H ⇥y
⇥
�l
�⇥ Z22
ZL
at boundaries
ZL,m = Zin,m+1
Impedance ... a sleeker way
ZL =�
Ex + E�x
Hy + H �y
⇥
0
=E2 + E�
2
E2 � E�2
Z2load impedance at z=0
input impedance at z=-l(for a λ/4 layer) Zin =
�Ex + E⇥
x
Hy + H ⇥y
⇥
�l
�⇥ Z22
ZL
at boundaries
ZL,m = Zin,m+1
input impedance at z=-l(for a λ/2 layer) Zin = ZL
Stack of λ/4 Layers
ZL =�
Ex + E�x
Hy + H �y
⇥
0
=E2 + E�
2
E2 � E�2
Z2load impedance at z=0
input impedance at z=-l(for a λ/4 layer) Zin =
�Ex + E⇥
x
Hy + H ⇥y
⇥
�l
�⇥ Z22
ZL
Zin,k = Z2k/ZL,k ⇥� ZL,k = ZT
...Zin,n = Z2
n/ZL,n ⇥� ZL,n = Zin,n+1...
E0 + E�0
E0 � E�0
Z0 = Zin ⇥� ZL = Zin = Zin,1
from the far side ....
Stack of λ/4 Layers
Zin,k = Z2k/ZL,k ⇥� ZL,k = ZT
...Zin,n = Z2
n/ZL,n ⇥� ZL,n = Zin,n+1...
E0 + E�0
E0 � E�0
Z0 = Zin ⇥� ZL = Zin = Zin,1
from the far side ....
ReflectivityR =
����E�
0
E0
����2
=����Zin � Z0
Zin + Z0
����2
AR-Coating ⇔ impedance matching Z0=Zin
Mirror ⇔ impedance mismatch: Zin=0 or Zin >> Z0
Multi-layer Stack�
E0
E⇥0
⇥=
⇤k⇧
l=1
Ml�1,l · M�/4
⌅· MnT ·
�ET
0
⇥
R =����(n1n3n5...nk�1)2nT � n0(n2n4...nk)2
(n1n3n5...nk�1)2nT + n0(n2n4...nk)2
����2
R =����(n1n3n5...nk)2 � n0nT (n2n4...nk�1)2
(n1n3n5...nk)2 + n0nT (n2n4...nk�1)2
����2
k odd
k even
Multi-layer Stack
k=2p+1 (odd)
k=2p (even)Zin = (Z2/Z3)2pZT
Zin = (Z2/Z3)2p Z22/ZT
R =����E�
0
E0
����2
=����Zin � Z0
Zin + Z0
����2
Summary & Outlook
Multi-Layer CoatingsAR Coatings / FPI ComparisonImpedance & multiple layers
Polarisationlinear, circular, elliptical
Polarisation opticsuni-axial crystalspolarising prismswave plates (λ/2 and λ/4)
Interference with polarised light
