Optical Spectroscopy with
dispersive Spectrometers
Basics  Building Blocks  Systems  Applications
by Wilfried Neumann
COLLECTION
Collected are
1) overlook of topics,
2) all parameters, indicators,
and symbols,
3) energy conversions,
4) formulas and
equations collected
1) Content / Topics of the Basics Pages 1  6
Index 


Topic 







Index 


Basics 1  Gratings 

1. 
0. 
2 

Introduction, Goal of the Basics 1 

1. 
1. 
0 

Grating functions, Basic equation F1 

1. 
1. 
1 

Free spectral range, equation F2 

1. 
2. 
0 

Dispersion by gratings, equation F3 

1. 
3. 
0 

Definition of spectral resolution, equation F4 & F5 

1. 
4. 
0 

Diffraction efficiency and polarization 

1. 
4. 
1 

Several dispersion element, kinds of gratings 

1. 
4. 
6 

The Prism, equation F6  F9 

1. 
5. 
0 

More grating features 

1. 
5. 
3 

Scattering effects 

1. 
5. 
4 

Grating ghosts 

1. 
5. 
5 

Shadowing and Diffusion 

1. 
5. 
4 

Coatings 




Basics 2  Spectrometer Concepts and Parameters 

2. 
0. 
0 

Introduction, Goal of the Basics 2 

2. 
1. 
1 

Basic principal of a spectrometer setup 

2. 
1. 
2 

Attributes of modular spectrometers 

2. 
2. 
1 

Littrow spectrometer, equation F11 and F13 

2. 
3. 
1 

Ebert spectrometer, equation F12, F14 and F15 

2. 
3. 
2 

Curved slits versus straight slits 

2. 
4. 
0 

CzernyTurner spectrometer, equation F12, F14 and F15 

2. 
4. 
0. 
1 
Angles inside a spectrometer 

2. 
4. 
1 

Imaging flat field and its correction 

2. 
4. 
6 

Influence of the internal angles on the wavelength 

2. 
5. 
1 

Vacuum spectrometers 

2. 
5. 
2 

Normal Incidence setup 

2. 
5. 
1 

SeyaNamioka setup 

2. 
5. 
1 

Grazing Incidence setup 

2. 
6. 
1 

Grating rotation and actuation 

2. 
7. 
0 

Aperture and light flux (luminosity), Omega 

2. 
7. 
2 

Impact of internal angles on the luminosity 

2. 
7. 
5 

calculations of fnumber vs Omega 

2. 
8. 
1 

Dispersion and calculation, equation F 18  F 21 

2. 
9. 
1 

Intensity distribution in the exit 

2. 
10. 
1 

Spectral resolution 

2. 
10. 
4 

Raleigh diffraction Limit, equation F 22 

2. 
10. 
5 

Resolution of a monochromator compared to a spectrograph 

2. 
10. 
6 

Numerical resolution, and the dependence on wavelength 

2. 
10. 
7 

Practical example on the optimization of resolution 

2. 
11. 
1 

Image quality and methods of correction, equation F 23  F25 

2. 
12. 
1 

False light, stray light, and contrast 

2. 
13. 
1 

Double pass, double and triple spectrometers 

2. 
14. 
1 

Efficiency behaviour and analysis 

2. 
14. 
2 

Energy transmission and bandwidth of single, double, and triple stage spectrometers 

2. 
14. 
3 

Effects of photon travelling time (Time of Flight) 

2. 
15. 
1 

Stability and thermal influence, equation F26 & 27 

2. 
16. 
1 

Spectral orders and filtering 

2. 
16. 
6 

Dispersion of a prism spectrometer, equation F 7, F9, F10 

2. 
17. 
1 

Light transfer by optical fibres 

2. 
18. 
1 

Collection of performance parameters of spectrometers 




Basics 3  Configurations of Monochromators and Spectrographs 

3. 
1. 
1 

Beam travel in EbertFastie and CzernyTurner configurations 

3. 
1. 
4 

Output wavelength in dependence of the source position 

3. 
1. 
5 

Dispersion behaviour over the output field of a spectrograph 

3. 
1 
7 

Strahlvergrößerung bei wechselnder Fokuslänge und Apertur 

3. 
2. 
1. 
1 
Principles of additive double spectrometers 

3. 
2. 
1. 
2 
Principles of subtractive double spectrometers 

3. 
2. 
2 

Modern additive double spectrometers 

3. 
2. 
3. 
2 
Modern subtractive double spectrometers 

3. 
2. 
3. 
3 
Mechanical filtering in double spectrometers 

3. 
2. 
5 

Triple stage spectrometers 

3. 
3. 
0 

The Prism spectrometer 

3. 
4. 
0 

Transmissions spectrometers 

3. 
5. 
0 

Echelle spectrometers 




Basics 4  Detectors for Monochromators and Spectrographs 

4. 
1. 
0 

Introduction, Goal of the Basics 4 

4. 
1. 
1 

The definition of work and power of light signals 

4. 
1. 
2 

Basic parameters of detectors, equation F 28 

4. 
2. 
1 

Photo tubes (PMT) 

4. 
2. 
4 

Photon counters 

4. 
2. 
5 

UV phototubes and scintillators 

4. 
3 


The principles of detector illumination 

4. 
4 


Channeltron and micro channel plate (MCP) 

4. 
5 


Image intensified PMT and single photon counter 

4. 
6 


Solid state detectors (semi conducters) 

4. 
6. 
1 

Planck´s radiation / Blackbody radiation 

4. 
6. 
2 

Background radiation, equation F29 

4. 
6. 
6 

Synchronized measurements (LockIn technique) 

4. 
6. 
9 

Tandem detectors / Sandwich 

4. 
6. 
10 

Solid state detector parameters 

4. 
6. 
11 

Illumination of small detector surfaces 

4. 
6. 
12 

Memory effects, recombination, hold time of semiconductor detectors 

4. 
6. 
13 

PIN and Avalanche diodes 

4. 
7 


Fibre optic coupling 

4. 
8 


Area detectors: CCD and Array, equation F 30 

4. 
8. 
6 

Time control / time steering, synchronization, shuttering, gating of CCD 

4. 
8. 
8 

Readout techniques of CCD 

4. 
8. 
9 

CCD/Array systems with image intensification 

4. 
8. 
10 
1 
Measurements in the µs regime 

4. 
8. 
10 
2 
Double pulse application 

4. 
9 


Other area detectors: CID, CMOS, NIR, IR, PSD 

4. 
10. 
1 

Exponential function, damping, filtering 

4. 
10. 
3 

The different definition of the bandwidth in electric versus optical systems 







Index 


Basics 5  Illumination  Transfer  Radiometry 

5. 
0. 
0 

Introduction, Goal of the Basics 5 

5. 
0. 
1 

The definition of work and power of light signals 

5. 
0. 
2 

The models of light distribution of different sources, collection 

5. 
0. 
3 

Definitions and nomenclature, equations F 32, F 33 

5. 
1. 
1 

Laser light sources and collection 

5. 
1. 
2 

Cone shaped light sources and collection 

5. 
1. 
3 

Spherical light sources, point sources, and collection 

5. 
1. 
4 

Diffuse, undirected light sources, integrating spheres 

5. 
1. 
4. 
2 
The collection of light from lamps 

5. 
1. 
5 

NIR radiators 

5. 
1. 
6 

IR sources 

5. 
2 


Optimization of a spectrometer, regarding the light source 

5. 
3. 


Transfer and coupling by fibre optics, equation F36 

5. 
3. 
1 

Transfer characteristics of fibre optics 

5. 
3. 
2 

Fibre optic parameters 

5. 
3. 
4 

Typical versions of fibre optic cables 

5. 
4. 


Optical Transfer and Coupling, F23C 

5. 
4. 
1 

Transfer and coupling by Fibre Optics only 

5. 
4. 
2 

Transfer and coupling via lens systems 

5. 
4. 
3 

Transfer and coupling through mirrors 

5. 
5 


Radiometry 

5. 
5. 
1 

Radiometry  measurement of the spectral radiant power (spectral radiant flux) 

5. 
5. 
2 

Radiometry  measurement of the spectral irradiance E, and the radiance L 

5. 
5. 
3 

Radiometric sample illumination 

5. 
5. 
3. 
4 
Radiometric calibration and validitation 

5. 
5. 
3. 
5 
Examples on radiometric measurements and illumination 







Index 


Basics 6  Applications A1 and A2 Transmission  Reflection 

A1. 
0. 
1 

Principles of the measurements of Absorption/Extinction and Reflection 

A1. 
1. 
1. 
1 
the optimum spectro photometer, and standard systems 

A1. 
1. 
1. 
3 
Spectro photometers with parallel detection 

A1. 
1. 
1. 
4 
a universal sample compartment for double beam spectro photometry 

A1. 
1. 
1. 
5 
Calibration and stray light definition 

A2. 
1 


Dynamic and kinetic measurements 







Index 


Basics 6  Applications A3 special Absorption Techniques 

A3. 
1. 


Atomic Absorption Spectroscopy AAS 

A3. 
2. 
1. 

the measurement of circular dichroism CD 

A3. 
2. 
2. 

the measurement of optical rotational dispersion ORD 

A3. 
3. 


special techniques for scattered transmission ST 

A3. 
4. 


photoacoustic (optoacoustic) spectroscopy PAS / OAS 







Index 


Basics 6  Applications C1 Calibration Methods 

C1. 
A. 


Calibration of the wavelength axis (x axis) 

C1. 
1. 


Definition of the angular position of dispersion elements 

C1. 
2. 


Control and driving system of a grating and a prism spectrometer 

C1. 
2. 
1. 

Grating spectrometer with linear drive and sine function 

C1. 
3. 


Grating spectrometer with rotational drive 

C1. 
4. 


Calibration of the field output 

C1. 
B. 


Calibration of the intensity axis (y axis) 

C1. 
5. 
1. 

Requirements for the portability of calibration data 

C1. 
5. 
2. 

Light sources for calibration 

C1. 
5. 
3. 

Intensity calibration step by step 







Index 


Basics 6 
Applications E1 Spectroscopic Ellipsometry and 

E1. 
0. 
1 

Applications of SE 

E1. 
0. 
2. 

the principal of SE 

E1. 
2. 


Calculations and equations of SE 

E1. 
3. 


When to prefer single wavelength/when spectral ellipsometry 

E1. 
5. 


SE with parallel detection 

E1. 
8. 


Examples of SE measurements 

E1. 
11. 


literature 

E1. 
13. 


SE extension towards the UV 

E1. 
14. 


SE extension towards the IR 







Index 


Basics 6  Applications E2 Emission Spectroscopy 

E2. 
0. 
1. 

Instrumental technology for Emission spectroscopy 

E2. 
0. 
2. 

typical atomic and plasma spectra 

E2. 
0. 
3. 

Echelle spectrometers for Emission spectroscopy 

E2. 
1. 


AES  Atomic Emission Spectroscopy 

E2. 
2. 


CL  Cathodo Luminescence Spectroscopy 

E2. 
3. 


ICP  inductive coupled Plasma Spectroscopy 

E2. 
4. 


FOES  Spark Emission Spectroscopy 

E2. 
5. 


LA  LaserAblation 

E2. 
5. 


LIBS  Laser induced Breakdown Spectroscopy 

E2. 
5. 


LIP  Laser induced Plasma Spectroscopy 

E2. 
5. 


LD  Laser Deposition 

E2. 
5. 


LD  LaserBeschichtung 


E2. 
6. 


PE  Plasma Etching 


E2. 
7. 


LS  Laser Emissions Spectroscopy 


E2. 
8. 


SolEm and StelEm  solar and stellar Emission 


E2. 
9. 


CombEm  Emission Methods at Flames and Explosions 








Index 


Basics 6  Applications L1 and L2 Luminescence  Fluorescence  Lifetime 

L1. 
0. 


Principles of the measurements of Luminescence, Jablonski diagram 

L1. 
0. 
2. 

requirements of luminescence spectroscopy 

L1. 
0. 
3. 

setup of static luminescence spectroscopy systems 

L1. 
0. 
5. 

methods of static luminescence spectroscopy 

L1. 
0. 
9. 

Calibration, stray light definition, methods to compare different systems 

L1. 
0. 
10. 

a research grade example realized 

L2. 
0. 


the definition of luminescence lifetime 

L2. 
0. 
1. 

setup of dynamic luminescence spectroscopy systems, lifetime systems


Index 


Basics 6  Application R2, Raman and Brillouin 



R2. 
0. 


Principle of Raman spectrometers 


R2. 
1. 


Requirements to Raman spectrometers 


R2. 
1. 
1. 

Beam travelling scheme and possible distortions 


R2. 
2. 


Spectral examples 


R2. 
2. 
1. 

Realization and setup, typical Laser wavelengths 


R2. 
2. 
3. 

how close to the Rayleigh line must the measurement reach? 


R2. 
2. 
3. 
1 
Single stage spectrometer with notch filter 


R2. 
2. 
3. 
2 
Double spectrometer 


R2. 
2. 
3. 
3 
Straylight, impact and discussion 


R2. 
2. 
3. 
4 
Brillouin spectrometer 


R2. 
2. 
3. 
4.1 
Echelle spectrometer and free spectral range 


R2. 
2. 
3. 
4.2 
History and alternative Brillouin spectrometers 


R2. 
2. 
3. 
5 
Triple stage spectrometer 


R2. 
2. 
3. 
6 
typical behaviour of straylight in several setups 


R2. 
3. 
1 

Raman and Fluorescence 


R2. 
3. 
2. 

NIR Raman 


R2. 
3. 
3. 
0 
UV Raman 


R2. 
3. 
4. 

Microscopy Raman 


R2. 
3. 
5. 

Resonance Raman, RR 


R2. 
3. 
6. 

Surface enhanced Raman, SERS 


R2. 
3. 
7. 

Coherent Antistokes Raman spectroscopy, CARS 





Basics 6  Application S1, Straylight 



S1. 
0. 


Nomenclature 


S1. 
1. 


Origin of Straylight 


S1. 
2. 
1. 

Impact of Straylight on discrete spectral Signals 


S1. 
2. 
2. 

Impact of Straylight on broadbanded spectral Signals 


S1. 
3. 


Analysis of Straylight 


S1. 
4. 


Minimizing the Impact of Straylight 






Status of April 2012, the Basics are complete 
2) Indicators and Symbols
a
angle of the light, illuminating the grating or
prism, with
respect to N
b
angle of the diffracted or refracted light, leaving the
disperser,
with respect to N
F
the median
grating angle, half the way between
a
and b,
required to calculate spectrometer dispersion/
F
radiant power /
flux
F
the phase
angle/phase shift in phase/modulation lifetime measurements
F
the angle of sample immumination in ellipsometry (SE)
d
inclusion angle of the light at the
disperser, originating from the lateral distance and width of the mirrors
d
phase angle or phase shift ellipsometry
(SE)
D
the imaginary part of ellipsometric data
e_{1}
angle of the grating impinging beam, in a symmetric system, it is half
d
e_{2}
angle of the beam leaving the grating, in a symmetric system, it is half
d
l
wavelength
t
the time constant
W
real and normalized aperture of a spectrometer, from
A / f^{2 }, also called light guide factor,
W
the normalized spread of a light beam, from
a / r^{2}, also called light guide factor, the numbers are identical
with steradiant sr
r
the complex result of ellipsometric data, leading itself to tg
Y and
cos D
Y
the real part of ellisometric data
w
the angular frequency
w
the normalized cone angle of illumination,
required to calculate light transfer factors, valid for spectrometer and
light source
A
the area
A
the light angle inside a prism, in a
reflecting prism the value A/2 is valid
A the Absorbance (Extinction) in
photometric absorption measurements
ADC
analogue to digital converter, also A/DC
B the bandwidth in a gaussian or similar distribution, like
spectroscopic peaks;
in optical systems, the fwhm value defines
B, in electrical systems the value 1/√2
(about 70,71%) is taken, both to international agreements
d
Deflection angle at the prism, must be identical
to
d
in a spectrometer
dB
deciBel, the logarithmic Value of attenuation (damping).
There are different interpretations for VoltagedB and PowerdB
D* the numeric capability of an IR detector
for low signals
e
Base of the natural logarithm, required for efunctions
el elbow,
created by the two beams at a mirror
e^{}
abbreviation for an electron
eV electron Volt, a measure of the energy of a photon
E_{(}_{l}_{) }
Irradiance of a light beam on a
normalized surface
f the
focal length
f the frequency
f_{c}
the angular frquency
fwhm full widht at half maximum
h height of slit
i_{1}
angle of the prism´s incident light, related to N
J work force, mechanical, electrical or
optical, mainly in Ws or Nm
k the grating constant, distance of the grating lines
k
the absorption coefficient of a material
K the thermal
dilatation coefficent
K Kelvin
L Luminosity, light flux in
spectrometers
L_{(}_{l)
}Radiance (beam density, fits the luminosity of
spectrometers)
LN liquid nitrogen
m the spectral order
m the modulation factor
in lifetime measurements by phase/modulation
MCP Micro Channel Plate, also miciro channel plate image
intensifier system
N the normal of a grating or prism
O_{1}
the basic
aberration, an additive distortion to image or focus
O_{ss}
more additive aberration, resulting from straight slits or straight area
reproduction in the exit
E
the additional aberration in a
spectrometer, resulting from the lateral angles, multiplicator
of O_{1}
w the median distance of a mirror to the
centre line / grating centre axis, leads to e
or d
P
the power, electrical or optical, in Watt
PMT
Photo Multiplier Tube, secondary electron multiplier, photo tube
PSD
Phase Sensitive Detector (n the Lockin), and Position Sensitive (counting)
Detector
Q quality
factor, with
X_{r} = real value
Q
the energy of radiationR
die numerische Auflösung, mit R_{r }reale Auflösung und_{ }R_{p}
theoretische Auflösung
r
the radius of curved slits, also the distance of the slit to the
instrument´s centre, only of sense in symmetric setup
r,
r_{p} und r_{s
}the absolute values of arallel and perpendicualr polarization
R numeric resolution, with R_{r }real resolution and
R_{p} theoretical resolution
R normalized reflectance of a
sample
ROI region
of interest, for the read out of an area detector
SNR signal to noise ratio, also S/NR
STD standard deviation
s
the constant of thermal diiffusion
sr steradiant, the spatial angle of a light
viewer or a light beam, compatible with
W
T
Temperature or thermal change
T
the normalized Transmission in photometric applications
w
the distance of the centre of
the mirror (collimating and focusing) to the spectrometer´s centre axis (from
e
or d)
W active grating width,
active mirror width
x the geometric dilatiation as function of thermal
change
y the geometric increase of the foacl
spot, as function of thermal change and dilatation
3) Conversion of Photon Energy
Conversions of
common energy meters in optical spectroscopy
All measures of energy
are derived from the energy of photons, the electron and the speed of light.
Speed of light in vacuum or air (rounded): c = 3 * 10^{8} m/s.
Basic energy of photons: Q = hc / l
= h n
The energy of a free electron: 1 eV = 1,602 * 10^{19} J
The most often used energy meter in optical spectroscopy is the wavelength
l,
represented in nanometers (nm), 1000 nm = 1 µm. In many books we also find the
Ångstroem
(Å),
10 Å
= 1 nm. If it comes to small units, the picometer is preferred, 1000 pm = 1 nm.
Via Q the wavelength l
is connected with the frequency
n,
n
= c /
l
.
In the infrared range (IR) the most common measure is the wavenumber
(cm^{1});
1 cm^{1} = [(10^{7}
/ nm) +0.5 ]
/ 10. A good approximation is
10.000 / cm^{1} = µm and vice versa.
In Raman spectroscopy the relative wavenumber is common. First the wavenumber position of the excitation source is calculated, and the spectra are referred to that point, leading to positive and negative (Stokes and Antistokes) cm^{1} spectra.
In vacuum
ultra violett (VUV) and material research we find the electron Volt eV;
1 eV = [(1,239,546.000
/ nm) +0.5 ] / 1000. An easy conversion is 1 eV = 1240 nm,
1000 nm (1 µm) = 0,8065 eV. The conversion from eV to cm^{1} : eV = cm^{1}
/ 8000.
For Lasers
and interferometers, n
is often specified in Hertz. Again the basis is the speed of light C,
with 3 * 10^{8} m/s. Through l
= 1/C the according wavelength of 1 Hz (1/s) is
l
= 3 * 10^{8} m. Regarding the median range of optical spectroscopy, we
prefer nm (10^{9} m) and GHz (10^{9} *
^{1}/s). The conversion
between frequency and wavelength is: fr = C /
l.
For example 1 µm: fr = (3 * 10^{8}
m/s) / (10^{6} m) = 3 * 10^{14} /s or 3 * 10^{5}
GHz, shortened to
1 nm = 300 / 10^{6} GHz and 10^{6} GHz = 300 / nm.
Transfer of small bandwidths from one unit to another happens by straight rule of three, like
D l / l = D n / n or D eV / eV = D cm^{1} / cm^{1} and so on
4A) Collection of all equations and formulas used. First they are in the order of
topics, red indices
show the point of discussion
The general grating function:
F1: m *
l = k *
(sin a
± sin
b)
(1.1.0)
with: m = spectral
order,
l =
wavelenth, k = grating constant, a
= angle of incidence light, rel. to N,
b =
angle of dispersed light after the grating, rel. to N,
N = grating normal
The grating function in a Littrow spectrometer: F11: m *
l = k *
2 sin a
(2.2.1)
Grating function in EbertFastie and
symmetric CzernyTurner instruments: F12: m *
l = k *
2 sin F
* cos e
(2.3.1)
with:
F
= the median angle at the grating ((a+b)/2), e
= the Ebert angle
Free Spectral Range (FSR):F2A: FSR = l / m, a more precise interpretation is F2B: l_{2} = l_{1} + (l_{1} /m) and FSR = l_{2}  l_{1 } (1.1.1)
The
angular dispersion after the grating: F3: m *
dl = k
* cos b
* db
or
db /
dl
= ( m / ( k * cos b
)) (1.2.0)
with:
dl
= small difference in wavelength after the grating,
db
=
small difference in angle after the grating
The
dispersion in a grating spectrometer: Angular dispersion
F18: f*
(db
/ dl)
= f * m / (k * cos b)
(2.8.1)
with: f = focal length
Calculation of the median grating angle, by Equation F19:
f
= arcsin ( l
/( 2 * k * cos e))
(2.8.1)
with: e =
in symmetric spectrometers use e,
in asymmetric ones use
d/2
With the F
found, F20 is applied for the dispersion:
Equation
F20: RLD = (cos (x + f)
* k) /( f * m)
(2.8.1)
For a quick, but rather precise, estimation, a simplifyed version can be used:
F21: RLD = l
/(2f * tan f)
(2.8.1)
with: f = median grating angle
Dispersion in symmetric, additive
double spectrometers (RLD) is modifyed
to F21DP: RLD =
l /(2(2f
* tan f))
(2.13.2)
General dispersion: Additive: F21DA: RLD =
l
/((2f * tan f)
+ (2f * tan f))
(2.13.4)
General dispersion: Subtractive: F21DS: RLD
= l
/((2f * tan f)
 (2f * tan f))
(2.13.4)
The numerical
Resolution: F4: R =
l / dl
(1.3.0)
The resolving power of a grating: F5: R_{p}
= m * W / k = m * W * ^{1}/mm
(1.3.0)
with: W = grating width
The measured resolution:
R_{r} =
l_{r}
/ dl_{r}
The Quality
Factor for all parameters: Q_{x} =
real value / theoretical value
(2.11.1)
i.e. for resolution:
Q_{r} = R_{r }/ R_{p}
(2.11.1)
One parameter for the real
resolution is the minimum slit width, the Raleigh Diffraction Limit
F22: m_{s} = (
l * f) / ( W * cos
b )
(2.10.4)
with: m_{s}
= minimum allowed slit widht
The general
correction factor for areas, in
two planes:
F13
: A_{iG} = A_{iM} * cos
a
* cos elh * cos elv
(2.2.2)
with:
A_{iG}
= actually used grating area,
A_{iM} = actually used mirror area, elh =
horizontal elbow angle, elv = vertical elbow angle
The area correction for EbertFastie and CzernyTurner spectrometers: for the
grating:
F14
: W_{iG} = W_{iM} * cos
a * cos
el_{in }
(2.3.2)
for the exit mirror:
F15:
W_{iMout} = W_{iG} * cos
b * cos
el_{out
} (2.3.2)_{
}with:
W_{iG}
= actually used grating area,
W_{iMout}
= actually used area of the exit mirror,
el_{out}
= horizontal elbow angle at the exit
Calcualtion of the light flux, the Luminosity:
F16A: L = A^{2}
* T * W,
^{ }(general
normalized aperture of an optical system)
(2.7.5)
^{
}
F16B:
W
= A_{g} / f^{2
}
( F16B is compatible with
F33)^{
}in detail:
F17:
L_{s} = T * A_{s} *
W
* (h_{D} * B)
(2.7.5)
with: A_{s}^{ }
=
area of entrance slit used, f = focal length.
A_{g}
= grating area, W is
the ratio of area / focal length^{2}
L = light flux factor
T = TransmissionEfficienicy of spectrometer
h_{D} = height of output slit or detecrtor
B = spectral bandwidth
Principal Aberrations, all calculations are valid
for estimation only:
F23A: O_{1}
= W * m_{s} /f^{2}
(2.11.1)
with: O_{1} = basic aberration,
the additve deformation of optical information in a single axial instrument
A reflecting spectrometer has two or even three axis. Increasing angles in a
spectrometer
(e,
d,
elbows)
lead to increasing aberrations.
Equation F23B introduces the multiplicator for those angles. The
calculation uses the component with the widest angle. Causion: The estimation is only for spherical optics. Instruments with corrected
optics, like toroidal mirrors, need other algorithms.
F23B:
E
= O_{1} * (1 + sin
d)
(2.11.1)
with:
E
= the deformation in a dual axial spectrometer,
d
= the internal full opening angle at the component with the widest
angle, like the elbows at a mirror, or the two
e at a grating. The
grating´s working angle
f plays no
role, because it is required for dispersion. But Coma
(2.11.6) originates from the working
angle of a grating or a prism.
The
slit heigth influences the total abberations, too.
For curved slits please use
F24; for straight slits please use F25::
F24:
H
= E * (1+
h/f)
(2.11.1)
with: H
= the total aberrations including the vertical factor in systems with curved slits, increasing the
distortion of the horizontal aberrations. It originates from the vertical
dimension of a curved slit or detector, h = the slit
height (curved slits only).
The estimation of spectrometers with straight slits
includes the radius, a curved slit would have, if used. It is the lateral
distance of the slit centre to the centre of the instrument. The additive aberration
resulting, is only valid for straightslitsystems. It is a value, which already
appears in the middle of the horizontal spread, as soon as the vertical position
departs from centre. See also 2.11.3 and graph 26, centre sketch.
F25:
O_{ss}
= E + (h^{2}/r)
(2.11.1)
with: O_{ss} =
the minimum aberration in a a two axial instrument, like a reflecting
spectrometer with straight slits
r = the radius of curvature of the slits, identical with the
distance slit to centre of instrument. Only valid for symmetric systems without
imaging correction. For instruments with nonspherical optics, a ray
tacing program is required.
The final abberations to be expected, are the result at the end of
the chain, found by F24 or F25.
Spectrometers with imaging correction may
have stronger distortions in the centre of the field, but way better values in
the outskirts of the xyframe.
Prism and Prism Spectrometer
The general equation for refraction is:
F6:
n1_{(}_{l)}
* sin a =
n2_{(}_{l)}
* sin b (1.4.6)
were:
n1
= Refractive Index (RI) in front of the
interface,
n2
= Refractive Index (RI) after the interface.
The minimum deflection of a prism:
F7: d_{min} =
[2
/ {sin
(n * sin A/2 ) }]
 A
(1.4.6,
2.16.6)
with:
d
= angle between incidence beam and refracted
beam,
A
= Prism angle in
transmission mode.
In a spectrometer, the incidence angle i_{1} changes, and the equation needs to
be extended to
F9:
(1.4.6, 2.16.6,
3.3.0)
with: A is the angle of the prism. For a reflecting prism, we find
A/2. But, as the prism will be passed twice, the double numeric value of A/2
is applied, leading to A again.
i_{1} is the incidence angle at the prism,
related to N at the front face.
To get the difference in deflection of two wavelength´s, F9 needs to be
differentiated or chained.
F9A:
dd
=
d (l2)

d
(l1)
(2.16.6)
Now, the dispersion can be calculated,
or by the help of
equation F10, we can directly define the dispersion of the system,
as Reciprocal Dispersion value:
F10: RD
= 1
/ [f
* (sin dd
/ dl)]
=
dl
/ (f *
sin dd)
(2.16.6)
with:
RD is the median reciprocal dispersion, in a spectrometer with focal
length f. It is clear, that RD is at no place linear.
Resolution of a prism:
F
8:
l/Dl =
b [dn/dl]
(1.4.6)
with b as the base width of the prism
The dilatation of
the focal plane as function of thermal changes:
Dilatation: F26: dx = K * 2f * dT
(2.15.1)
with: K = Thermal Coefficient, T = Temperature
the resulting
increase of the focal spot:: F27: dy = dx / n
(2.15.1)
were: n = fnumber of the spectrometer
Detection:
The Signal/Noise ratio
F28: SNR = (SB) / N
(4.1.3)
with:
S the median value of the signal
B the median of the Background value
N the value of the standard deviation of the noise amplitude
The limiting
capability of an IR detector
F29:
D* = (SNR *
D
f ^{½})
/ P * A ^{½}
(4.6.3)
with
D* the detectivity,
SNR the signal/noise ratio measured,
D
f ^{½ }
square root of the bandwidth b, P is the
optical power transmitted to the detector in W, and A ^{
½ }
is the square root of the
detector area.
CCD readout in standard mode:
The transfer und readout time
F30: t_{Readn} =
(SL * t_{SL}) + (SR * t_{SR}) * (hb * t_{ADC})
(4.8.2.1)
with
t_{Readn}
the total time required to read the CCD content
in normal mode
SL
the number of vertical lines
t_{SL}
the time to tranfer one line to
the next (vertical or parallel shift)
SR
the number of horizontal pixels in
the register
t_{SR}
the time required to shift from
one register position to the next (horizontal or read
shift)
t_{ADC}
the time for ADC cylce, with
hb
the number of data units to be converted  ADC cycles,
(if single data poins or gouped/binned data makes no difference here; it
is assumed the the
data storage time inside the computer does not require extra time)
The attenuation, also called damping, of
measurement signals and power amplitudes:
F31A: dB = 20 log_{10} (U/U_{0})
(4.10.2) (called voltagedB)
For Signals, like from detectors and amplifiers,the interpretation of the
"voltage calculation" ist used:
dB = 20 log_{10} (U/U_{0}). An attenuation of 3 dB means
factor 0.86071.
F31B: dB = 10
log_{10} (P/P_{0})
(4.10.2) (called powerdB)
For
Power calculations,
mainly in Watt, the interpretation of "power calculation" takes place:
dB = 10 log_{10} (P/P_{0}), leading to an attenuation of
3 dB for a factor of
0.74082.
The reason to dicriminate is, that the "voltage calculation" is based on one
parameter only (like v), while "power calculation" describes a product (like
photons, Watt), based on a multiplication. In the brackets U resp. P are
the output values, while U_{0} and P_{0} are the input.
Illumination, Integration Spheres, and Radiometry
F23C:
F1/O1
= F2/O2
(the general rule for optical transfer and object
reproduction) (5.4)
with F1 resp. F2, the focal distances, and O1 resp. O2, the size of the
object resp. the reproduction of the object
F32:
L_{(}_{l)
}
or L_{e(}_{l)}
= (F
*
W)
/( A *
dl
) in [µW/(sr * mm^{2} * nm)]
(Radiance
oder spectral beam density)
(5.03)
allows the calculation of the
densitiy of spectral power L_{e(}_{l)}
in W, mW, or µW
F
is the Radiant Power, also called Radiant Flux, available
W is the normalized angle of illumination, resulting from illuminated area and
its distance from the light source,
A is the illuminated area in
m^{2} or cm^{2 } or mm^{2}, and
dl
is the wavelength interval in
nm
F33:
Steradian_{
}
: sr = a / r^{2}
(normalized
illumination angle for the radiance / spectral beam density)
(5.03
and 2.7.5)
also called Light Guide Factor,
( F33 is compatible
with F16B)
Output irradiance of an integration sphere:
F34:
E_{e(l)} =
F_{
e(l)}
*
{ R / [p * A_{s}
* { 1 [ R *(1 – f )]}] in [W / (cm^{2} * nm)]
(Irradiance
in the output area of an integration sphere)
(5.1.4.1);
with
F_{
e(l)}
the spectral radiant power available in the sphere
R the coefficient of
reflectance of the sphere at the wavelength monitored
A_{s}
the total inner area of the sphere
f the sum of all open areas of the sphere  all input and output
areas without reflection
F35: L_{ e(l)}
= E_{e(l)}
*
W
in [W / (
sr * cm^{2}
* nm )] (Radiance
of a diverting beam)
(5.1.4.1);
with
E_{e(l)}
the irradiance at the beam origin
W
the light guide factor
Fibre Optics
F36: n_{0}
sina
= ( n^{2}_{2}
– n^{2}_{1}
) ^{1/2 }
(Acceptance
angle of an optical fibre)
(5.3.1)
with
n_{0} the refrative index (RI) of the
environment, normally air
a
the acceptance angle of the fibre
n_{2}
the RI of the cladding material of the fibre
n_{1}
the RI of the core material of the fibre
Application Oriented Formulas:
F37:
A = log_{10} [(e_{0}
– BG) / (e_{1} – BG)]
(Absorbance
[Extinction]
of an absorbing sample [LambertBeer])
(6.A1.0.2)
with
e_{0}
the optical entrance signal in
front of the sample
e_{1} dthe
optical output signal after the sample
BG is the socalled Background, which includes all electic and environmental
signals of the detection and electronic channel.
Atomic Absorption (Application A3.1):
For compensated AAS applies F37AA: A = log_{10} ([(e_{0} –
N)(BG – N)] / [(e_{1} – N)(BG – N)])
with
e_{0 }
the optical
signal with "empty" solution (no sample material)
e_{1 }
the optical
signal with "loaded" solution (includes sample material)
N_{ }
the dark signal (the
instrumental background)
BG_{ }
the AA specific
background signal, which is the Absorption in the vicinity of the element
specific spectral line (called Background in AAS)
F38: R = [(e_{1}
– BG) / (e_{0} – BG)]
(Value
of Reflectance in photometry)
(6.A1.0.3)
with
e_{0}
the optical entrance signal in
front of the sample
e_{1} dthe
optical output signal after the sample
BG is the socalled Background, which includes all electic and environmental
signals of the detection and electronic channel.
R often is also presented als percentage value.
Polarization
(Application L1):
F39, the degree of polarization: P = [(I_{p} – I_{s})
/ [(I_{p} + I_{s})]
F40, the Anisotropy r
= [(I_{p} – I_{s}) / [(I_{p} + 2I_{s})]
with I_{p} =
parallel plane of polarization
and I_{s} = the
perpendicular plane
At totally polarized light or illumination with parallel polarized light (I_{p})
we find I_{s} = 0, that leads to P = r = 1. If the orientation is I_{s} = 1,
accordingly the same rules apply. For perfectly depolarized light (I_{p} = I_{s})
we find P = r = 0. All
intermediate states result in P and r > 0 and < 1.
Luminescence Lifetime
(Application L2), detected by phase/modulation
measurement:
For the calculation of the phase angle, equation F41
F41: tan F
= w
*
t_{p
}
For the calculation of the modulation factor, equation F42
F42: m = [ 1 + w^{2}
*
t^{2}_{m}
]^{1/2
}whereas
F is the resulting phase
angle,
w is the circular frequency
of modulation,
t is the lifetime,
m is the resulting modulation factor.
Spectroscopic Ellipsometry (SE) (Application
E1):
Measuremt data are based on the Fresnel equations.
F43:
,
also presented as
with
r_{p} and r_{s},_{
}the absolute values of parallel and perpendicular polarisation
d,
the phase shift or phase position
r,
the complex result, which in turn leads to
tg Y and
cos D führt.
The illumination angle at the sample carries the sign F.
The complex value
r, in a system with Polarizer  Sample 
Compensator  Analyzer, is defined by
F44:
Incorporating A, C, P, the angles of die Winkel von Polarizer, Compensator, Analyzer,
resp. Almost all SE measurements are satisfied by acquiring the rael part only (without
Compensator).
An SE instrument with rotating polarizer, no compensator, and with a programed analyzer,
creates the measured value from:
F45:
wheras after the data acquisition, the coefficients
a and
b are reduced
by Hadamard algorithms, which lead to tg Y
and cos D.
F46:
4B) Collection of all equations and formulas used.
Here they are in numerical order with marking of
topics, and
red indices
showing the point of discussion
F1: m *
l = k *
(sin a
± sin
b)
(Grating Function)
(1.1.0)
F2A: FSR =
l
/ m,
a more precise interpretation is
F2B:
l_{2}
= l_{1}
+ (l_{1}
/m) and FSR = l_{2}
 l_{1 }(Free Spectral
Range)_{ }
(1.1.1)
F3: m *
dl = k
* cos b
* db
or
db /
dl
= ( m / ( k * cos b
)) (Angular
Dispersion)
(1.2.0)
F4: R =
l / dl (General Resolution)
(1.3.0)
F5: R_{p}
= m * W / k = m * W * ^{1}/mm
(Grating
theoretical Resolution)
(1.3.0)
R_{r} =
l_{r}
/ dl_{r}
(real Resolution) (1.3.0)
Q_{x} =
measured value / theoretical value (Quality
Factor)
(2.11.1)
Q_{r} = R_{r }/ R_{p} (Quality
Factor)
(2.11.1)
F6:
n1_{(}_{l)}
* sin a =
n2_{(}_{l)}
* sin b
(Prism
Basic
Function)
(1.4.6)
F7: d_{min}
=
[2
/ {sin
(n * sin A/2 ) }]
 A
(Prism
min
Deflection)
(1.4.6,
2.16.6)
F
8:
l/Dl =
b [dn/dl]
(Prism
theoretical
Resolution)
(1.4.6)
F9:
(Deflection
in Prism Spectrometers)
(1.4.6, 2.16.6,
3.3.0)
F9A:
dd
=
d (l2)

d
(l1) (Dispersion
after prism)
(2.16.6)
F10: RD
= 1
/
[f
* (sin dd
/ dl)]
=
dl
/ (f *
sin dd)
(Dispersion
in Prism Spectrometers)
(2.16.6)
F11: m *
l = k *
2 sin a (Grating
in Littrow Spectrometer)
(2.2.1)
F12: m *
l = k *
2 sin F
* cos e
(Grating in EbertFastie Spectrometer)
(2.3.1)
F13
: A_{iG} = A_{iM} * cos
a
* cos elh * cos elv (Area
Correction)
(2.2.2)
F14
: W_{iG} = W_{iM} * cos
a * cos
el_{in }
(Width
Correction)
(2.3.2)
F15:
W_{iMout} = W_{iG} * cos
b * cos
el_{out
}
(Output
Width
Correction)
(2.3.2)_{
}
F16A: L = A^{2}
* T * W
(general Luminosity = Light
Flux)
(2.7.5)
F16B:
W
= A_{g} / f^{2
}(normalized aperture of a spectrometer, for
light flux)
(2.7.5),
F16B is compatible
with F33^{
}F17:
L_{s} = T * A_{s} *
W
* (h_{D} * B)
(spectrometric Luminosity = Light
Flux)
(2.7.5)
F18: f*
(db
/ dl)
= f * m / (k * cos b)
(Angular dispersion in Grating Spectrometer)
(2.8.1)
F19:
f
= arcsin ( l
/( 2 * k * cos e))
(Median
Grating Angle)
(2.8.1)
F20: RLD = (cos (x + f)
* k) /( f * m) (Dispersion
Calculation, fine)
(2.8.1)
F21: RLD = l
/(2f * tan f) (Dispersion
calculation, more rough)
(2.8.1)
F21DP: RLD =
l /(2(2f
* tan f)) (RLD
in additive Double Spectrometers)
(2.13.2)
F21DA: RLD =
l
/((2f * tan f)
+ (2f * tan f)) (RLD
in additive Double Spectrometers) (2.13.4)
F21DS: RLD
= l
/((2f * tan f)
 (2f * tan f)) (RLD
in subtractive Double Spectrometers)
(2.13.4)
F22: m_{s} = (
l * f) / ( W * cos
b ) (Raleigh Diffraction Limit) (2.10.4)
F23A: O_{1}
= W * m_{s} /f^{2} (Aberration,
the additive Distortion in a single axial Instrument) (2.11.1)
F23B:
E
= O_{1} * (1 + sin d)
(Multiplication Factor by the internal opening Angle)
(2.11.1)
F23C:
F1/O1
= F2/O2
(the general rule for optical transfer and object
reproduction) (5.4)
F24:
H
= E * (1+
h/f) (Multiplication
factor
in a System with curved Slits)
(2.11.1)
F25:
O_{ss}
= E + (h^{2}/r) (Sum
of aberrations
with straight Slits)
(2.11.1)
F26: dx = K * 2f * dT
(Dilatation Shift / Thermal Change)
(2.15.1)
F27: dy = dx / n
(Dilatation
Shift / Thermal Change / Focal Spot
increase)
(2.15.1)
F28: SNR = (SB) / N
(Signal/Noise
ratio) (4.1.3)
F29: D* = (SNR *
D
f ^{½})
/ P * A ^{½}
(Limiting
capability of an IR detector)
(4.6.3)
F30: t_{Readn} =
(SL * t_{SL}) + (SR * t_{SR}) * (hb * t_{ADC})
(CCD transfer und readout time, standard mode)
(4.8.2.1)
F31A: dB = 20 log_{10} (U/U_{0}) (attenuation,
also called damping, of measurement signals = voltagedB)
(4.10.2)
F31B: dB = 10
log_{10} (P/P_{0}) (attenuation,
also called damping, of power amplitudes = powerdB)
(4.10.2)
F32:
L_{(}_{l)
}
or L_{e(}_{l)
}
: mW/(sr * cm^{2} * nm) (Radiance
or spectral density)
(5.03)
F33:
Steradian_{
}
:
W
= sr = a / r^{2}
(Normalized
angle of illumination or Light Guide Factor)
(2.7.5, 5.03),
( F33 is compatible with F16B)
F34:
E_{e(l)} =
F_{
e(l)}
*
{ R / [p * A_{s}
* { 1 [ R *(1 – f )]}] in [W / (cm^{2} * nm)]
(Irradiance
in the output area of a integration sphere)
(5.1.4.1);
F35: L_{ e(l)}
= E_{e(l)}
*
W
in [W / (
sr * cm^{2}
* nm )] (Radiance
of a diverting beam)
(5.1.4.1);
F36: n_{0}
sina
= ( n^{2}_{2}
– n^{2}_{1}
) ^{1/2 }
(Acceptance
angle of an optical fibre)
(5.3.1)
F37:
A = log_{10} [(e_{0}
– BG) / (e_{1} – BG)]
(Absorbance
[Extinction]
of an absorbing sample [LambertBeer])
(6.A1.0.2)
F37AA: A = log_{10} ([(e_{0} –
N)(BG – N)] / [(e_{1} – N)(BG – N)])
Atomic Absorption (Application
A3.1)
with Background Compensation
F38: R = [(e_{1} – BG) / (e_{0} – BG)]
(Value
of Reflectance in photometry)
(6.A1.0.3)
Polarization (Application L1):
F39, the degree of polarization: P = [(I_{p} – I_{s})
/ [(I_{p} + I_{s})]
F40, the Anisotropy r
= [(I_{p} – I_{s}) / [(I_{p} + 2I_{s})]
with I_{p} =
parallel plane of polarization
and I_{s} = the
perpendicular plane
At totally polarized light or illumination with parallel polarized light (I_{p})
we find I_{s} = 0, that leads to P = r = 1. If the orientation is I_{s} = 1,
accordingly the same rules apply. For perfectly depolarized light (I_{p} = I_{s})
we find P = r = 0. All
intermediate states result in P and r > 0 and < 1.
Luminescence Lifetime
(Application L2), detected by phase/modulation
measurement:
For the calculation of the phase angle, equation F41
F41: tan F
= w
*
t_{p
}
For the calculation of the modulation factor, equation F42
F42: m = [ 1 + w^{2}
*
t^{2}_{m}
]^{1/2
}whereas
F is the resulting phase
angle,
w is the circular frequency
of modulation,
t is the lifetime,
m is the resulting modulation factor.
Spectroscopic Ellipsometry (SE) (Application
E1):
F43:
or apperaing as
F44:
F45:
F46:
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Spectroscopy with dispersive Spectrometers
Basics  Building Blocks  Systems  Applications " are reserved by
Wilfried Neumann, D88171 WeilerSimmerberg.
Status April 2012