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


Index

 

 

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

 

Czerny-Turner 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

 

Seya-Namioka 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 f-number 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


Index

 

 

Basics 3 - Configurations of Monochromators and Spectrographs

3.

1.

1

 

Beam travel in Ebert-Fastie and Czerny-Turner 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


Index

 

 

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 (Lock-In 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 semi-conductor 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

 

Read-out 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, C-MOS, 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.

 

 

photo-acoustic (opto-acoustic) 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
Polarization Measurement (SE)

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.

 

 

F-OES - Spark Emission Spectroscopy

E2.

5.

 

 

LA - Laser-Ablation

E2.

5.

 

 

LIBS - Laser induced Breakdown Spectroscopy

E2.

5.

 

 

LIP - Laser induced Plasma Spectroscopy

E2.

5.

 

 

LD - Laser Deposition

E2.

5.

 

 

LD - Laser-Beschichtung

 

 

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 set-up, 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.

typical behaviour of straylight in several set-ups

 

 

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

 

 


Index

 

 

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)
the imaginary part of ellipsometric data
e1
 angle of the grating- impinging beam, in a symmetric system, it is  half d
e2  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 / f2 , also called light guide factor,
W   the normalized spread of a light beam, from a / r2, 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/D-C
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 Voltage-dB and Power-dB
D*  the numeric capability of an IR detector for low signals
e   Base of the natural logarithm, required for e-functions
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
    the focal length
    the frequency
fc     the angular frquency
fwhm   full widht at half maximum

h     height of slit
i1   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
  the thermal dilatation coefficent
   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
O1    the basic aberration, an additive distortion to image or focus
Oss  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  O1
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 Lock-in), and  Position Sensitive (counting) Detector

Q     quality factor, with Xr =  real value
Q     the energy of radiationR   die numerische Auflösung, mit Rr reale Auflösung und Rp 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,   
rp und rs the absolute values of arallel and perpendicualr polarization
R    numeric resolution, with Rr real resolution and  Rp 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/N-R
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 * 108 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 =
[(107 / 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 * 108 m/s. Through l = 1/C the according wavelength of 1 Hz (1/s) is
l = 3 * 108 m. Regarding the median range of optical spectroscopy, we prefer nm (10-9 m) and GHz (109 * 1/s). The conversion between frequency and wavelength is: fr = C / l. For example 1 µm: fr = (3 * 108 m/s) / (10-6 m) = 3 * 10-14 /s or 3 * 105 GHz, shortened to
1 nm = 300 / 106 GHz and 106 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 Ebert-Fastie- and symmetric Czerny-Turner 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: l2 = l1 + (l1 /m) and FSR = l2 - l1     (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: Rp = m * W / k  = m * W * 1/mm     (1.3.0)
with:  W
= grating width

The measured resolution:
  Rr = lr / dlr 

The Quality Factor for all parameters: Qx = real value / theoretical value    (2.11.1)
i.e. for resolution: Qr =  Rr / Rp     
(2.11.1)

One parameter for the real resolution is the minimum slit width, the Raleigh Diffraction Limit  F22: ms = ( l * f)  / ( W  * cos b )   (2.10.4)
with:  
ms = minimum allowed slit widht 

The general correction factor for areas, in two planes: F13 : AiG = AiM * cos  a * cos el-h * cos el-v    (2.2.2)
with: 
AiG = actually used grating area, AiM = actually used mirror area, el-h = horizontal elbow angle, el-v = vertical elbow angle
The area correction for Ebert-Fastie- and Czerny-Turner spectrometers: for the grating:
F14 : WiG = WiM * cos  a * cos elin   (2.3.2)
for the exit mirror:
  F15: WiM-out = WiG * cos  b * cos elout    (2.3.2)
with
WiG = actually used grating area, WiM-out = actually used area of the exit mirror, elout = horizontal elbow angle at the exit

Calcualtion of the light flux, the Luminosity:
  F16A: L = A2  * T * W,   (general normalized aperture of an optical system)   (2.7.5)

F16B: W = Ag / f2     ( F16B is compatible with F33)
in detail: F17:
Ls = T * As * W * (hD * B)    (2.7.5)
with: 
As  = area of entrance slit used,    f = focal length.
Ag =
grating area, W is the ratio of area / focal length2
L
= light flux factor
T = Transmission-Efficienicy of spectrometer
hD = height of output slit or detecrtor
B = spectral bandwidth
 

Principal Aberrations, all calculations are valid for estimation only: 
F23A:
O1 = W * ms /f2
   (2.11.1)
with:  O1   =  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 =  O1 * (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 straight-slit-systems. 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: Oss =
E + (h2/r)   (2.11.1)
with:
  Oss  = 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 non-spherical 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 x-y-frame.

Prism and Prism Spectrometer
The general equation for refraction is:  F6n1(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:  dmin = [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  i1 changes, and the equation needs to be extended to

F9:   
Formula F9 Prism     (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.
i1 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
= f-number of the spectrometer

Detection:
The Signal/Noise ratio
F28:   SNR = (S-B) / 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 read-out in standard mode:
The transfer- und read-out time

F30: tRead-n = (SL * tSL) + (SR * tSR) * (hb * tADC)   (4.8.2.1)
with
tRead-n  
the total time required to read the CCD content in normal mode
SL
       the number of vertical lines
tSL
        the time to tranfer one line to the next (vertical or parallel shift)
SR
        the number of horizontal pixels in the register
tSR
        the time required to shift from one register position to the next (horizontal or read shift)
tADC
       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 log10 (U/U0)   (4.10.2)  (called voltage-dB)
For Signals, like from detectors and amplifiers,the interpretation of the "voltage calculation" ist used:   
-dB = 20 log10 (U/U0). An attenuation of  -3 dB means factor
0.86071.
F31B:  -dB = 10 log10 (P/P0)    (4.10.2)    (called power-dB)
For  Power calculations, mainly in Watt, the interpretation of "power calculation" takes place:
 -dB = 10 log10 (P/P0), 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 U0 and P0 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 Le(l) = (F * W) /( A * dl ) in [µW/(sr * mm2 * nm)]  (Radiance oder spectral beam density)  (5.03)
allows the calculation of the densitiy of spectral power Le(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 m2 or cm2  or mm2, and
dl  is the wavelength interval in  nm
F33: 
 Steradian : sr = a /  r2   (
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
:  Ee(l) = F e(l) { R / [p * As * { 1- [ R *(1 – f )]}]  in [W / (cm2 * 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) =  Ee(l) * W   in  [W / ( sr * cm2 * 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:   n0 sina =  ( n22 – n21 ) 1/2    (Acceptance angle of an optical fibre)  (5.3.1)
with 
n0   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 = -log10 [(e0 – BG) / (e1 – BG)]   (Absorbance [Extinction] of an absorbing sample [Lambert-Beer])  (6.A1.0.2)
with
e0
the optical entrance signal in front of the sample
e1
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 F37-AA: A = -log10 ([(e0 – N)-(BG – N)] / [(e1 – N)-(BG – N)])
with
e
the optical signal with "empty" solution (no sample material)
e 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 = [(e1 – BG) / (e0 – BG)]   (Value of Reflectance in photometry)  (6.A1.0.3)
with

e0
the optical entrance signal in front of the sample
e1
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 = [(Ip – Is) / [(Ip + Is)]
F40, the Anisotropy   r = [(Ip – Is) / [(Ip + 2Is)]
with Ip = parallel plane of polarization
and Is = the perpendicular plane
At totally polarized light or illumination with parallel polarized light (Ip) we find Is = 0, that leads to P = r = 1. If the orientation is  Is = 1, accordingly the same rules apply. For perfectly depolarized light (Ip = Is) 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  * tp
For the calculation of the modulation factor, equation F42
F42:  m = [ 1 + 
w2  * t2m ]-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 rs, 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

F
1:    m * l = k * (sin a ± sin b)  
(Grating Function) (1.1.0)
F2A:  FSR = l / m, a more precise interpretation is  F2B: l2 = l1 + (l1 /m) and FSR = l2 - l1    (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: Rp = m * W / k  = m * W * 1/mm    (Grating theoretical Resolution)  (1.3.0)
Rr = lr / dlr    (real Resolution (1.3.0)

Qx = measured value / theoretical value   (Quality Factor)   (2.11.1)
Qr =  Rr / Rp  
 (Quality Factor)    (2.11.1)

F6
n1(l) * sin a = n2(l) * sin b    (Prism Basic Function)  (1.4.6)

F7:  dmin = [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:   Formula F9 Prism   (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 Ebert-Fastie Spectrometer)      (2.3.1)
F13
: AiG = AiM * cos  a * cos el-h * cos el-v  (Area  Correction) (2.2.2)
F14
: WiG = WiM * cos  a * cos elin     (Width  Correction (2.3.2)
F15: WiM-out = WiG * cos  b * cos elout     (Output Width  Correction)  (2.3.2)
F16A: L = A2  * T *
W 
(general Luminosity  = Light Flux)   (2.7.5)   
F16B: W = Ag / f2          (normalized aperture of a spectrometer, for light flux)   (2.7.5), F16B is compatible with F33
F17:
Ls = T * As * W * (hD * 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: ms = (
l * f)  / ( W  * cos b )  (
Raleigh Diffraction Limit)  (2.10.4)
F23A: O1 = W * ms /f2  (Aberration, the additive Distortion in a single axial Instrument)  (2.11.1)
F23B: E =  O1 * (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: Oss = E + (h2/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 = (S-B) / 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: tRead-n = (SL * tSL) + (SR * tSR) * (hb * tADC (CCD transfer- und read-out time, standard mode)     (4.8.2.1)
F31A:   -dB = 20 log10 (U/U0)  (attenuation, also called damping, of measurement signals = voltage-dB)  (4.10.2)
F31B:  -dB = 10 log10 (P/P0(attenuation, also called damping, of power amplitudes = power-dB)   (4.10.2)
F32:   L(l) or Le(l) : mW/(sr * cm2 * nm)    (Radiance or spectral density)  (5.03)
F33:   Steradian : W = sr = a /  r2   (Normalized angle of illumination or Light Guide Factor)  (2.7.5, 5.03), ( F33 is compatible with F16B)
F34:  Ee(l) = F e(l) { R / [p * As * { 1- [ R *(1 – f )]}]  in [W / (cm2 * nm)] 
(
Irradiance in the output area of a integration sphere)   (5.1.4.1);
F35
:  L e(l) =  Ee(l) *
W   in  [W / ( sr * cm2 * nm )] (Radiance of a diverting beam)   (5.1.4.1);
F36:   n0 sina =  ( n22 – n21 ) 1/2    (Acceptance angle of an optical fibre)  (5.3.1)
F37: A = -log10 [(e0 – BG) / (e1 – BG)] (Absorbance [Extinction] of an absorbing sample [Lambert-Beer])  (6.A1.0.2)
F37-AA: A = -log10 ([(e0 – N)-(BG – N)] / [(e1 – N)-(BG – N)]) Atomic Absorption (Application A3.1) with Background Compensation
F38
: R = [(e1 – BG) / (e0 – BG)]  
(Value of Reflectance in photometry)  (6.A1.0.3)
Polarization (Application L1):
F39, the degree of polarization:  
P = [(Ip – Is) / [(Ip + Is)]
F40, the Anisotropy   r = [(Ip – Is) / [(Ip + 2Is)]
with Ip = parallel plane of polarization
and Is = the perpendicular plane
At totally polarized light or illumination with parallel polarized light (Ip) we find Is = 0, that leads to P = r = 1. If the orientation is  Is = 1, accordingly the same rules apply. For perfectly depolarized light (Ip = Is) 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  * tp
For the calculation of the modulation factor, equation F42
F42:  m = [ 1 + 
w2  * t2m ]-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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Status April 2012