Simulation and Design of a stray light measurement system according to the concept of BSDF (Bidirectional Scattering Distribution Function)

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Simulation and Design of a stray light measurement system according to the concept of BSDF (Bi-

directional Scattering Distribution Function)

Master’s thesis in the field of Photonics by

Florian Huber (Dipl.-Ing.)

Hochschule München

Munich University of Applied Sciences

Department Physikalische Technik / Feinwerk- und Mikrotechnik

1st Assesor: Prof. Dr. Roths 2nd Assesor:

Supervisor: Dipl.-Phys. Kampf, Kayser Threde GmbH

Periode: 01.03.2011 – 31.08.2011

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Acknowledgment

I would like to express my thanks to Kayser-Threde GmbH, particularly to my supervisor Mr.

Dirk Kampf for his useful comments and proofreading.

I would also like to say thank you to Mr. Johannes Kolmeder, Mr. Markus Glier, Mr. Sven Gutruf, Mr. Amir Mottaghibonab and Dr. Michael Hart, for their notice and cooperation.

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Kurzfassung

Primäres Ziel der hier vorgestellten Arbeit ist der Aufbau und der Test eines flexiblen Mess- standes zur Erfassung des Streulichtverhaltens konstruktiver und optischer Oberflächen. Die Proben werden mit einer Laserquelle bestrahlt. Je nach Oberflächenbeschaffenheit streuen die beleuchteten Flächen unterschiedlich stark in verschiedene Richtungen des Raumes ab.

Die sogenannte Streufunktion oder BSDF (Bidirectional Scattering Distribution Function) gibt den Zusammenhang von abgestrahlter Leuchtdichte (Strahldichte) zur einfallenden Beleuch- tungsstärke (Bestrahlungsstärke) wieder. Die BSDF ist abhängig von der Richtung, aus der die Probe bestrahlt wird, so dass sich eine Vielzahl von Streugraphen für eine Probenoberfläche ergeben kann. Der Aufbau, realisiert als Mehr-Achs-Goniometer mit motorisiertem Aufbau, ermöglicht eine dreidimensionale Erfassung des Streulichtes.

Die Flexibilität des Aufbaus wird durch optische Schienen erreicht, so dass die Winkelauflö- sung durch Veränderung des Abstandes erhöht werden kann.

In dieser Arbeit ist zunächst die automatisierte Vermessung in einer Ebene (Azimuth) realisiert. Ein in LabView erstelltes Steuerprogramm, stellt eine GUI (Graphical User Interface) mit den grundlegenden Steuerfunktionen bereit.

Die Messeinheit wird mit dem optischen Simulationstool „Zemax“ simuliert. Dazu wird der Aufbau mit allen relevanten Komponenten im Rechner nachgestellt. Die verwendete Soft- ware führt auf Grundlage einstellbarer mathematischer Streumodelle Monte Carlo Simula- tionen durch. In dieser Arbeit wird das sog. Abg-Modell verwendet. Die aus den Monte-Carlo Simulationsdaten extrahierten Streufunktionen werden mit dem jeweils zugrundeliegenden analytischen Modell auf Übereinstimmung überprüft.

Dem Aufbau des Messstandes folgen eine Kalibrierung der Winkel und erste Messungen dif- fus reflektierender Flächen. Eine Plausibilitätsbetrachtung anhand der theoretischen BSDF

„lambertscher“ Oberflächen wird der Messung vorangestellt.

Die Untersuchung des Streuverhaltens weiterer Materialien und dem Anpassen von Modell- kurven an die gemessenen Kurven schließt die Arbeit ab.

Schlüsselwörter: BSDF, ABg–Modell, Streulicht, Simulation, Messsystem

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Abstract

The main goal of the present thesis is the design, assembly and test of a flexible measurement system in order to determine scattering characteristics of optical surfaces. The samples are illuminated with an adequate coherent laser source. The scattering characteristics depend on the respective surface conditions. The so-called “Bidirectional Scattering Distribu- tion Function” (BSDF) gives a relation between scattered radiance and incidence irradiance depending on all incident directions. This means that one sample has many different scatter curves.

Four independent motorized axes form a Multiple-Axes-Goniometer in order to get data about stray light from all spatial directions. Therefore, the sample can be irradiated from each spatial direction. The usage of optical rails allows removing and adding optical parts in a flexible way. The observation radius can easily be changed as well.

In this thesis, the measurement is at first, set up automatically in one plane (azimuth).

For this purpose a control program, which provides a Graphical User Interface with the basic control functions, is developed in LabView.

In order to verify the assembly, an analysis with all relevant components of the setup is car- ried out with the simulation tool „Zemax”. The simulation, based on “Monte–Carlo–

Methods”, executes the simulation with selectable mathematical stray light models. In this thesis, the used model is the so called “ABg–Model”. The stray light curves, extracted from the Monte-Carlo-Simulation, are compared consensual with the appropriate underlying ABg- Model.

After assembling and programming the measurement setup, the angles of the system are calibrated. The first functional performances of the design are executed with diffuse reflect- ing surfaces like lambertian surfaces. A plausibility consideration with the theoretical BSDF of such lambertian samples is preceded.

An examination of the scattering characteristics of further materials and the fitting of model data (ABg – model) on the measured curves completes this thesis.

Keywords: BSDF, ABg – model, scatter light, simulation, measurement setup

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List of Figures

Figure 1: Rayleigh – Scattering (Photo taken in West USA) ... 9

Figure 2: Illustration of radiometric quantities ... 10

Figure 3: Energy transmission between two surface elements ... 12

Figure 4: Illustration of the kinds of BSDF ... 13

Figure 5: Dependent quantities with regard to the BSDF ... 14

Figure 6: Surface elements irradiated with different irradiance ... 15

Figure 7: Surface elements irradiated with different incident angles ... 15

Figure 8: Derivation of the Harvey – Shack model ... 17

Figure 9: ABg–Model with A = 1 · 10^-2; B = 3 · 10^-4 and g = 3 ... 18

Figure 10: BSDF measurement system according to [12] ... 19

Figure 11: Measured intensity ... 22

Figure 12: Different types of surfaces ... 25

Figure 13: particle distributions (MIL–standard) [6] ... 27

Figure 14: Predicted BRDFs at λ=10 µm on contaminated mirrors [17]¹; Left hand side: log-log-plot [6]² ... 28

Figure 15: Predicted BRDFs at λ=5 µm, 10µm and 20µm on particles deposited on mirrors [17] ... 28

Figure 16: BRDFs at λ=0,633 µm, 1,15 µm, 3,39 µm and 10,6 µm on particles deposited on mirrors [6]... 29

Figure 17: Definition of surface roughness [6] ... 30

Figure 18: Equivalent Cleanliness level (compared to roughness) as function from the standard β-β0 plot ... 31

Figure 19: Scattering in the “Zemax-Simulation” ... 34

Figure 20: Simulation Detection system; Left: Side View; Right front view ... 35

Figure 21: Detector image ... 37

Figure 22: Image of the detector sphere... 37

Figure 23: Interpretation of the elevation angle. ... 38

Figure 24: Data interpretation of simulation model 1 ... 38

Figure 26: Parameter used to calculate the BSDF ... 39

Figure 26: Simulation and analytic data with set of parameter 1 (semi logarithmic plot) ... 40

Figure 27: Extract from the simulation setup ... 40

Figure 28: Simulation and analytic data (1) with corrected angle (semi logarithmic plot) ... 41

Figure 29: Simulation and analytic data (1) with corrected angle (logarithmic plot) ... 41

Figure 30: Detector with higher resolution ... 42

Figure 31: Simulation and analytic data (1) (logarithmic plot) with an additional set of data ... 43

Figure 32: Simulation and analytic data of the second set of parameter (semi logarithmic plot) ... 44

Figure 33: Simulation and analytic data of the second set of parameter (logarithmic plot) ... 45

Figure 34: Simulation data with fitted curve (purple color) ... 46

Figure 35: Simulation data in |β - β0| – plot (2) with fitted curve (purple color) ... 46

Figure 36: Simulation of lambertian scattering with fitted curve (purple) and model curve (red) ... 47

Figure 37: Pixel size as function of azimuth angle ... 48

Figure 38: BSDF Simulation (4) with model curve (red) ... 49

Figure 39: BSDF Simulation (4) with Fitting curves (purple) ... 49

Figure 40: Product tree of the BSDF measurement system ... 51

Figure 41: System parts of the BSDF measurement system ... 51

Figure 42: Measurement setup: Side view ... 52

Figure 43: Measurement setup: Top view; The coordinate systems are described later more accurate ... 52

Figure 44: Two possible goniometer configurations ... 53

Figure 45: Useful coordinate systems ... 53

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Figure 46: Sample surface vector ... 53

Figure 47: Goniometer part of the setup ... 54

Figure 48: Mounting of post P150/M within rotation stage M1 (8MR190-90-4247) (Right hand side) ... 55

Figure 49: Left side: XYR1/M; Center: M3, M4 (8MR191-28); Right hand side M2 (8MR174-11-20) ... 55

Figure 50: Sample mounting ... 56

Figure 51: Photo from the irradiation Unit ... 57

Figure 52: Irradiation Unit ... 57

Figure 53: Expander with aperture ... 58

Figure 54: PDA100EC ... 59

Figure 55: DAQ NI USB-6009 ... 60

Figure 56: Detector PDA connected with NI USB 6009 ... 60

Figure 57: Detection system (Detector with lens) ... 61

Figure 58: Irradiated spot enlarges with rotating the sample ... 62

Figure 59: Optical throughputs 𝜹𝜹 in an image optic ... 63

Figure 60: In the setup used optics ... 64

Figure 61: Elements on a graphical user interface ... 66

Figure 62: Power display ... 68

Figure 63: GUI of the control program ... 68

Figure 64: Parameter input fields in the GUI... 70

Figure 65: State-Event-matrix with finite state diagram ... 71

Figure 66: Finite state diagram of the BSDF control system ... 72

Figure 67: Finite state machine in LabView ... 73

Figure 68: Setup in order to measure laser power... 74

Figure 69: Power measurement via time ... 76

Figure 70: deviation of laser power from the average as function from the time ... 76

Figure 71: Schematic of coordinate systems ... 76

Figure 72: Calibration of the goniometer ... 77

Figure 73: Incident and returned wave on white paper (zero–angle–calibration) ... 78

Figure 74: Angle calibration with translation stage – last step ... 78

Figure 75: Setup of the first measurements... 79

Figure 76: Samples ... 80

Figure 77: Surface with lambertian characteristics ... 80

Figure 78: BSDF data of Al2O3 coating as function of the scatter angle ... 82

Figure 79: Same figure as Figure 78 but with interpolated data ... 82

Figure 80: Same figure as illustrated in Figure 79 but as logarithmic plot ... 83

Figure 81: Graph from Figure 79 with error bars ... 85

Figure 82: Optic from chapter 6.5 ... 85

Figure 83: BSDF measured with optics ... 86

Figure 84: Goniometer distance changes at each graph ... 86

Figure 85: BSDF of Al2O3 coating with receiver optics at different distance ... 87

Figure 86: Spot sizes from left to right: ∅ = 10, 15, 20 mm ... 87

Figure 87: BSDF of Al2O3 – Irradiated spot enlarges from 10 mm to 20 mm spot size ... 88

Figure 88: BSDF of Al2O3 – Irradiated spot enlarges from 10 mm to 20 mm spot size ... 88

Figure 89: BSDF of the sample “Infragold” ... 89

Figure 90: BSDF of the black paper ... 89

Figure 91: BSDF of the diffuser DG10-600 ... 90

Figure 92: BSDF of diffuser DG10-1500)... 90

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List of Tables

Table 1: Comparison of different detection techniques ... 21

Table 2: Extract from the simulation output file ... 36

Table 3: Extract from the simulation output (flat detector) ... 42

Table 4: Parts of the BSDF-Measurement system ... 50

Table 5: Abstract from datasheet with regard to PDA100-EC ... 59

Table 6: Gain factors of PDA100-EC ... 59

Table 7: BSDF Parameter... 69

Table 8: Extract of the protocol file ... 70

Table 9: Events, Action and States in the control program ... 71

Table 10: Measured and calculated power ... 75

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Abbreviations

ADC Analogue-digital-converter

ASTM American Society for Testing and Materials BRDF Bidirectional Reflection Distribution Function BSDF Bidirectional Scattering Distribution Function BTDF Bidirectional Transmission Distribution Function DAC Digital-analogue-converter

DAQ Data acquisition CL Cleanliness Level CW Continuous Wave

DLR Deutsches Zentrum für Luft- und Raumfahrt DUT Device Under Test

EnMAP Environmental Mapping and Analysis Program ESA European Space Agency

GUI Graphical User Interface LTI Linear Time-Invariant NEP Noise Equivalent Power NI National Instruments PSD Power Spectral Density RMS Root Mean Square TIS Total Integrated Scatter

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Introduction

1.1 Structure and aim of this work

Kayser-Threde-GmbH (KT), a medium-sized enterprise, is working in collaboration with other companies and institutes (Max-Planck-Institut, ESA, DLR...) on the development and execu- tion of research projects, with the aim to explore the space. Applications of their business are in optics, telematics, crash tests, data acquisition and space missions [1].

For this purpose, optical instruments (telescopes, cameras, spectrometers, ect.) with high resolution and quality are necessary. The study of scattering effects is an important aspect in the department “Optical Systems” of Kayser-Threde GmbH.

One of many projects, in which KT is involved, is the Environmental Mapping and Analysis Program (EnMAP). The main objective of this project is to investigate a wide range of ecosys- tem parameters, agriculture, forestry, soil and geological environments, such as coastal zones and inland waters. KT is herein responsible for developing the sophisticated Hyper Spectral Instrument, integration and test of the satellite. The accomplishment of the launch phase belongs to their responsibility as well [2]. In this consortium, stray light acts an important part as well.

Stray light is all about us and helps to look about the environment. By contrast, stray light in engineering can lead to incalculable and undesirable effects. Stray light in optical instruments influence and falsify pictures such as stars or planets, which shall be observed. Scat- tering effects influence the quality of pictures, which become perceivable by reduced image contrast and sensitivity. Disruptive scattering effects in astronomical instruments have their origin in remote light sources, which are in the field of view. This remote light is interacting with instrument surfaces and leads to scattering of light. It is an understandable fact that this can lead to problems, if weak luminous objects are observed in the presence of bright light sources. Stray light from such a bright source outshines the observing object. In accurate radiometric measurements, the scattered light is one of the worst problems too [3].

This thesis discusses the measurement of stray light with reference to material surfaces. The measurement is based on the physical concept of BSDF (Bidirectional scattering distribution function) which has achieved an international standard to characterize materials, respective to their optical properties. The objective is the development and verification of a measure-

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ment setup in order to get representative repeatable scatter measurement data of relevant materials in optics.

First, some theoretical aspects regarding the concept of BSDF, optical geometry and radiometry in the field of stray light shall be summarized. In the optics community, often used models to describe stray light, are the so called „Harvey-Shack- and the Abg-Model“. The theory behind these models is explained.

As a second step, the general requirements with restrictive theoretical assumptions and considerations regarding the measurement are discussed.

Before assembly, some simulations of the setup with the simulation software tool “Zemax”

are done. These simulations are necessary in order to verify the setup. Anymore, the stray light plots, achieved from the Monte-Carlo simulations, are compared with the analytic ABg- model parameters.

In the third part, the setup is designed and assembled with reference to the requirements.

The setup is programmed with the common graphical programming language “LabView”.

The program enables to control all motors by graphical user interface (GUI) and provides automated measurements, at which light, reflected in the azimuth plane, is examined.

In the last step, the systems angles are calibrated. The setup is tested and verified if all requirements such as automatically and correctly generated BSDF data are fulfilled. The first measurement is done with a white (lambertian-like) surface with known theoretical BSDF.

An error estimate is done. A measurement with evaluation of further materials completes the work.

1.2 Stray light as physical phenomenon

In principal stray light is close to us, all time. All things, such as tables, landscapes, peoples etc., which can perceived by our eyes, being not a light source or radiates by it selves, can only be seen, because of radiation (electromagnetic waves) from light sources interacts with their surfaces. This interaction leads to redirection, absorption or transmission of light, which is now called “Stray light”. Each stray light beam has its origin in these phenomenons.

In a more physical view, light can be described as waves as other physical phenomenons such as acoustic noise, matter waves, water waves et cetera. If wave fronts impact on any kind of matter, these wave fronts will be changed with reference to amplitude and phase.

The superposition of the reflected waves leads to interference and diffraction in which both items are physically the same. The difference is only in the number of the considered waves.

With interference it is normally meant, that only two or a few different waves are super-

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posed, whereas diffraction is the same with a huge amount of waves (Principle of Huygens) [4]. So, stray light is a general term for diffraction and interference. For these manifold and complex superimposed waves, which have their origins in interaction with matter, it is often not possible to give a determined formalism for the intensity of the scattered light.

In the range of imaging and transmissive optics, scatter has its origin in surface roughness, differences in refraction indices or molecular and particulate impurities. When surface struc- tures are formed by regular geometries such as optical gratings, the scattered wave fronts can in principal be described by Fourier formalism although this can be difficult. In this case, the resulting matter is denoted as diffraction image. A theory which determines the diffraction matter is the “Fresnel-Kirchhoff’sche Diffraction Theory“ [5]. However, in most cases, scatter appears in complex statistical pattern [6]. In this case, stray light is described with statistical methods, empiric models or semi-empirical models. It can be said, there is a simple unrealistic deterministic nature of scattering and a realistic complex nature of scattering.

Theoretical models, which are executed as computerized algorithms, can simulate the propagation and distribution of light and help the designer to optimize an optical system for the designated application. The used model, the repetition rate and the numerical resolution limit always the truth and quality of this kind of analysis. Components, which are not directly related to the optics, for example structuring elements, are often not simulated. Therefore, the simulation can never replace the experience of the designer.

Figure 1: Rayleigh – Scattering (Photo taken in West USA)

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2 Theoretical aspects regarding the BSDF

2.1 Radiometry

Before the concept of BSDF can be described, some important radiometric items have to be defined and declared.

Radiometry is the field in technical optics in which electromagnetic rays are quantitative detected. In this context, the similar field of photometry deals with the perceived brightness by human eyes and is particularly interesting in the field of measuring light sources. Therefore, a special function, called luminosity function is defined. Apart from that, radiometry and photometry are analogue terms. The used physics behind radiometry and photometry are geometrical optics. Here, only the field of radiometry is considered. The following definitions are well known.

The radiant flux or radiant power Φ is defined as an amount of electromagnetic energy per time interval. Figure 2 left and side shows a surface element of a spacious radiation source.

γ

M

dAM dΩ^M

dΦ ^sphere

ρ

A

M1

γ

M2

γ φM

Sphere

Figure 2: Illustration of radiometric quantities

𝛾𝛾_𝑀𝑀 is the angle to the normal of the surface element 𝑑𝑑𝑑𝑑_𝑀𝑀, at which 𝑑𝑑Ω𝑀𝑀 is the solid angle element. The index “𝑀𝑀” relates to an emitting surface and the index “𝐸𝐸” is relates to a receiving surface.

Taking an infinite flux element 𝑑𝑑Φ divided through an infinite surface element of the radiation source, the result is a physical quantity called radiant Excitance or Emittance .

𝑀𝑀 = 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝑀𝑀�𝑊𝑊

𝑚𝑚²� (1)

The radiant excitance 𝑀𝑀 (𝑥𝑥𝑀𝑀, 𝑦𝑦𝑀𝑀, 𝑧𝑧𝑀𝑀) does depend upon the position of a spatial surface source.

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Analogue to the excitance there is a physical quantity called Irradiance 𝐸𝐸(𝑥𝑥𝐸𝐸, 𝑦𝑦_𝐸𝐸, 𝑧𝑧_𝐸𝐸) which describes an infinite flux hitting the receiver surface 𝑑𝑑𝑑𝑑𝐸𝐸.

𝐸𝐸 = 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝐸𝐸�𝑊𝑊

𝑚𝑚²� (2)

A further definition is called Radiant intensity 𝐼𝐼(𝜙𝜙, 𝜃𝜃), which describes the infinite flux per infinite solid angle. The intensity is an integral property of the light source and is measured in the far field, so that the source can be considered as a point source. The intensity depends on the direction from which the source is observed but not from the distance

𝐼𝐼 = 𝑑𝑑𝑑𝑑 𝑑𝑑𝛺𝛺𝑀𝑀�𝑊𝑊

𝑠𝑠𝑠𝑠� (3)

Including the dimensions of the source and their surface dependent brightness, the so called Radiance 𝐿𝐿(𝑥𝑥𝐸𝐸, 𝑦𝑦_𝐸𝐸, 𝑧𝑧_𝐸𝐸, 𝜙𝜙, 𝜃𝜃) describes the propagation properties. The radiance is the infinite intensity referred to a surface element 𝑑𝑑𝑑𝑑𝑀𝑀.

𝐿𝐿 = 𝑑𝑑𝐼𝐼

𝑐𝑐𝑐𝑐𝑠𝑠(𝛾𝛾𝑀𝑀)𝑑𝑑𝑑𝑑𝑀𝑀 = 𝑑𝑑²𝑑𝑑

𝑑𝑑𝑑𝑑𝑀𝑀𝑑𝑑𝛺𝛺𝑀𝑀𝑐𝑐𝑐𝑐𝑠𝑠(𝛾𝛾𝑀𝑀) =

𝑑𝑑²𝑑𝑑

𝑑𝑑𝑑𝑑𝐸𝐸𝑑𝑑𝛺𝛺𝐸𝐸𝑐𝑐𝑐𝑐𝑠𝑠(𝛾𝛾𝐸𝐸) � 𝑊𝑊

𝑚𝑚²𝑠𝑠𝑠𝑠� (4)

The solid angle Ω [1 𝑠𝑠𝑠𝑠⁄ ] is the analogue spatial term to the radian in plane geometry and is defined as surface on a sphere divided through the square of the radius. In general, the solid angle is calculated by an integral over the two spatial angles:

𝛺𝛺[1 𝑠𝑠𝑠𝑠⁄ ] = 𝑑𝑑𝑆𝑆𝑆𝑆ℎ𝑒𝑒𝑠𝑠𝑒𝑒

𝜌𝜌² = � � 𝑠𝑠𝑠𝑠𝑠𝑠(𝛾𝛾_𝑀𝑀)𝑑𝑑𝜙𝜙𝑑𝑑𝛾𝛾𝑀𝑀 𝜙𝜙_𝑀𝑀1

𝜙𝜙_𝑀𝑀1 𝛾𝛾_𝑀𝑀2

𝛾𝛾_𝑀𝑀1 (5)

The cos-factor belongs implicitly to the solid angle considering radiance and is called “projected solid angle” (Ω ∙ cos(γ)).

The radiance depends upon the position on the surface element and the observation angle.

Radiance 𝐿𝐿 is the most exact description of a radiation source in geometric optics.

If applicable, further dimensions such as wavelength, polarization or time can be added. In these cases, the physical values can be defined as spectral or polarization values such as spectral radiance 𝐿𝐿𝜆𝜆(𝑥𝑥_𝐸𝐸, 𝑦𝑦_𝐸𝐸, 𝑧𝑧_𝐸𝐸, 𝜙𝜙, 𝜃𝜃, 𝜆𝜆).

With these definitions, the fundamental law of radiation can be expressed. The radiation law describes the energy exchange between two surface elements.

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One of these elements (𝑑𝑑𝑑𝑑_𝑀𝑀) is the radiation source and the other one the receiver (𝑑𝑑𝑑𝑑_𝐸𝐸).

The radiation flux, from 𝑑𝑑𝑑𝑑𝑀𝑀 to 𝑑𝑑𝑑𝑑𝐸𝐸 is then given by:

𝑑𝑑²𝑑𝑑 = 𝐿𝐿𝑑𝑑𝑑𝑑_𝑀𝑀𝑐𝑐𝑐𝑐𝑠𝑠(𝛾𝛾_𝑀𝑀)𝑑𝑑𝛺𝛺_𝑀𝑀 (6)

The flux can exactly be described from the point of sight, in which radiation is collected by the receiver surface:

𝑑𝑑²𝑑𝑑 = 𝐿𝐿𝑑𝑑𝑑𝑑_𝐸𝐸𝑐𝑐𝑐𝑐𝑠𝑠(𝛾𝛾_𝐸𝐸)𝑑𝑑𝛺𝛺_𝐸𝐸 (7) This leads to the well-known radiation law:

𝑑𝑑²𝑑𝑑 = 𝐿𝐿𝑑𝑑𝑑𝑑𝐸𝐸𝑐𝑐𝑐𝑐𝑠𝑠(𝛾𝛾𝐸𝐸)𝑑𝑑𝑑𝑑𝑀𝑀𝑐𝑐𝑐𝑐𝑠𝑠(𝛾𝛾𝑀𝑀) 1

𝑠𝑠² (8)

Equations (6) and (7) are identical and describe the same. Therefore, a geometrical factor called Etendue 𝛿𝛿 is defined. With 𝑠𝑠[𝑚𝑚] as the distance between the emitting surface element and the receiving surface element.

dAM

dAE

dΩM

γM ^γ^E

γ

^M

γ

_E

dAM dA_E

r

Figure 3: Energy transmission between two surface elements

𝑑𝑑²𝛿𝛿 = 𝑑𝑑𝑑𝑑_𝐸𝐸𝑐𝑐𝑐𝑐𝑠𝑠(𝛾𝛾_𝐸𝐸)𝑑𝑑𝛺𝛺_𝐸𝐸 = 𝑑𝑑𝑑𝑑_𝑀𝑀𝑐𝑐𝑐𝑐𝑠𝑠(𝛾𝛾_𝑀𝑀)𝑑𝑑𝛺𝛺_𝑀𝑀 (9)

For one beam, the Etendue, known as optical throughput, is a conserved quantity. If the radiation flux is observed in an optical non-dispersive system, the Etendue remains also constant. The Etendue is a quality criterion for optical systems. An optical system shall be designed so that the Etendue is maximized. From the radiation law we can get the more specific radiation square law:

𝐸𝐸 = 𝐼𝐼(𝛾𝛾_𝑀𝑀)𝑐𝑐𝑐𝑐𝑠𝑠(𝛾𝛾_𝐸𝐸) 1

𝑠𝑠² (10)

As the radiance dropped from this formula, the square law is used in order to measure the intensity of light sources. Since 𝐼𝐼 is an integral property of the light source, the square law is only applicable, if distance 𝑠𝑠 is large enough relative to the dimensions of the source. There are literature regarding these definitions: [7], [5], [8], [9]

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2.2 The concept of BSDF

The so called BSDF (Bidirectional scattering distribution function) is a numerical function, which depends on four parameters – two input angles and the two output angles in the usual spherical coordinate system. The BSDF describes the spatial distribution of the scattered light and is a generic term for transmissive, absorptive and reflective light scatter. The general BSDF can be separated into BRDF (Bidirectional Reflection Distribution Function) and the BTDF (Bidirectional Transmission Distribution Function).

BTDF BRDF BRDF & BTDF

Figure 4: Illustration of the kinds of BSDF

Scatter light, which depends on incident angle, is captured in transmission as well as in reflection. The complexity of the „Distribution body“ can strongly vary. The used physics in the measurement setup is solely geometric optics and neglects therefore effects, which normally have to be described by wave optics.

Going into detail, a lot of model and theories about stray light can be found. The concept of BSDF is a practical way to get usable scatter data from measurements.

As said, the BSDF is a function of four angles and the BSDF can be noted as 𝐵𝐵𝑆𝑆𝐵𝐵𝐵𝐵 = 𝑓𝑓_𝑆𝑆(𝜙𝜙_𝑠𝑠, 𝜃𝜃_𝑠𝑠, 𝜙𝜙_𝑠𝑠, 𝜃𝜃_𝑠𝑠). The BSDF and its nomenclature is derived by Nicodemus 1977

[10]. According to Nicodemus, the scattered radiance element is in general, proportional to the incident flux element. He assumed at first a more general function 𝑆𝑆 after he came to the more practical BSDF. He assumed a proportionality between incident flux and scattered irradiance:

𝑑𝑑𝐿𝐿_𝑠𝑠 = 𝑆𝑆𝑑𝑑𝑑𝑑_𝑠𝑠 = 𝑆𝑆𝑑𝑑𝐸𝐸_𝑠𝑠𝑑𝑑𝑑𝑑_𝑠𝑠

The proportionality constant 𝑆𝑆 [1 𝑠𝑠𝑠𝑠 ∙ 𝑚𝑚²⁄ ] can initially depend on the radiation incident angle, scattering angle, the location of the striking point and the location of the scattering point. It is not a fact, that the point, at what an incident ray strikes the surface, is the same point from which the appending ray leaves the surface. Scattering can of course be inside

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the material. In [10] 𝑆𝑆 = 𝑆𝑆(𝜙𝜙𝑠𝑠, 𝜃𝜃_𝑠𝑠, 𝑥𝑥_𝑠𝑠, 𝑦𝑦_𝑠𝑠, 𝜙𝜙_𝑠𝑠, 𝜃𝜃_𝑠𝑠, 𝑥𝑥_𝑠𝑠, 𝑦𝑦_𝑠𝑠) is described as a function of six parameters. Nicodemus named this function BSSDF (Bidirectional Scattering-Surface Distribution Function)

/ /

;

s

s s s

s s

P L I θ φ

Ω

x y z

θi θ^r

φr

φi

/ /

;

i

i i i

i i

P L E θ φ

Ω

Figure 5: Dependent quantities with regard to the BSDF

Ω: Solid angle [1 𝑠𝑠𝑠𝑠⁄ ] 𝐸𝐸: Irradiance [𝑊𝑊 𝑚𝑚²⁄ ]

𝑃𝑃: Radiation power [𝑊𝑊] 𝜃𝜃: Elevation angle [°]

𝐿𝐿: Radiance [𝑊𝑊 𝑚𝑚²𝑠𝑠𝑠𝑠⁄ ] 𝜙𝜙: Azimuth angle [°]

Index 𝑠𝑠: Input (Incident) parameters Index 𝑠𝑠: Output (Scattering) parameters

The interaction processes of radiation on the surface such as diffraction, absorption, transmission must not be considered in detail. The concept of BSDF is a way to capture scattered light by measurement techniques. In order to make the general BSSDF more practical, some simplifications are necessary. Nicodemus assumed that the surface is irradiated by constant irradiance 𝐸𝐸 and that the incident direction is considered constant over the surface. Further he assumed that the surface is isotropic (no direction dependency) or homogeneous (struc- tures throughout the surface are uniform). So 𝑆𝑆 leads to the BSDF 𝑓𝑓𝑠𝑠.

𝑆𝑆(𝜙𝜙𝑠𝑠, 𝜃𝜃𝑠𝑠, 𝑥𝑥𝑠𝑠, 𝑦𝑦𝑠𝑠, 𝜙𝜙𝑠𝑠, 𝜃𝜃𝑠𝑠, 𝑥𝑥𝑠𝑠, 𝑦𝑦𝑠𝑠) ⟹ 𝑓𝑓𝑠𝑠(𝜙𝜙𝑠𝑠, 𝜃𝜃𝑠𝑠, 𝜙𝜙𝑠𝑠, 𝜃𝜃𝑠𝑠)

For the scattered radiance, he got finally 𝑑𝑑𝐿𝐿𝑠𝑠 = 𝑓𝑓𝑠𝑠(𝜙𝜙𝑠𝑠, 𝜃𝜃𝑠𝑠, 𝜙𝜙𝑠𝑠, 𝜃𝜃𝑠𝑠)𝑑𝑑𝐸𝐸𝑠𝑠 with 𝑓𝑓𝑠𝑠 as the noted BSDF. The incident irradiance 𝐸𝐸𝑠𝑠 can be expressed as integral of 𝐿𝐿𝑠𝑠 over the projected solid angle (𝐸𝐸𝑠𝑠(𝜙𝜙𝑠𝑠, 𝜃𝜃𝑠𝑠) = ∫ 𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃𝛺𝛺 𝑠𝑠)𝑑𝑑𝛺𝛺𝑠𝑠 ⟹ 𝑑𝑑𝐸𝐸𝑠𝑠(𝜙𝜙𝑠𝑠, 𝜃𝜃𝑠𝑠) = 𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃𝑠𝑠)𝑑𝑑𝛺𝛺𝑠𝑠)

→ 𝑓𝑓_𝑠𝑠(𝜙𝜙_𝑠𝑠, 𝜃𝜃_𝑠𝑠, 𝜙𝜙_𝑠𝑠, 𝜃𝜃_𝑠𝑠) =𝑑𝑑𝐿𝐿𝑠𝑠(𝜙𝜙𝑠𝑠, 𝜃𝜃𝑠𝑠, 𝜙𝜙𝑠𝑠, 𝜃𝜃𝑠𝑠)

𝑑𝑑𝐸𝐸_𝑠𝑠(𝜙𝜙𝑠𝑠, 𝜃𝜃_𝑠𝑠) = 𝑑𝑑𝐿𝐿𝑠𝑠(𝜙𝜙𝑠𝑠, 𝜃𝜃𝑠𝑠, 𝜙𝜙𝑠𝑠, 𝜃𝜃𝑠𝑠)

𝐿𝐿_𝑠𝑠(𝜙𝜙𝑠𝑠, 𝜃𝜃_𝑠𝑠)𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃𝑠𝑠)𝑑𝑑𝛺𝛺𝑠𝑠 (11)

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Using this definition, the scattered radiance is given by:

𝐿𝐿𝑠𝑠 = 𝐿𝐿𝑠𝑠� 𝑓𝑓𝑠𝑠𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃𝑠𝑠)𝑑𝑑𝛺𝛺𝑠𝑠 (12)

This definition with infinite angles and surfaces cannot be measured in reality. In each measurement, only finite apertures and therefore finite surfaces and angles can be used. The definitive physical quantities regarding irradiation, intensity and radiance can also be measured within finite spacing and time intervals. A sensible determination must be made over which the value stays constant. A real measured value is therefore always an integral value and can be considered as an average value. This “Average” BSDF is defined according to [10]

as:

→ 𝑓𝑓̅𝑠𝑠 = 𝐿𝐿𝑠𝑠

𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃𝑠𝑠)𝐿𝐿𝑠𝑠𝛺𝛺𝑠𝑠 (13) The BSDF 𝑓𝑓̅𝑠𝑠 can be transformed into the following shapes.

→ 𝑓𝑓̅𝑠𝑠 = 𝐿𝐿𝑠𝑠

𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃𝑠𝑠)𝐿𝐿𝑠𝑠𝛺𝛺𝑠𝑠 = 𝑀𝑀𝑠𝑠

𝐸𝐸𝑠𝑠𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃𝑠𝑠)𝛺𝛺𝑠𝑠 = 𝑃𝑃𝑠𝑠

𝑃𝑃𝑠𝑠𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃𝑠𝑠)𝛺𝛺𝑠𝑠 (14)

These assumptions by Nicodemus are not always valid (Stover, John C. “Optical Scattering - Measurement and Analysis” [11]). The profiles of radiation sources, even in the case of using a laser (Gauss-profile), are not complete homogeneous.

The definition of the BSDF asks for parallel and homogeneous light beams. Furthermore, the light source shall be monochromatic. A look to the kind of irradiation shall be taken in Fig. 6 and Figure 7.

Figure 6: Surface elements irradiated with different irradiance

Figure 7: Surface elements irradiated with different incident angles

In Figure 6, the sample is irradiated with a light source whose irradiation profile is not homogeneous. The surface pattern isn’t changed via irradiation area, so each surface element generates the same scatter pattern but with different intensities. Summed up the BSDF’s of

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all surface elements the integrated value must be the same as without special profile and so BSDF measurements are independent from the irradiation profile.

In Figure 7, the sample is irradiated with a light source, whose rays have different angles of impact. In principal, these rays can cause different BSDF curves, since the BSDF depends on the incident angle. We will see that this fact can completely be neglected if a laser source is used. Furthermore, scatter light depends on the used wavelength. In the case of a wide- band source, the BSDF is a mix of different BSDF-curves. Therefore, initially one wavelength shall be used.

Scatter depends on the polarization condition too. Looking at the reflected glare light, hori- zontal polarization component often dominates the vertical polarization. In a normal background light of all polarization conditions are available in equal parts [11]. When light is scattering the amplitude, polarization condition and direction is changed.

According to [11] a more practical definition is the so-called “cos-corrected BSDF”:

→ 𝑓𝑓̅_𝑠𝑠 = 𝑃𝑃𝑠𝑠

𝑃𝑃_𝑠𝑠𝛺𝛺_𝑠𝑠 (15)

However, with reference to a given surface and absolute measurements the cos-factor cannot simply be removed. The BSDF is originally defined as proportional factor in order to describe the scattered radiance and is derived from the radiation law. Therefore, we have to include this factor; otherwise, it isn’t a definition in radiometric terms. In the literature, the BSDF is mostly defined as illustrated in equation 14. In the context of BSDF-measurements there is a value defined which give information about the whole scatter without specular and incident radiation. This value is named “Total integrated scatter” and is the integral of the BSDF over the half-space projected solid angle of the irradiated “Device under Test (DUT)” [11] .

→ 𝑇𝑇𝐼𝐼𝑆𝑆 = � � 𝑓𝑓�_𝑠𝑠

𝜋𝜋�2

0 2𝜋𝜋

0

(𝜃𝜃𝑠𝑠, 𝜙𝜙_𝑠𝑠, 𝜃𝜃_𝑠𝑠, 𝜙𝜙_𝑠𝑠)𝑠𝑠𝑠𝑠𝑠𝑠𝜃𝜃𝑠𝑠𝑐𝑐𝑐𝑐𝑠𝑠𝜃𝜃_𝑠𝑠𝑑𝑑𝜙𝜙_𝑠𝑠𝑑𝑑𝜃𝜃_𝑠𝑠 (16)

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3 A common scatter model used in the context of BSDF

The two used models in the context of BSDF are the so-called “Harvey-Shack-Model” and the

“ABg-model” [6]. The ABg-Model is used in chapter “Simulation”. Figure 8 shows a schematic from the model geometry.

y

θs

β^ β^0

ri ^r^s



r0

θ0

x z

Figure 8: Derivation of the Harvey – Shack model

The model describes the BSDF as linear-shift invariant function, which mean that the model illustrates the BSDF as function from the difference between specular and scattered rays.

The following description illustrates the context. The projected vectors in scatter- and specular direction are:

𝛽𝛽⃑ = 𝑠𝑠⃑_𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠(𝜃𝜃_𝑠𝑠); 𝛽𝛽⃑0 = 𝑠𝑠⃑₀𝑠𝑠𝑠𝑠𝑠𝑠(𝜃𝜃₀)

β�⃑ is the vector in scatter direction while β�⃑0 is the vector in specular direction. θs is the general scatter angle and θ0 the scatter angle in specular direction. The vectors r⃑ are unit vectors in Cartesian coordinates. If the BSDF data are plotted as a function of the sion �𝛽𝛽⃑ − 𝛽𝛽⃑0�, the BSDF can be illustrated in the specular range very well. The ABg-Model is written through the Parameter A, B and g:

𝐵𝐵𝑆𝑆𝐵𝐵𝐵𝐵_{𝑑𝑑𝐵𝐵𝐴𝐴}�𝛽𝛽⃑, 𝛽𝛽⃑₀� = 𝑑𝑑

𝐵𝐵 + |𝑠𝑠⃑_𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠(𝜃𝜃_𝑠𝑠) − 𝑠𝑠⃑₀𝑠𝑠𝑠𝑠𝑠𝑠(𝜃𝜃₀)|^𝐴𝐴

The next step, which is made, is to transform the more general form, in a form without vectors since the more simple form is mostly enough to acquire the scattering characteristics.

𝐵𝐵𝑆𝑆𝐵𝐵𝐵𝐵𝑑𝑑𝐵𝐵𝐴𝐴(𝜃𝜃𝑠𝑠, 𝜃𝜃0) = 𝑑𝑑

𝐵𝐵 + |𝑠𝑠𝑠𝑠𝑠𝑠(𝜃𝜃𝑠𝑠) − 𝑠𝑠𝑠𝑠𝑠𝑠(𝜃𝜃0)|^𝐴𝐴 (17) The Harvey-Shack-Model [6] looks similar:

𝐵𝐵𝑆𝑆𝐵𝐵𝐵𝐵𝐻𝐻𝐻𝐻𝑠𝑠𝐻𝐻𝑒𝑒𝑦𝑦 −𝑆𝑆ℎ𝐻𝐻𝑐𝑐𝑎𝑎(𝜃𝜃_𝑠𝑠, 𝜃𝜃₀) = 𝑏𝑏₀�1 + �𝑠𝑠𝑠𝑠𝑠𝑠(𝜃𝜃𝑠𝑠) − 𝑠𝑠𝑠𝑠𝑠𝑠(𝜃𝜃0)

𝐿𝐿 �

2

�

𝑆𝑆 2⁄

(18)

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The parameter 𝑆𝑆 lies normally between −0,5 and −2. If 𝑆𝑆 is smaller as −2, the optic material is sometimes named super polished, which is comparable with an RMS-roughness smaller than 10 Angstróm (𝜎𝜎𝑠𝑠𝑚𝑚𝑠𝑠 < 10 𝑑𝑑). Both models can be transformed into one another [6]:

Abg  Harvey - Shack: 𝑏𝑏0 =^𝑑𝑑_𝐵𝐵; 𝑆𝑆 = −𝐴𝐴; 𝐿𝐿 = 𝐵𝐵^{1 𝐴𝐴}^� Harvey - Shack  Abg: 𝑑𝑑 = 𝐵𝐵𝑏𝑏0; 𝐴𝐴 = −𝑆𝑆; 𝐵𝐵 = 𝐿𝐿^−𝑆𝑆

Most of the common design- and simulation software has adopted, like “Zemax” the ABg- Model. In order to get a better orientation about the parameter and their influence, a simple LabView program is designed. In Figure 9 a plot of the ABg-Model can be seen. On this example the specular reflection amounts to 20°.

The graph to the right is easy to understand. The scatter angle is related to the sample surface so that the specular reflection is seen at 20°. On the left side, the BSDF is illustrated as double logarithmic plot. The flat range in the graph now characterizes the specular reflection. The longer this area the broader the specular reflection peak on the right side. The left graph is an illustration independent from the incident angle. These graphs are used in the simulations and experiments later in this thesis.

peak knee

slope

peak

Figure 9: ABg–Model with A = 1 · 10^-2; B = 3 · 10^-4 and g = 3

In the models the characteristics change with the parameter 𝑑𝑑, 𝐵𝐵 and 𝐴𝐴. The parameter 𝑑𝑑 causes an offset in the ordinate. The slope and the peak broadening change with 𝐴𝐴. Parame- ter 𝐵𝐵 changes only the peak amplitude and the slope remains constant. The ABg- and Har- vey-Shack parameter can be used to estimate the total “Total Integrated Scatter” (TIS) by using equation (16). A conversation formula can be found in the Appendix D: Derivations.

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4 Requirements and discussions

4.1 General Requirements

In SPIE 3141, 220-231 (1997) [12], a similar setup is described in order to carry out scattering measurements (Figure 10)

Figure 10: BSDF measurement system according to [12]

In Figure 10, the laser radiation passes a power stabilizer in order to stabilize the power via measurement time. A chopper in relation with Lock-in amplifier avoids disturbances by background light and a polarizer with 𝜆𝜆 2⁄ -waveplate generates linear polarized light. The lens with pinhole acts as low-pass-filter. The sample is irradiated and the receiver detects the reflections.

Now, the requirements regarding the setup must be determined. It shall be possible that the sample can be irradiated from all spatial directions. Furthermore, the goniometer shall be designed flexible enough so that scatter light can be detected from all directions. The system must be automatable, which makes a special kind of goniometer with some motorized rotation stages necessary. The system shall be designed with optical rails so that, optical elements can be mounted and removed in a simple way. The scatter functions are only considered in reflection (BRDF) and not in transmission (BTDF). A sample holder allows a multi- tude of different samples. After the laser, a beam expander with adjustable aperture should enlarge the beam diameter, so that the irradiated area can be varied.

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The setup has to be as compact enough as it stays transportable and can be assembled at an optical table. For this specification, the setup shall not exceed one meter in diameter. All mechanical parts should be black in order to avoid unwanted scatter light.

In this thesis instead of a Lock-in amplifier, one kind of optics in front of the detector is used in order to improve the signal and avoid scatter from the background.

With these assumptions, the general requirements can be listed and summed up:

• Four-axes-goniometer, controlable via Graphical User Interface (GUI)

− enables the sample to be irradiated from different spatial directions

− Enables the detector to detect scatter light from different spatial directions

• Automatic measurement in plane (azimuth direction) (10 values / second)

• BSDF-type (BRDF – Bidirectional Reflection Distribution Function) (BRDF=BSDF)

• Suitable for assembly on an optical table with dimensions of (1000 x 600) mm

• Different mountable samples (Colors, Diffusers, Mirrors) with diameter ∅ < 50 𝑚𝑚𝑚𝑚

• Flexible assembly with adjustable elements (radius, detector, source, samples)

• Black anodized parts in order to avoid disturbances from background

• An Effective detector optic to avoid background stray light

• Expandable with Polarizer and Lock-In-Amplifier

4.2 Preliminary considerations as to detection of light

In order to detect the scattering characteristics, there are in principal more possibilities.

The first possibility is to image the test surface via objective at a CCD-Chip (Charge coupled device). The CCD gets the whole information about the brightness and therefore about the radiance related to a specific direction. With a calibrated system, the radiance can be directly measured. Each point on the sample is imaged on the CCD. That means, that the irradiation distribution of the source must also be measured with a CCD-Chip [11].

The CCD can also be used without optic in this case the signal to noise ratio is lower because of “collected light” is lower. For it the resolution in solid angle is higher.

The CCD-Chip can also be replaced by a diode as a basic measurement technique. In Table 1 a comparison of the different detection techniques can be seen.

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Advantages Disadvantages CCD with

image optic • No disturbing light from background

• Surface dependent radiance can directly be measured with a calibrated system

• High resolution in scatter angle

• Extensive assembly

• Extensive measurement

• Much data

• Lower resolution in solid angle

• Better signal to noise ratio

CCD • High resolution in solid angle

• Extensive assembly

• Disturbing light from background

• Much data

• Worse signal to noise ratio Diode with

image optic • No disturbing light from background

• Basic assembly

• Simple assembly

• Surface dependent radiance cannot be detected

• Lower solid angle resolution

Diode • High solid angle resolution

• Basic assembly

• Manageable data

• Disturbing light from background

• Worse signal to noise ratio

Table 1: Comparison of different detection techniques

4.2.1 Measurement according to the inverse square law

The next point, to be discussed, concerns the dimensions and the distance of the detector with which a proper measurement system shall be provided.

Comparing the intensity distribution of a light source with the BSDF, the conclusion can be drawn that BSDF is in principal the same as intensity. The only difference is that the BSDF represents an angle dependent factor, which is multiplied with the incident power flux.

𝑃𝑃𝑠𝑠 = 𝑓𝑓�𝑃𝑃𝑠𝑠 𝑠𝑠𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃𝑠𝑠)𝛺𝛺𝑠𝑠 ⟺ 𝑑𝑑𝑑𝑑 = 𝐼𝐼0𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃𝑠𝑠)𝑑𝑑𝛺𝛺𝑠𝑠 (19)

The radiation “Inverse square law” is strictly speaking valid for point sources only. If the square law is applied to expanded sources, a failure in measurement is generated. There is a so-called radiometric limit distance defined for which the failure is lower than a given per- centage.

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Assuming a lambertian source, the failure can be estimated by the following formula, which is derived from the radiation law 𝐸𝐸 = 𝐼𝐼(𝛾𝛾𝑀𝑀)𝑐𝑐𝑐𝑐𝑠𝑠(𝛾𝛾_𝐸𝐸)𝑠𝑠⁻²:

𝐼𝐼_{𝑇𝑇𝑠𝑠𝑇𝑇𝑒𝑒}

𝐼𝐼_{𝑀𝑀𝑒𝑒𝐻𝐻𝑠𝑠} = ∫ 𝐿𝐿𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃𝑠𝑠)𝑑𝑑𝑑𝑑𝑠𝑠

𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃1𝐸𝐸)𝛺𝛺0𝜌𝜌²𝐸𝐸 = ∫ 𝐿𝐿𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃𝑠𝑠)𝑑𝑑𝑑𝑑𝑠𝑠

𝛺𝛺10𝐸𝐸𝜌𝜌² = 𝐿𝐿𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃_𝑠𝑠)𝑑𝑑𝑠𝑠

𝛺𝛺10𝐸𝐸𝜌𝜌² (20)

The intensity with a lambertian source and a radiation angle of 𝜃𝜃𝑠𝑠 = 0 amounts to:

𝐼𝐼𝑇𝑇𝑠𝑠𝑇𝑇𝑒𝑒 = 𝐿𝐿 ∙ 𝑑𝑑𝑠𝑠 = 𝐿𝐿 ∙ 𝑅𝑅²𝜋𝜋

While the measured intensity according to the inverse square law amounts to

𝐼𝐼𝑀𝑀𝑒𝑒𝐻𝐻𝑠𝑠 = 𝜌𝜌²∫ 𝐿𝐿𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃𝐸𝐸)𝑑𝑑𝛺𝛺𝐸𝐸 = 𝜌𝜌²∫ 𝐿𝐿𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃𝐸𝐸)𝑠𝑠𝑠𝑠𝑠𝑠(𝜃𝜃𝐸𝐸)𝑑𝑑𝜃𝜃𝐸𝐸𝑑𝑑𝜙𝜙 = 𝜌𝜌²𝐿𝐿𝜋𝜋 ∙ 𝑠𝑠𝑠𝑠𝑠𝑠²(𝜃𝜃𝐸𝐸) =^𝜌𝜌_𝑅𝑅²₂^{𝐿𝐿𝜋𝜋𝑅𝑅}_+𝜌𝜌₂²

Figure 11: Measured intensity

The ratio between true intensity and measured intensity is then given as:

𝐼𝐼_{𝑇𝑇𝑠𝑠𝑇𝑇𝑒𝑒}

𝐼𝐼_{𝑀𝑀𝑒𝑒𝐻𝐻𝑠𝑠} = 𝐿𝐿 ∙ 𝑅𝑅²𝜋𝜋 𝜌𝜌²𝐿𝐿𝜋𝜋 𝑅𝑅𝑅𝑅² + 𝜌𝜌² ²

=𝑅𝑅²+ 𝜌𝜌²

𝜌𝜌² (21)

In addition, the relative failure with reference to the true value is given by:

𝐼𝐼𝑇𝑇𝑠𝑠𝑇𝑇𝑒𝑒 − 𝐼𝐼𝑀𝑀𝑒𝑒𝐻𝐻𝑠𝑠

𝐼𝐼𝑇𝑇𝑠𝑠𝑇𝑇𝑒𝑒 = 𝐿𝐿 ∙ 𝑅𝑅²𝜋𝜋−𝜌𝜌²𝐿𝐿𝜋𝜋 𝑅𝑅𝑅𝑅²+ 𝜌𝜌² ²

𝐿𝐿 ∙ 𝑅𝑅²𝜋𝜋 = 1 − 𝜌𝜌²

𝑅𝑅²+ 𝜌𝜌² (22)

With a failure lower than 1 % it follows for the ratio 𝑠𝑠𝐻𝐻𝑚𝑚𝑆𝑆𝑠𝑠𝑒𝑒 𝑠𝑠𝐻𝐻𝑑𝑑𝑠𝑠𝑇𝑇𝑠𝑠 𝐴𝐴𝑐𝑐𝑠𝑠𝑠𝑠𝑐𝑐𝑚𝑚𝑒𝑒𝑔𝑔𝑒𝑒𝑠𝑠 𝑠𝑠𝐻𝐻𝑑𝑑𝑠𝑠𝑇𝑇𝑠𝑠⁄ :

1 −_𝑅𝑅₂^𝜌𝜌_+𝜌𝜌² ₂ < 1 % → _0,99¹ − 1 <^𝑅𝑅_𝜌𝜌²₂ → 0,01 <^𝑅𝑅_𝜌𝜌

The result shows, that with a relative failure of 1 % in intensity measurements, the goniometer radius has to be 10 times larger than the sample radius!

ρ

θE

As

R Sample

Detector

𝑠𝑠𝑠𝑠𝑠𝑠²𝜃𝜃_𝐸𝐸 =_𝑅𝑅₂^𝑅𝑅_+𝜌𝜌² ₂

(31)

For lambertian sources, the limit distance has to be more than ten times of the radius of the source. For high-directed sources, the limiting distance can be more than ten times.

This relation should be also valid in the case of BSDF measurements. The intensity can be written in terms of BSDF 𝑓𝑓�: 𝑠𝑠

𝑃𝑃𝑠𝑠 = 𝑓𝑓�𝑃𝑃𝑠𝑠 𝑠𝑠𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃𝑠𝑠)𝛺𝛺𝑠𝑠 ⟹ 𝑓𝑓�𝑃𝑃𝑠𝑠 𝑠𝑠𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃𝑠𝑠) = 𝑃𝑃_𝑠𝑠

𝛺𝛺𝑠𝑠 = 𝐼𝐼(𝜃𝜃𝑠𝑠) ⟹ 𝑓𝑓� = 𝐼𝐼𝑠𝑠 (𝜃𝜃_𝑠𝑠)

𝑃𝑃𝑠𝑠𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃𝑠𝑠) (23) Putting 𝐼𝐼𝑇𝑇𝑠𝑠𝑇𝑇𝑒𝑒 (numerator of equation (20)) into equation (23) the ratio between measured BSDF and true BSDF (𝑓𝑓�� to 𝑓𝑓𝑠𝑠_{𝑇𝑇𝑠𝑠𝑇𝑇𝑒𝑒} ��) leads to: 𝑠𝑠𝑀𝑀𝑒𝑒𝐻𝐻𝑠𝑠𝑇𝑇𝑠𝑠𝑒𝑒

𝑓𝑓𝑠𝑠_{𝑇𝑇𝑠𝑠𝑇𝑇𝑒𝑒}

��

𝑓𝑓𝑠𝑠𝑀𝑀𝑒𝑒𝐻𝐻𝑠𝑠𝑇𝑇𝑠𝑠𝑒𝑒

�� =

∫ 𝐿𝐿𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃𝑠𝑠)𝑑𝑑𝑑𝑑𝑠𝑠

𝑃𝑃_𝑠𝑠𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃_𝑠𝑠) 𝑃𝑃_𝑠𝑠 𝑃𝑃𝑠𝑠𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃𝑠𝑠)𝛺𝛺𝑠𝑠

=

𝑃𝑃_𝑠𝑠𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃_𝑠𝑠) 𝑀𝑀_𝑠𝑠 𝐸𝐸𝑠𝑠𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃𝑠𝑠)𝛺𝛺𝑠𝑠

=

𝑃𝑃_𝑠𝑠𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃_𝑠𝑠) 𝑃𝑃_𝑠𝑠

𝑑𝑑𝑠𝑠

𝑃𝑃𝑠𝑠 1

𝑑𝑑_𝑠𝑠𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃_𝑠𝑠) 𝑑𝑑𝜌𝜌^{𝐵𝐵𝑒𝑒𝑔𝑔}²

=

𝑃𝑃_𝑠𝑠𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃_𝑠𝑠) 𝜌𝜌² 𝑃𝑃_𝑠𝑠

𝑑𝑑𝐵𝐵𝑒𝑒𝑔𝑔

𝑃𝑃𝑠𝑠𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃1 𝑠𝑠)

⟹ 𝑓𝑓��𝑠𝑠_{𝑇𝑇𝑠𝑠𝑇𝑇𝑒𝑒}

𝑓𝑓𝑠𝑠𝑀𝑀𝑒𝑒𝐻𝐻𝑠𝑠𝑇𝑇𝑠𝑠𝑒𝑒

�� =∫ 𝐿𝐿𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃𝑠𝑠)𝑑𝑑𝑑𝑑𝑠𝑠

𝐸𝐸_𝑠𝑠𝜌𝜌² = 𝐿𝐿𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃𝑠𝑠)𝑑𝑑𝑠𝑠

𝐸𝐸_𝑠𝑠𝜌𝜌² (24)

This result is identical with equation (20) and results in 𝑓𝑓�� 𝑓𝑓𝑠𝑠_{𝑇𝑇𝑠𝑠𝑇𝑇𝑒𝑒} ��𝑠𝑠𝑀𝑀𝑒𝑒𝐻𝐻𝑠𝑠𝑇𝑇𝑠𝑠𝑒𝑒 = (𝑅𝑅²+ 𝜌𝜌²) 𝜌𝜌⁄ . The ² true intensity and BSDF is about this factor higher than the measured value!

f_s_True

��; f��: True and measured BSDF [1 sr]sMeasure ⁄ Pi; Ps: Power, Flux [𝑊𝑊]

θs, θE: Scatter direction [°] L: Irradiance [W sr ∙ mm⁄ ²] As; ADet: Sample surface and Detector surface [𝑚𝑚𝑚𝑚²] ρ: radius of gonoimeter [mm²] E_s; M_s: Irradiance, Excitance [W mm⁄ ²]

Looking at a coherent light source, like a laser it is obvious that there are some deviations because of the not fulfilled radiometric square law. Only the scatter light aside of the laser spot can provide a relevant value regarding the BSDF.

It was shown, that with 𝜌𝜌 = 10𝑅𝑅 the failure is 1 %. A photo sensor does this far field technique. There is also a near field technique, which can be done by CCD cameras [13]

If the surface is placed close to the detector, the test surface would appear as spatial source and with it, the detector would detect an integrated value over the visible angle. In the case of highly specular reflectance, as in mirrors, the limiting distance cannot be maintained.

Therefore, only measurements aside specular directions are valid.

(32)

4.2.2 Estimation of the dynamic range

A practical detector must be sensitive enough in order to detect the reflections from the sample. A material with very low reflectance 𝑅𝑅 = 4,5 ∙ 10⁻⁴ is a kind of material with vertical arranged nanotubes from carbon [14], [15]. This reflectance shall serve as reference value in order to estimate the irradiance (power) in a distance ρ from the sample.

Now it is assumed the laser has a total power of 1 𝑚𝑚𝑊𝑊, the reflectance 𝑅𝑅 amounts to 0,045 %, and the reflection shall be measured at a distance of ρ = 500 𝑚𝑚𝑚𝑚. These are realistic values with regard to a mobile and manageable BSDF-measurement system. The black sample shall reflect the laser radiation uniformly (lambertian reflection) into the half- space.

The constant reflected radiance from the surface with reflectance 𝑅𝑅 into the half-space (projected solid angle Ω𝑠𝑠 = 𝜋𝜋) is given as:

𝐿𝐿𝑠𝑠 = 𝑅𝑅 𝑃𝑃_𝑠𝑠

𝜋𝜋𝑑𝑑𝑠𝑠 = 𝑅𝑅𝐸𝐸_𝑠𝑠 𝜋𝜋 The flux (Power 𝑃𝑃𝑠𝑠), detected by the detector is then given as:

𝑃𝑃𝑠𝑠 = 𝐿𝐿𝛺𝛺𝐵𝐵𝑒𝑒𝑔𝑔𝑑𝑑𝑠𝑠𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃𝑠𝑠) = 𝑅𝑅 𝑃𝑃_𝑠𝑠

𝜋𝜋𝑑𝑑_𝑠𝑠𝛺𝛺𝐵𝐵𝑒𝑒𝑔𝑔𝑑𝑑𝑠𝑠𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃𝑠𝑠) = 𝑅𝑅𝑃𝑃_𝑠𝑠 𝜋𝜋

𝑑𝑑_{𝐵𝐵𝑒𝑒𝑔𝑔}

𝑠𝑠² 𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃𝑠𝑠)

In this equation, the solid angle was approximated as detector surface divided trough the distance from the sample. The detected irradiance can then be calculated as:

𝐸𝐸_𝑠𝑠 = 𝑅𝑅1 𝜋𝜋

𝑃𝑃_𝑠𝑠

𝑠𝑠²𝑐𝑐𝑐𝑐𝑠𝑠(𝜃𝜃_𝑠𝑠) (24) The largest irradiance is detected with 𝜃𝜃𝑠𝑠 = 0°:

𝐸𝐸𝑠𝑠 = 4,5 ∙ 10⁻⁴1 𝜋𝜋

1 𝑚𝑚𝑊𝑊

(500 𝑚𝑚𝑚𝑚)² = 0,57 µ𝑊𝑊 𝑚𝑚𝑚𝑚²

In the case of power is detected at a large angle such as 𝜃𝜃𝑠𝑠 = 89° the irradiance is just only

𝐸𝐸𝑠𝑠 = 4,5 ∙ 10⁻⁴1 𝜋𝜋

1 𝑚𝑚𝑊𝑊

(500 𝑚𝑚𝑚𝑚)²𝑐𝑐𝑐𝑐𝑠𝑠(89°) = 0,01 µ𝑊𝑊 𝑚𝑚𝑚𝑚²

Simulation and Design of a stray light measurement system according to the concept of BSDF (Bidirectional Scattering Distribution Function)