Introduction
The light generated by a LASER LED in the figure above may be used to detect an excessive inclination or height of a closure, and also the filling level of a beverage in a transparent container. Just two in a multitude of possible electromagnetic measurements with a specific purpose, or inspections. The industrial inspection equipments using electromagnetic waves in a wide frequency spectrum are:
- useful for quality assessment in many branches of industry, an example relevant for us being all Foods and Beverages,
- operated in the Frequency or Time domains,
- using a frequency spectrum from 30 kHz - 300000 THz, namely from Ultrasounds to Gamma-Rays (see figure below).
Gathering Information by the em Waves
Information about a beverage, packaging, foodstuff or pharmaceutical, can be gathered from its interaction with electromagnetic (em) waves. The information may be stored in the:
- amplitude,
- phase,
- polarisation,
- angular distribution of energy transportation,
- spectral characteristics.
When retrieved from the wave, certain material properties may be determined indirectly. When compared to direct material analysis, the indirect methods require calibration and result quite prone to interferences from undesired sources. On the other hand, however, they allow the determination of features inaccessible by direct methods, such as non-destructive material testing, deep penetration depth or an high measurement speed. Non-destructive testing and high measurement speed are clearly vital features during the Automated Quality Assurance of Food, Pharma and Beverage Packagings of all packagings and materials. Measurements conditioned by Production Machinery nominal speed.
Projection of a quantum state-vector | ψ⟩ into a vector subspace S by a projector P(S ). Here shown the projection onto a ray corresponding to | ψm⟩, with which it makes an angle θ. The probability for this transition to occur is cos2θ. But, a von Neumann Measurement is not an individual rather an entire set of possible projections, onto a complete orthogonal set of rays of the Hilbert space being measured (
abridged by Jaeger/2009)
Can we really know everything about these objects just by the outcomes of the automated analytical measurements? There is more than an obstacle preventing us from doing this. With reference to the figure above, according to the postulates of quantum mechanics the state of whatever physical system, whatever its size, mass or chemical constitution, is represented by a vector | ψ⟩ in a complex vector space H . Complex-valued vector space named Hilbert space. Vector spaces do not admit any preferred basis, and because of this reason the postulates of quantum mechanics are independent of any such basis. But, a vector space containing all possible states for an object, has plenty of structures. As an example, the position in the physical space implies a preferred basis. Also, each type of particle or collection of them, comes with its own base in the Hilbert space. In the case of particles, the total space is calculated by taking tensor products between copies of the space associated to each particle. Spaces which cannot be inferred by the measurements’ outcomes after considering that similar outcomes would also be obtained after a rotation of the state vector in the state space,by a unitary transformation. What before shows that the Information about the position basis and the tensor product structure is not encoded in the outcomes of the Measurements. Also, (see figure below) we can only ask whether the system is in a small subset of a possible state. Those particular states for which the property we measure is well-defined.
The complex vector Hilbert space H of Quantum Mechanics, depicts a multitude of possible eigenstates | ψ⟩ related to what is realized as a single observation O at Time t0. Themselves based in subspaces of H and each of them encoding a definite history of a physical system being measured. Any property we choose to observe, is well-defined only for a small subset (blue colour) of any of the possible states. Far from there, the superposed terms cancels each other (O.C. Stoica in eds. A. Aguirre, B. Foster, Z. Merali/2015)
Definition meaning stability of the measured signal and amplitude over the Detector’s or Observer’s own threshold sensitivity. Knowingly, if the initial conditions of a measurement or observation could be specified completely with infinite precision, the solution should be unique. But, a Detector (or, Observer) “questions the Object” about just one of its physical properties. After establishing the correlation of Observer or Detector and object, the “answer” is encoded in their resulting superposition:
- positive answer, i.e., an Information state “1“ or “yes”, if a specific physical property results well-defined in a small subset of a possible state;
- negative answer, i.e., an Information state “0“ or “no”, on the opposite.
on practice, … bits of Information. At any instant of Time, there are coexisting realities compatible with those answers. Looking the fine-detail, the question selects a subset among all solutions of the Schrödinger equation. But the possible answers to the question are limited by the amount of the remained solutions. Our observations allow us to specify only partially the initial conditions, i.e. those at time t0.
That’s why there are more possible solutions at time t2, say more possible measurement outcomes. What precedes was yet partially described with other words and formalism (Minkowski, 1908) with respect to the simultaneity of Events as seen by Detectors or Observers in reciprocal movement. One of the many effects of the fact that each 4D point comprises an infinity of 3D points and that all objects, atoms, pixels, or Observers are sets of 4D points.
Long time ago it was understood that the measurement of any physical property of one and the same object, could present results completely different. An example in the figure at right side, showing the world-tube of an object, i.e., an atom of Silicium part of a pixel in the array of a CMOS imaging Detector, before and after the branching related to the interaction with an impinging photon. After the interaction the same atom detects in different bases simultaneous low and high grey levels, depending on what of the tilted 3D hypersurfaces (∑0 or ∑1) is the base for the projection of the quantum state-vector |ψ⟩. By General Relativity it is known that they coexist multiple (potentially, an infinity) hyperplanes ∑i interposed between the planes shown ∑0, ∑1, inclined at different angles. Implying the commonly observed measurement fluctuations in a wide spectrum of eigenvalues, causing the repeatability error.
Aside of these obstacles, an ideal measurement requires an infinite time available to establish the correlation between the object to be measured and the Environment, including Detector or Observer. Why an infinite time? One of the ways to answer this fundamental questions is that correlations between the two systems Detector and Object (i.e., an atom of Silicium in a pixel and a photon reflected by a container) is:
- progressively established during interaction,
- happening in a Domain including the multitude of correlations with the Environment,
- proportional to the natural logarithm (ln t) of the interaction time t.
A measurement is the collective name for the set of correlations established during the interaction between an Object and a Detector. Both typically rich of substructure and microstates; and with the Detector yet correlated to its causally related Environment before the interaction.
Each measurement selects a subset among all solutions of the Schrödinger equation. But the possible answers to the question are limited to the remained solutions. Our measurements allow us to specify only partially the initial conditions, i.e., those at time t0. That’s why they coexist additional possible solutions at time t2 ( O.C. Stoica in eds. A. Aguirre, B. Foster, Z. Merali/2015)
“A measurement is the collective name for the set of correlations established during the interaction between an object and a Detector”
Extending the correlation (or, interaction) Time, reduces the space of the solutions of the Schrödinger equation describing the object-Detector superposition, as hinted by the figure above. An observation well-known to the Instrumentation Engineers acting in the Maintenance Departments. Accustomed to witness the inverse relation of the fluctuations of the measurements’ amplitudes and measurement duration time. Then, with reference with the figure above, the observed fluctuations of a measurement started at time t0, shall show a dispersion progressively sharpening at t1, t2,… What, from a:
- Classic perspective, was tentatively explained in terms of increased measurement reproducibility. Circular explanation, without any added explanatory power;
- Quantum perspective, is the effect of the selection of a subset of all the coexisting solutions of the Schrödinger equation. A selection of histories.
Thus, correlation Time acts asymptotically selecting a single history where object and Detector coexist, for t → ∞. Synthesizing, no measurement technology may proceed to a total retrieval of Information by macroscopic objects. This introduction paves the way to answer a practical question. What is the the optimal amount of Detectors to scan the Hilbert space correspondent to the correlation established between object and its Environment ?
A Detector “questions” the object about any of its physical properties. After establishing the correlation between Detector (or, Observer) and an Observable physical property of the object, the “answer” is encoded in their resulting superposition: positive answer, i.e. an Information state “1“ or “yes”, if a specific physical property results well-defined in a small subset of a possible state; negative answer, i.e. an Information state “0“ or “no”, on the opposite
1-parameter Measurements
The amounts of water in gaseous phase (foam-phase) and liquid-phase in a closed bottle can be considered unrelated physical properties. Vectors' orthogonality reflects their measurements' independence. All High Frequency Fill Level Inspections treating foaming beverages are sensible to a long row of other variables, in the graphic above simplified for reasons of clarity and synthesized as: “environment”. Here, as environment is meant the complex vector superposition of the projections along that direction of many other terms. Environmental temperature, environmental humidity, container speed, filling pressure, off-axis in millimetres of the HF Fill Level Inspection bridge with respect to the bottle, etc. The High Frequency bridge's final measurement, the complex-valued vector Impedance Z, can be represented as a function of the vectorial superposition of foam-phase, liquid-phase and many others without relation with the bottle’s capacity. As a consequence, the module of the vector Impedance Z results fluctuating, originating the Inspection’s widely observed False Positives and False Rejects
The first application of electromagnetic wave interaction with matter involved measurement of amplitude changes at a single frequency caused by material properties. It is the approach still today adopted by many of the High Frequency Fill Level Inspections. Inspection methods finely detailed in other pages of this web site. High Frequency Fill Level inspections particularly prone to parasitic reflections and amplitude instabilities, as explained in this web page. This approach was soon supplemented by single frequency phase measurements, in order to avoid distortions through amplitude instabilities or parasitic reflections. Of course, the 1-parameter measurements by definition, require dependence only on one variable in the measured process and sufficient stability of all other conditions. If that is not the case, the 1-parameter measurement fails. As an example, the performances of the High Frequency fill level inspection (defects' detection ratio and false rejects' ratio) applied to a statistically significative population of containers, are heavily conditioned by the constancy of several factors, as shown and detailed in the remarks to the figure above.
The main contributions, with a specific value for each one inspected container (they all random variables whose superposition is itself a random variable), accounting for nearly all of the performances' values established during containers' High Frequency fill level inspection, are:
- ambient temperature, causing changes on the dielectric permittivity ε of the beverage, being this last in-the-ambient, then following ambient temperature tendencies;
- ambient relative humidity of the air, causing biased results arising by a secondary contribution (air features its specific relative dielectric permittivity different than that of water), added to the one of medium (the liquid in the container), whose Capacity of electrons is measured by the High Frequency fill level inspection bridge;
- containers' speed, preventing a tilted upper surface of the liquid in the container;
- physical properties of the beverage, i.e., density and kind of foaming, due to the Mixer and Carbonator Machines;
- chemical properties of the beverage, i.e., molecular structure, foaming;
- physical properties of the container, i.e., container height, diameter, shape, density, etc.;
- chemical properties of the container, i.e., container molecular structure;
- filler machine conditions, i.e., quality of the Filler Valve and of the filling setup during production;
- foaming, affecting the beverage in the neck and head space of the container.
Visibly, too many different conditioning factors making the 1-parameter Fill Level Inspection an unavoidable generator of false positives and false negatives (false rejects and non-rejected low-filled containers) we all know. 1-parameter electromagnetic measurements can be advantageously complemented by other methods, to remove disturbances by undesired sources. The effect of temperature is frequently eliminated by an additional temperature sensor. A case explored in the fine details in another page of this web site with reference to the High Frequency Fill Level inspection.
2-parameters Measurements
In the majority of the cases, the filling level of these bottles has been measured by mean of a 2-parameters technology at ~ 21 MHz
2-parameters measurement are now the state of the art. In the 2-parameters methods they are simultaneously determined at a single frequency the measurements’ changes deriving by two separate channels, for the:
- amplitude,
- phase,
arising by a material exposed to the electromagnetic field. It is the case of some commercially available High Frequency Fill Level inspections present in the Full Bottle Inspectors and devoted to Foaming Beverages. An application we explore in its fine-details in another page of this website. The single frequency, in the case of the High Frequency Fill Level inspections, is typically ranging ~ 21 MHz. In the figure below an example referred to X-Rays photons determination of energy and polarisation. Alternative approaches to the acquisition of amplitude and phase, are the acquisition of the:
- permittivity,
- loss factor,
of a material, or the:
- shift of the resonant frequency,
- deterioration of the quality factor,
of a resonator. This way, as an example, they become possible the simultaneous determination of:
- material density,
or the:
- mass,
- moisture content.
Example of measurement in 2 Sample spaces of the properties (Energy, Polarization) of an object. Yellow and gray colour are a couple of X-ray Silicon Imaging Detectors. A Compton scattering Event in the upper detector implies a first measurement of Energy. By the measured value, it is determined the maximum scattering angle and thus drawn a hollow cone in which a coincidence Event can occur. If a photon absorbed in the lower detector corresponds to a scattering in the upper both in time and space, its polarisation can be found. Two measurements of Energy, both random variables, result in X-ray's total Energy and Polarisation angle ( Muleri, et al./2012)
Pharma and Food Measurements
The line of reasoning and methods listed until now, valid for all Automated Quality Assurance Controls for the industries of:
- Food,
- Beverage,
- Pharmaceuticals,
diverge for the Pharmaceutical and Food Packagings, when what has to be measured is the amount of moisture. Knowingly, many of the dry Foods and Pharmaceuticals keep their initial qualitative characteristics only if the moisture is nil or under some specified limits. For these materials they exist two methods adopting microwaves rather than ionising radiations, allowing:
- density-independent moisture measurement, capable to measure the moisture content within an accuracy of fractions of a percent;
- mass-independent moisture measurement.
Meaning that the common radiometric (i.e., by mean of Gamma-Rays) mass gauges, requiring activities and calibrations of Maintenance Technicians, can be avoided jointly with the any negative influence of the Gamma-Rays radiation on materials such as foodstuffs. 2-parameter measurements adopt as interactant the microwaves. Because of this basic reason, they are satisfactory only in environments and for materials, where the material's:
- conductivity is small,
- dielectric losses dominate.
This because the effective permittivity and the dielectric loss factor already constitute
a set of 2-parameters, which can be determined by a 2-parameter microwave measurement. If for example an ionic conductivity is included in the effective loss factor, as it is often the case in practise, one additional microwave measurement is required.
Frequency spectrum of the complex-valued vector Dielectric Permittivity ε. Water dipolar bonds relaxation evidenced by yellow colour at the frequency of 21 MHz adopted by electronic inspectors’ High Frequency (HF) fill level inspection. The red and blue curves represent the real and imaginary components of the complex vector Permittivity
3-parameters Measurements
When these notes are being written, the 3-parameter determination is gaining ground worldwide as a superior level of inspection. For example, the third parameter can be acquired by adding a measurement at another frequency. However, since the dielectric properties change quite slowly with frequency (see figure above, at right side) a certain care has to be exercised, so to choose a frequency sufficiently different than the original one.
Multi-parametric Measurements
In real applications, what interferes is not just the the conductivity of a material, rather also the additional contributions from several other constituents, affecting the permittivity spectrum. The rationale of the dielectric multi-parameter measurements lies in taking account of that situation, allowing the simultaneous determination of a larger set of unknowns. The key point lies in the way to correlate the direct measurement variables like amplitudes and phases at various frequencies and the desired information about the properties of the material, thus calibrating the multi-measurement instrument. The solution consists in a physical model of the interrogated material, allowing the direct computation of the desired relations. As an example, models of this kind have been developed in the past for some idealized compositions of constituents, by applying electromagnetic analysis. Also, they exist empirical approaches, but very often it is experienced that a model satisfactorily working in a context, fails for another even though it only differs marginally from the first. The key point is that the majority of the materials are too complex to be described accurately by a simplified models.
Statistical Evaluation
A recently proposed solution evaluates the statistical properties of the matter. The dielectric spectrum, when recorded across a wide bandwidth at a limited number of sampling frequencies, contains a vast amount of Information, although in a subtle manner, not yet fully understood by physical models. A well visible example in the figure below. It shows the effect over a dielectric of Signals’ frequencies in the wide range (0 - 34) GHz. Wide deviations out of what expected by Classical Electrodynamics are registered in the range (27 - 34) GHz.
The behaviour expected by Classic Electrodynamics for the simplest electric components is visibly violated when increasing the Signals' frequency. In this example a dielectric spectrum. When recorded across a wide bandwidth at a limited number of sampling frequencies, it contains a vast amount of Information although in a subtle manner, not yet fully understood by physical models. In the example, Signals’ frequencies ranging (27 - 34) GHz correspond to wide deviations out of what expected by Classical Electrodynamics.
The measurement instrument has to undergo a learning phase, when the instruments is in contact with samples of materials of yet established properties. The recorded measurement values represent a section of the dielectric spectrum. Then, a calibration curve can be extracted by applying suitable Statistics methods of Multivariate analysis. Later, the unknown materials can be evaluated using that calibration curve. This modern approach delivers valuable results with measurement values recorded in the:
- frequency domain,
- time domain,
measurements. It is actually mainly adopted in the radio frequency (RF) and microwave domains, with wide open space fur future extension and improvements.
Links to other pages:
- Electromagnetic Measurements of Food and Beverages: an Information Retrieval problem
- Near-Field 3D
- Physics of Triggering
- Fill level inspection tech: a TCO point of view
- What Detectors detect ?
- Electronic Inspectors’ nonclassic components
- What Today Is a "Root Cause"
- Measures in a Decohering Environment