Juergen Popp

The Bouguer-Beer-Lambert law is frequently applied in spectroscopy and spectrophotometry. Often it is assumed that it accurately describes the observed changes induced by light-matter interactions and properly reflects the physical phenomena at play. For most cases, however, this is not true, and issues can arise when the Bouguer-Beer-Lambert law is applied uncritically. In this short article, we will comment on its fundamental limits and their physical background.

The Bouguer-Beer-Lambert (BBL) law has one big flaw, and that is its name. A more appropriate denomination, in our opinion, would be the Ideal Absorption law. In analogy to the ideal gas law, everybody would be well aware of the fact that it is just a (sometimes very good) approximation. Consequently, we would be much more cautious when applying it. Also, authors of basic textbooks and university teachers would spend more effort to explain its limits. Since completely renaming a well-known relation is usually not a good idea, we will call it the “BBL Approximation” in this article, where we will make you familiar with its limitations and pitfalls.

For truly understanding the BBL approximation, it is helpful to have some knowledge about its origin. When Bouguer and Lambert found the expression (1,2):

they were studying the absorbing properties of the atmosphere. 

 is the intensity of the light before and 

 the intensity after the traverse through the atmosphere, 

, a constant. As a side note, in the original expression the natural logarithm came up, but since it was much easier at that time to compute decadic logarithms, and since both differ just by a constant, the decadic logarithm was used later on.

Now, what are the limitations of equation 1? In general, it is always a good start to recall the particular use case—the experimental conditions (see Figure 1). If we acquire a spectrum of the atmosphere, the light source and the detector would actually be within the medium that is investigated. Consequently, we are not dealing with transmission of light through a sample, but with light that travels within a medium. Accordingly, in contrast to transmission, there is no reflection, since there is no interface between the incidence medium and the sample, as the sample is in fact the incidence medium. In addition, there is no interface between the sample and an exit medium, because the sample is also the exit medium in this special case. So, again, reflection at this second interface also does not occur. If a sample is put in the light path that has well-defined interfaces perpendicular to this path, then the light transmitted into the sample is not only reflected at the second interface and traveling back, but a part is then reflected at the first interface. This part is then traveling along the original light path until it hits the second interface where again a part is reflected and traveling back (3). These multiple reflections are not considered in equation 1. If the thickness of the sample is the same over the whole area where the light beam hits it, then something happens which is not at all taken into account in equation 1. Light behaves as a wave and the forward and backward traveling waves interfere with each other. Depending on the thickness of the sample, the wavelength of the light and the refractive index of the sample at that wavelength, the light intensity fluctuates (4). If the interference is destructive, the intensity will be lower than expected and higher for constructive interference. Accordingly, for such samples one would not obtain a straight line, if the intensity was plotted over the thickness, but it would in fact fluctuate. This is of great importance for interpreting infrared (IR) absorbance spectra, since the described effects will affect band intensities in corresponding spectra. In addition, band shapes can change and shifts can occur. Typical samples, where this can happen, are films on IR transparent substrates like CaF

, ZnSe, or Si (see Figure 2). The latter two often lead to distinct interference fringes which boldly remind you that equation 1 is just a first approximation (5). For CaF

, visible fringes are absent, but interference effects are still at play, although more subtly (6). Thin films on metals are another case where the wave nature of light leads to pronounced effects: one measures the reflectance, but often obtains spectra very similar to transmission experiments. Constructive and destructive interference is in particular strong for such samples, and bands can even occur in spectral regions where there is no absorption of the material the film is made of (7,8). Many methods have been devised to deal with interference effects. Unfortunately, most of them just aim at removing the fringes visually without considering their physical origin. Consequently, other interference-based changes remain in the spectra, which makes their correct interpretation difficult. To achieve more than cosmetics, a wave optics-based approach is required (5,6,8).

FIGURE 1: Schematic display of the different conditions of the experiments as performed (a) by Bouguer and Lambert and (b) the way spectroscopy and spectrophotometry are usually performed today.

FIGURE 2: Comparison of the influence of sample geometry, measurement technique and substrate on absorbance spectra. Bottom shows light interactions with different substrates. Adapted with permission from reference (9).

A common misbelief is that interpreting equation 1 in the following, different way, can solve the issues mentioned above in general:

 is the transmittance of the sample, and 

 is the transmittance of the substrate. Where does equation 2 come from? It actually reflects the way spectrophotometry is being performed. Accordingly, 

 is the transmittance of the absorbing solute in a non-absorbing solvent and 

 is the transmittance of the neat non-absorbing solvent. However, when certain conditions are met, equation 2 will actually work. As long as the cuvette is thick and has thickness inhomogeneities, these will cause the interference effects to average out (10). Furthermore, as long as the refractive index of the solution is not very different from that of the neat solvent, the way light is multiply reflected is approximately the same, and corresponding effects cancel out as well. Finally, the refractive index of the solvent should be as small as possible, but values up to around 1.5 are all right, if equation 2 is used, in contrast to equation 1, where the refractive indices are assumed to stay close to 1 as those of gas mixtures do (11). So far, we did not discuss the BBL approximation in the form which is commonly used in textbooks (12,13).

 is the molar absorption coefficient and 

 is the molar concentration. Before we discuss the limits of the right side of equation 3, a word of caution is necessary: It seems that using the molar concentration is somewhat arbitrary, and one can use mass or weight equally well. This is a fallacy caused by the assumption that this right part is empiric. Actually, it is not, as recently has been shown (14). It must be a concentration based on the number of molecules in a certain volume. This means that amount or volume fractions for close to ideal mixtures are equally acceptable, but not weight or mass fractions. Overall, you should keep in mind that the concentrations must be very small. Not only because of changing chemical interactions that can lead to changing material properties—this is nowadays a very common misconception. In fact, if you put a dye in different solvents with which it does not chemically interact, it nevertheless changes its color. The reason is that light interacts with matter and polarizes it, which can lead to color changes. The degree of polarization depends on the environment of a molecule (15) and, thereby, the color changes (16). As a consequence, 

 is a constant only for neat substances. As long as you keep the concentration small, a dye molecule will only be influenced by the solvent, but once the concentration becomes too large, the dye molecule is also influenced by others of its kind. If you are interested in larger concentrations or even in liquid or solid mixtures (in gases there is no problem at lower pressures due to the larger intermolecular distances), and you have made sure that chemical interactions do not occur or did not change, we recommend focusing on comparably weak bands (17), because the lower the transition moment, the smaller the polarizability. Focusing on a single band caused by a weak transition can be much more accurate than chemometric methods using the full spectrum (18).

By the way, sometimes you read that shadowing of one molecule by another is a problem at higher concentrations. In our opinion, this is nothing but a hoax, since molecules are so small compared to the wavelength that light acts as a wave and not as a ray.

Coming back to the experiments conducted by Bouguer, Lambert, and Beer, there is one more aspect, we have not discussed yet. Gases as well as diluted solutions are microhomogeneous. This means that if you use a microscope operating at the same frequencies, wavelengths, or wavenumbers and inspect your sample, it will look exactly the same everywhere. If this is not the case, you will be in trouble, if you want to apply the BBL approximation (19,20). As an example, consider a polymer like polytetrafluoroethylene which has pores with diameters in the micrometer range. This might be all right in the terahertz range, because, due to the long wavelengths, it feels an effective refractive index. If the wavelengths are smaller, than light that travels within a pore is not absorbed at all. On the other hand, while entering and leaving the pore, it is reflected, leading to scattering. Depending on the microstructure and the wavelength, one or the other effect will dominate. But, since the microstructure depends, for example, on the thickness of the polymer, simply determining the absorbance and dividing it by the thickness will not lead to the same result as equation 1 would imply.

Certainly, this short article could only touch superficially on this topic. A more detailed discussion of these and other limits can be found in a recent review by us (9). For solutions of some of the related problems, we suggest to consult books about wave optics and dispersion theory (4,21,22).

(2) Lambert, J. H.; Anding, E. Lambert’s photometrie: (photometria sive de mensura et gradibus luminis, colorum et umbrae). (1760). Heft 3, Tl. 6 und 7 -

(3) Airy, G. B. On the Phenomena of Newton’s Rings token formed between two transparent Substances of different refractive Powers. 

The London and Edinburgh Philosophical Magazine and Journal of Science

rinciples of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light; 

(5) Mayerhöfer, T. G.; Pahlow, S.; Hübner, U.; Popp, J. Removing Interference-Based Effects from Infrared Spectra – Interference Fringes Re-Revisited. 

(6) Mayerhöfer, T. G.; Pahlow, S.; Hübner, U.; Popp, J. CaF2: An Ideal Substrate Material for Infrared Spectroscopy? 

(7) Mayerhöfer, T. G.; Popp, J. The Electric Field Standing Wave Effect in Infrared Transflection Spectroscopy. 

(8) Mayerhöfer, T. G.; Pahlow, S.; Hübner, U.; Popp, J. Removing Interference-Based Effects from the Infrared Transflectance Spectra of Thin Films on Metallic Substrates: A Fast and Wave Optics Conform Solution. 

(9) Mayerhöfer, T. G.; Pahlow, S.; Popp, J. The Bouguer-Beer-Lambert Law: Shining Light on the Obscure. 

(10) Hawranek, J. P.; Neelakantan, P.; Young, R. P.; Jones, R. N. The Control of Errors in I.R. Spectrophotometry—III. Transmission Measurements Using Thin Cells. 

(11) Mayerhöfer, T. G.; Mutschke, H.; Popp, J. Employing Theories Far beyond Their Limits—The Case of the (Bouguer-) Beer–Lambert Law. 

(14) Mayerhöfer, T. G.; Popp, J. Beer’s Law Derived from Electromagnetic Theory. 

(15) Lorentz, H. A. Ueber die Beziehung zwischen der Fortpflanzungsgeschwindigkeit des Lichtes und der Körperdichte. 

(16) Spange, S.; Mayerhöfer, T. G. The Negative Solvatochromism of Reichardt‘s Dye B30 – A Complementary Study. 

(17) Mayerhöfer, T. G.; Ilchenko, O.; Kutsyk, A.; Popp, J. Beyond Beer’s Law: Quasi-Ideal Binary Liquid Mixtures. 

(18) Mayerhöfer, T. G.; Ilchenko, O.; Kutsyk, A.; Popp, J. Infrared spectroscopy of quasi-ideal binary liquid mixtures: The challenges of conventional chemometric regression. 

(19) Jones, R. N. The Absorption of Radiation by Inhomogeneously Dispersed Systems. 

(20) Mayerhöfer, T. G.; Popp, J. Beyond Beer’s Law: Spectral Mixing Rules. 

Wave Optics in Infrared Spectroscopy - Theory, Simulation and Modelling;

 are with the Leibniz Institute of Photonic Technology (IPHT), in Jena, Germany, and the Institute of Physical Chemistry and Abbe Center of Photonics at Friedrich Schiller University, in Jena, Germany. Direct correspondence to: 

Articles

The Bouguer-Beer-Lambert law has its limitations and it doesn't always properly reflect the physical phenomena at play. This article examines the law's limitations. 

Understanding the Limits of the Bouguer-Beer-Lambert Law

	This article highlights the use of optical molecular spectroscopy in combination with artificial intelligence (AI) methods in the framework of process analytical technology (PAT). Specifically, we focus on the combination of Raman, infrared (IR), mid-IR, and near-infrared (NIR) spectroscopy and AI methods such as machine learning, deep learning, and chemometrics for PAT. The combination of these molecular spectroscopy methods in combination with tailor-made AI data analysis methods has the potential to revolutionize PAT in the upcoming years.	 

	Innovations in quality assurance and process control are of essential importance for the coming decades given the immensely growing world population, the development of demography, and the increasing demands of people on products and services. Quality assurance is subject to constantly increasing requirements and is essential for every individual as well as for companies for economic reasons. As a result of industrial processes one expects to see high safety, high and stable product quality, high yields, low consumption of resources, reduced influence of the variability of raw materials as well as long product durability. In recent years, new developments in the field of process analytics have been continuously observed.

	The quality by design (QbD) concept plays a decisive role in this development (1). This concept is based on the fact that a high-quality level of end products cannot be achieved by testing the products alone, but by an intelligent design of the entire manufacturing process into which the products are incorporated. This approach requires a comprehensive understanding of the entire manufacturing process. If one is informed about all relevant characteristics along a production process, one gets a better picture of the quality of a product or intermediate product than by certifying the final product- which is still generally accepted today as a measure of quality. The technology that makes QbD possible is process analytical technology (PAT). PAT is an approach for the improved control of production processes. Although originally promoted by the U.S. Food and Drug Administration (FDA) for use in the pharmaceutical sector, the PAT initiative is also gaining in importance for related industries, such as the food industry. If PAT is moved further forward along the entire production process, deviations from specified guideline values and possible risks in production and manufacturing processes can be detected and eliminated immediately. 

	PAT distinguishes between off-line, at-line, on-line and in-line measurement methods (1). In off-line measurements, samples are taken from the process and analyzed in a laboratory environment that is clearly separated from the industrial plants. Off-line analysis therefore always shows significant delay times between the detection of irregularities and the counteraction. In at-line measurements, the sample is taken from the process stream and analyzed with analytical equipment located in the immediate vicinity of the industrial plants. This process significantly reduces the reaction time for countermeasures. In on-line measurements, the samples are not completely removed from the process stream, but are temporarily separated, such as via a bypass system that transports the sample directly through the on-line measuring device, where the sample is analyzed in the immediate vicinity of the industrial processing and then recombined with the process stream. When using in-line instruments, the sensor is immersed directly into the process stream and remains in direct contact with the unaltered material stream. Increasingly, the need for in-line process analytics is growing and with it the availability of appropriate measurement technologies. From the large variety of analytical methods, optical molecular spectroscopy is one of the most promising approaches, which can best be integrated in-situ within an in-line process control.	 

	In order to be able to intervene quickly in the event of changes in the process that lead to a reduction in the quality of the end product, powerful in-situ analytical approaches are required. In this context, optical spectroscopic methods in particular are becoming increasingly important for in-line process monitoring. Optical spectroscopic methods are usually nondestructive and provide continuous information that can be used for process control. Therefore, optical spectroscopic methods are becoming more and more important in PAT. Depending on the measurement problem, a wide range of wavelength ranges and phenomena of light-matter interaction can be used, such as:	

ultraviolet-visible (UV-vis) spectroscopy (electronic excitation)

fluorescence spectroscopy (absorption and emission of electrons)

vibrational spectroscopy (NIR or mid-IR absorption and Raman spectroscopy)

laser induced breakdown spectroscopy (LIBS).

	All these spectroscopic techniques differ in molecular selectivity, sensitivity, readiness for use in PAT as a result of special sampling conditions, and so on. Vibrational spectroscopic methods are characterized by a particularly high molecular selectivity, because they provide direct information about characteristic molecular vibrations and thus about the molecular structure. A vibrational spectrum can therefore be seen as a kind of characteristic “molecular fingerprint” of a molecular species, which enables the identification of organic, inorganic, or biological components. Absorption in the mid-IR range excites fundamental vibrations that can be directly assigned to the basic building blocks in a molecule. NIR spectroscopy, on the other hand, measures vibrational overtones and combination bands and is therefore less selective (no easy direct interpretability) and also less sensitive as compared to mid-IR absorption spectroscopy. Despite these disadvantages, the main advantage of NIR spectroscopy compared to mid-IR spectroscopy is that no sample preparation (such as dilution) is required even at higher concentrations. It is important to emphasize that both NIR and mid-IR spectroscopy are very sensitive to water absorption, which limits a broad applicability of these methods in aqueous environments, such as for the investigation of fermentation processes. For the analysis of aqueous systems Raman spectroscopy can be advantageous compared to NIR and mid-IR spectroscopy. In recent years the potential of Raman spectroscopic methods for PAT has been increasingly recognized (1–3). The advantages of Raman spectroscopy are an unprecedented high specificity and a variety of applications. Raman spectroscopy is a noninvasive technique where no special sample preparation is required. Solid, liquid, and gaseous samples as well as transparent and nontransparent samples or samples with inhomogeneous surface properties can be measured. Limitations to the applicability of Raman spectroscopy result from the possible occurrence of an interfering fluorescence background and the low signal yields in comparison with, for example, mid-IR or NIR absorption or fluorescence spectroscopy. In recent years, however, decisive technical progress has been made to overcome these limitations, in particular through the realization of highly efficient spectrographs and the implementation of novel detector technologies. These advances have led to a significant reduction in acquisition times down to the ms range (4,5).	 

	In recent years, new developments in the field of PAT have been continuously observed using optical spectroscopic methods. Especially in the field of chemical and pharmaceutical analysis, optical and spectroscopic methods are used in on-line process monitoring (2,3,6). It can be stated that especially the interest in NIR and Raman spectroscopy-given their potential for fast and nondestructive analysis of samples in different physical states (liquids, solids, pastes, and so on) and their capability to be used on-line or in-line-as potential PAT analysis methods has increased significantly. Although NIR spectroscopy is still predominant in the pharmaceutical industry, Raman spectroscopy is becoming more widespread and has already proven its feasibility for the analysis of critical quality characteristics throughout the drug manufacturing process (2,3). 

	Worldwide, the demand for high-quality and safety in the food supply are also constantly increasing. Thus, a safe and high-quality food supply can only be guaranteed if new possibilities in analytics and holistic process control are generated. However, food analysis presents PAT with extreme challenges. In comparison to the chemical and pharmaceutical industries, in the field of the food industry, the sample compositions are much more complex. They are mixtures of different substance classes. 

	Furthermore, the materials are often soft, variable in size, brittle, or slippery. They are extremely susceptible to changes in pressure, temperature, and humidity, making noninvasive, fast methods for process analysis essential. In recent years, technological developments have already been made that make it possible to replace laboratory-based methods with on-line process control. These are mainly based on fluorescence or IR or NIR spectroscopy (7–9). Furthermore, the potential of Raman spectroscopic methods has been increasingly recognized. This potential has been successfully demonstrated using the example of determining the content of glucose, fructose, and sucrose in commercially available soft drinks (10). In the field of drinking water monitoring, Raman-based methods for the in-line detection of smallest amounts of chemicals and pharmaceutical impurities were applied (11).  

	Furthermore, surface-enhanced Raman spectroscopy (SERS) was applied for the detection of prohibited dyes as well as azo dyes in food (12,13). These representative examples show that optical spectroscopic methods and in particular Raman spectroscopy have enormous potential for on-line process analysis in food production.

	To extend the analytical potential with respect to selectivity, sensitivity, and so on, it can be very advantageous to combine several of the above mentioned optical spectroscopic contrast mechanisms in a multimodal approach (14). From a practical point of view, besides the need for high sensitivity and selectivity, the price-to-performance ratio and the simplicity of application of optical spectroscopic methods are of particular importance. The availability of compact, high-performance spectrometers including detectors as well as easy-to-operate compact light sources-and especially lasers with acceptable operating times-play a decisive role. 

	An essential feature of optical process spectrometers is the coupling of probes or the use of flow cells. Using fiber-optic coupled probes, it is possible to spatially separate spectrometer and application in the process and thus to look directly into individual processes. Fiber-optical sensors help to record data at critical points. As the “sensory organs of process control technology,” they enable quality-optimized automation of processes. In this context, particularly great progress has been made in recent years in the development of high-performance, tailor-made Raman probe concepts, mainly for biomedical applications (15–17), which are also of high relevance for in-line process analysis. In summary, technological developments have been made in the last few years in the areas of light sources, spectrometers, detectors and fiber probes, which will make it possible to replace laboratory-based methods with on-line process control in the future.

	Essential challenges for a continuous quality and safety monitoring of the processes as well as the resulting products and materials are seen in the still missing automation of process analytical techniques and here especially in the automated analysis of the recorded spectral measurement data. This automated analysis of the spectral data is accomplished by data analysis pipelines, which combine standardization and correction procedures with chemometric techniques and machine learning models (18,19). The overall aim of these data pipelines is the translation of the optical or spectral data into meaningful higher information (see Figure 1). In PAT this higher information might by the concentrations of given substances, the quantification of the sample composition or the prediction of a sample property like “good”or “bad.”	 

	 To perform this translation of the experimentally recorded spectral data, standardization and correction procedures are utilized first. For Raman spectroscopy a de-spiking (20), a wavenumber, and spectrometer calibration (21), a background correction (22) and a normalization are performed. These sequential steps correct for corrupting effects and standardize the Raman data, while similar preprocessing steps are done for other optical technologies like IR or NIR spectroscopy. After the data is standardized, machine learning models such as deep learning (23) or chemometric models (19) are utilized for the translation of the preprocessed data into meaningful information. As mentioned above, this higher information might be concentration estimates of substances or compounds of the sample or prediction of sample attributes like “good” and “bad.” Depending on the choice of higher lever information, either regression or classification models are utilized. Most often for spectral measurements, chemometric techniques are utilized whereas for image-based measurements, deep learning methods, like convolutional neuronal networks, are chosen for the analysis. All of these models are optimized based on training data and afterward the models are evaluated for their generalization performance (24). Thereafter, the models can be used for the prediction of unknown test data (with the estimated generalization error) and the prediction can be utilized for process analytics in chemical, pharmaceutical, and food industry.	 

	This contribution provides a short overview on the use of optical molecular spectroscopy such as Raman, IR, mid-IR, NIR spectroscopy as process analytical technology (PAT). An essential step to successfully apply these measurement methods as PAT is their combination with artificial intelligence methods like machine learning, deep learning, and chemometrics in order to translate the spectroscopic information into meaningful higher information such as quantification of the sample composition or the prediction of a sample property. This combination is done by tailor-made data pipelines and is briefly explained for the example of Raman spectroscopy. Overall, this contribution highlights the great potential of optical molecular spectroscopy in combination with artificial intelligence for PAT.	 

	Financial support of the EU, the ”Thüringer Ministerium für Wirtschaft, Wissenschaft und Digitale Gesellschaft,” the ”Thüringer Aufbaubank”, the Federal Ministry of Education and Research, Germany (BMBF), the German Science Foundation, the Fonds der Chemischen Industrie, and the Carl-Zeiss Foundation are greatly acknowledged.	 

Kandelbauer, M. Ruhe, and R. W. Kessler, in J. Popp, V.V. Tuchin, A. Chiou, and S.H. Heinemann, Eds., 

Handbook of Biophotonics Vol. 3: Photonics in Pharmaceutics, Bioanalysis and Environmental Research (

Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, Germany, 2011), pp. 3–70.

K.A. Esmonde-White, M. Cuellar, C. Uepmann, B. Lenain, and I.R. Lewis, 

Nagy, A. Farkas, E. Borbás, P. Vass, Z.K. Nagy, and G. Marosi, 

Krafft, I.W. Schie, T. Meyer, M. Schmitt, and J. Popp, 

Schie, J. Rüger, A.S. Mondol, A. Ramoji, U. Neugebauer, C. Krafft, and J. Popp, 

Process Analytical Technology: Spectroscopic Tools and Implementation Strategies for the Chemical and Pharmaceutical Industries 

(Wiley, Hoboken, New Jersey, 2nd Ed., 2010).

Niemöller and D. Behmer, in J. Irudayaraj and C. Reh, Eds., 

 (Wiley Blackwell, Ames, Iowa, 2008), Chapter 4.

Hitzmann, R. Hauselmann, A. Niemoeller, D. Sangi, J. Traenkle, and J. Glassey, 

K. Ilaslana, I. Hakki Boyaci, and A. Top, 

Z. Li, M.J. Deen, S. Kumar, and P.R. Selvaganapathy, 

S. Hea, W. Xiea, W. Zhanga, L. Zhang, Y. Wang, X. Liu, Y. Liu, and C. Du, 

Spectrochim Acta A Mol. Biomol Spectrosc. 

V. Peksa, M. Jahn, L. Štolcová, V. Schulz, J. Proška, M. Procházka, K. Weber, D. Cialla-May, and J. Popp, 

T. Meyer, M. Schmitt, B. Dietzek, and J. Popp, 

Krafft, S. Dochow, I. Latka, B. Dietzek, and J. Popp, 

Latka, S. Dochow, C. Krafft, B. Dietzek, and J. Popp, 

Cordero, I. Latka, C. Matthäus, I. W. Schie, and J. Popp, 

Ryabchykov, S. Guo, and T. Bocklitz, in J. Popp, Ed., 

Micro-Raman Spectroscopy Theory and Application 

S. Guo, O. Ryabchykov, N. Ali, R. Houhou, and T. Bocklitz, in 

Comprehensive Chemometrics Molecular Sciences and Chemical Engineering 

Ryabchykov, T. Bocklitz, A. Ramoji, U. Neugebauer, M. Foester, C. Kroegel, M. Bauer, M. Kiehntopf, and J. Popp, 

T. Bocklitz, T. Dörfer, R. Heinke, M. Schmitt, and J. Popp, 

Pradhan, S. Guo, O. Ryabchykov, J. Popp, and T. W. Bocklitz, 

Guo, T. Bocklitz, U. Neugebauer, and J. Popp, 

 is the head of the Research Department of Photonic Data Science, at the Leibniz Institute of Photonic Technology, in Jena, Germany. 

 is an assistant professor at the Institute of Physical Chemistry and Abbe Center of Photonics at Friedrich Schiller University Jena, Germany. 	 

 is the scientific director of the Leibniz Institute of Photonic Technology, Jena, Germany, and also holds the chair for Physical Chemsitry at the Friedrich Schiller University Jena, Germany. Direct correspondence to: 

In celebration of Spectroscopy’s 35th Anniversary, leading experts discuss important issues and challenges in analytical spectroscopy.

Juergen Popp

Articles

Understanding the Limits of the Bouguer-Beer-Lambert Law

Optical Molecular Spectroscopy in Combination with Artificial Intelligence for Process Analytical Technology