Stochastic and corpuscular theory of (polarized) light

Abstract

Both the corpuscular theory of light and the theory of stochastic processes are well known in the literature. However, they are not systematically used together for description of optical phenomena. There are optical phenomena, such as the well-known three-polarizer experiment or other phenomena related to polarization of light, which have never been quantitatively and qualitatively explained using the concept of quantum of light (photon). The situation changed in 2022 when stochastic memoryless and independent (IM) process formulated within the framework of the theory of stochastic processes was introduced. It is suitable for determination of probability (density) functions characterizing interaction (transmission or reflection) of individual photons with optical elements on the basis of experimental data. The process has memoryless (Markov) property, and it is assumed that the interactions of individual photons with an optical system are independent. Formulae needed for analysis of data in the context of polarization of light are derived. An example analysis of the three-polarizer experiment is performed and numerical result of the probability (density) functions are determined. These original results were missing in the literature. The results imply that the possibilities of the corpuscular theory of light to describe optical phenomena can be significantly extended with the help of stochastic IM process and the theory of stochastic processes in general.

1 Introduction

The question of whether light is corpuscular or wave in nature is very old and has been debated in the literature for centuries. Sometimes the corpuscular theory and the wave theory are understood as “either-or” and sometimes as “both-and” (wave–particle dualism). We will not attempt to answer this question once and for all. We will only focus on the possibilities of the corpuscular theory of light itself.

I. Newton tried to explain a number of phenomena with light by using the idea that light consists of particles (corpuscles). However, determination of properties of these particles varied greatly with time; useful historical references and many comments to different models of light quanta (photons) are in [1]. The interaction of light with matter is still the subject of extensive scientific research. One of many important discoveries related to particle of light concerned its energy. M. Planck in 1900 [2, 3] introduced quantization of energy of electromagnetic radiation to describe spectrum of black-body radiation. A. Einstein in 1905 [4] developed further this quantum hypothesis. He basically introduced modern concept of particle of light photon—and used it to explain the photoelectric effect. Photon has been successfully used for explanation of many other observed phenomena related to light since the beginning of the twentieth century. It has led to numerous technological advances which further demonstrate in various ways the usefulness of the concept of particles of light (photons). There is now whole field called photonics [5,6,7,8] which began with the invention of the laser in 1960 and which further demonstrate the practical applicability of photons.

However, there are phenomena in optics which have never been described both quantitatively and qualitatively using the corpuscular idea of light. This concerns experiments with polarization of light. Interesting properties of light related to polarization may be easily demonstrated with the help of the well-known and discussed three-polarizer experiment. When incident unpolarized light passes through two crossed linear polarizers, then the intensity of light is strongly reduced (nearly no light passes through). However, when another polarizer is placed between the two polarizers, then some light can pass through the sequence of the three polarizing filters. The intensity of light behind the third polarizer in the sequence is maximal when the second polarizer has polarizing axis oriented at $45^{\circ }$ relative to the polarizing axis of the first polarizer (and the third polarizer). This experimental result, which can be easily reproducible and observed by naked eye, has attracted attention of many researches as it is clear that the effect cannot be explained by only absorption of some photons from the beam by the polarizers; the polarizers must change also a property of the transmitted photons (called polarization). Moreover, both the effects must depend on the orientation of the axes of the polarizers. Similar effects can be observed when light is transmitted through other polarization-sensitive elements.

The situation changed significantly in 2022 when stochastic independent and memoryless (Markov) process (IM process) formulated within the theory of stochastic processes was introduced in [9]. The independence is related to assumed independence of outcomes (realization) of an experiment. It has opened completely new possibilities to describe a whole range of (not only) particle phenomena that have not yet been described particle-wise. IM process can describe under the two assumptions: particle decays, motion of particles when their initial conditions are specified statistically, particle–matter interactions and particle–particle collisions in unified way [9].

Several statistical descriptions of optical phenomena involving photons have been, implicitly or explicitly, based on the two assumptions. Description based on IM process can be, therefore, reduced to these relatively simple descriptions. One may ask how IM process (the theory of stochastic processes in general) can help to describe optical phenomena (such as the three-polarizer experiment) which have not yet been described by using the corpuscular theory of light and statistics. The theory of stochastic processes has very wide applications. Many stochastic processes having various properties are studied in the literature. However, optics is one of the fields where it has been used only partially or not at all. We will try to show that the theory of stochastic processes can be very useful in explaining and describing phenomena in optics.

This paper is structured as follows. The statement of the problem is discussed in greater detail in Sect. 2. Section 3 briefly summarizes measurement of number of photons (beam intensities) transmitted through a sequence of $M$ optical elements in dependence on their rotations and the results concerning linear polarizers. Contemporary statistical theoretical approaches used for description of polarization of light and light in general are discussed in Sect. 4. They contain several limitations that in many cases make them unsuitable for analysis of experimental data and achieving the stated goals. These limitations can be removed using stochastic IM process (or other suitable process formulated within the theory of stochastic processes) which is explained in Sect. 5. IM process describing transmission of individual photons through $M$ polarization-sensitive elements (including, but not limited to, linear polarizers) is formulated in Sect. 6. Several ideas of M. V. Lokajíček [10] will be developed further. Probability (density) functions characterizing transmission of individual photons through optical elements can be used for definition of properties of the beam and various types of optical elements. Definitions related to polarized light and various types of polarization-sensitive elements are discussed in Sect. 7. An example data corresponding to transmission of light through three linear polarizers are analyzed with the help of IM process in Sect. 8 where numerical results are shown, too. The example analysis of data in Sect. 8 leads to probability (density) functions which are not determined uniquely. What to do in similar situations is discussed in Sect. 9. Summary and concluding remarks are in Sect. 10.

2 Problem statement

Let us consider a sequence of $M$ optical elements (they may or may not be polarization sensitive), as shown in Fig. 1. This is arrangement of many experiments. (In general, a net of optical elements can be consider.) The interaction of a photon through i-th element ($i=0,\ldots ,M-1$) can be considered stochastic (random) process (a photon may or may not pass through given element, it can be reflected, etc.). We may ask how to determine probability (density) functions characterizing interaction of photons with the individual elements in the sequence. We will commonly use the term transmission to refer to any interaction of a photon (beam) with an optical element.

According to [9] (see Eq. (63) therein), probability of transition of a system from given state to another given state can be factorized into three conditional probabilities (assuming Markov property and independence of outcomes of an experiment). In the context of optics, the probability of transmission of given input photon state through i-th sensitive optical element to given output photon state can be factorized into three conditional probabilities, i.e., three probabilistic effects can be distinguished:

• 1. an incoming photon before interacting with i-th optical element has its state specified by a probability density function $\rho _{S,i}$;

• 2. an incoming photon in given state may or may not be transmitted through i-th element (characterized by a probability function $P_{T,i}$ being function of the incoming photon state, not the output photon state);

• 3. if the photon in given state is transmitted through the element then its state may or may not change (described by a probability density function $\rho _{C,i}$ being function of both the incoming and the outgoing photon state).

Both incoming and outgoing photon states are in general described by random and non-random variables. Optical elements can be characterized by random and non-random variables, too.

One may ask how the probability (density) functions $\rho _{S,i}$, $P_{T,i}$ and $\rho _{C,i}$ characterizing transmission of individual photons with given i-th optical element can be determined on the basis of experimental data. It seems that it has never been done systematically.

Fig. 1

Scheme of sequential arrangement of many optical systems (experiments with light). A beam of photons is passing through control surfaces $\Sigma _{i}$ ($i \in (0,1, \dotsc , M)$) which define $M$ transport segments, each of them containing an optical element

3 Measurement

It is an experimental goal to measure the beam characteristics before and after interaction with a given optical element (“sample”). Important experimental data about the sample can be obtained when input (diagnostic) beams of various properties and the properties of the output beams are measured, see Sect. 2 in [9], i.e., response of an element to various inputs can be measured. In general transmission of a photon through, an optical element depends on random and non-random variables. State of input and output beam can be characterized by quantity called density of photon states which depends on the random and non-random variable. Density of states will be defined later in Sect. 5.1. In many cases, it is not possible to measure it as a function of all the random and non-random variables.

Let us now assume that only the number of photons before and after an interaction with each optical element in a sequence can be measured as a function of non-random variables. This is discussed in Sect. 3.1 in the case the non-random variables correspond to the rotations of the elements (a common case when measuring polarization-sensitive elements). In Sect. 3.2, one can find the dependences in the case of sequence of linear polarizers.

3.1 Measurement of number of photons passing through sequence of optical elements in dependence on their rotations

3.1.1 Experimental setup

Let us consider photon beam passing through a sequence of $M$ optical elements, see Fig. 1. Control surfaces $\Sigma _i$ ($i \in (0,1, \dotsc , M)$) can be introduced. Each optical element is placed between two neighboring control surfaces. These control surfaces do not correspond to physical surfaces of the optical elements, but to places where characteristics of the beam can be measured. (They will be called surfaces for short.)

3.1.2 Number of photons

Let $N_{i}$ be the number of photons which passed through i-th surface $\Sigma _i$. The number of photons passed through i-th optical element, i.e., $N_{i+1}$ ($i=0,\ldots ,M-1$ in this case), may in general depend on various random and non-random variables characterizing states of the optical elements and states of photons. Let us assume, for the sake of simplicity, that it depends only on rotation of the i-th optical element, the rotations of all the other optical elements placed in front of it and the number of initial particles $N_{0}$. We may introduce vector $\vec {\alpha }$ having components $\alpha _{i}$ ($i=0,\ldots ,M-1$) and representing rotations of all the individual optical elements. For the sake of simplicity, we introduce rotation of an optical elements characterized by only one parameter (non-random variable), but two other parameters fully specifying rotation of given element in space could be introduced, too. It is useful to introduce a convention that if a function depends on $\vec {\alpha },$ then it may depend only on some of its components.

Transmittance $T_{i}(\vec {\alpha })$ of i-th optical element ($i=0, \ldots , M-1$) is standardly defined as

$$\begin{aligned} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad T_i(\vec {\alpha })=\left\{ \begin{array}{lll} \frac{N_{i+1}(\vec {\alpha })}{N_{i}(\vec {\alpha })} &{}\quad \text {if }N_{i}(\vec {\alpha })\ne 0 &{}\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad\,\, \mathrm{(1a)}\\ 0 &{}\quad \text {otherwise}&{}\qquad \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad\,\, \mathrm{(1b)} \end{array}\right. \end{aligned}$$

It characterizes how much i-th optical element transmits light in dependence on its rotation and rotations of all the preceding optical elements.

Remark 3.1

The number of photons transmitted through an optical element may in general depend on time. In some cases, this experimental information must be taken into account to correctly interpret the experimental results. However, in many cases (e.g., in the case of linear polarizers which will be discussed below) the measured quantities often do not depend on time. (Numbers of transmitted photons corresponding to the initial number of photons $N_0$ are counted behind each optical element in a sequence independently of the time when the photons were transmitted.) Time variable will not be of our interest in the following.

3.1.3 Number of photons and beam intensity

Assumption 3.1

(Energy of photons) Photons have the same energy when they pass through surface $\Sigma _{i}$ for all $i=0,\dotsc ,M$.

Sometimes intensity of a particle beam defined as energy incident on a surface per unit of time and per unit of area (i.e., having units of $Js^{-1}m^{-2}=Wm^{-2}$)¹ is measured in an experiment. The intensity of light beam passing through surface $\Sigma _i$ may be denoted as $I_{i}(\vec {\alpha })$ ($i=0,\ldots ,M$), see Fig. 1. The intensity $I_0$ is the initial intensity of the beam. If the intensity $I_{i}$ and the number of photons $N_{i}$ correspond to the same surface area and time interval and the photons have the same energy (see assumption 3.1), then it holds (if $N_{i}(\vec {\alpha })) \ne 0$, $i=0,\ldots ,M-1$)

$$\begin{aligned} \frac{N_{i+1}(\vec {\alpha })}{N_{i}(\vec {\alpha })}&= \frac{I_{i+1}(\vec {\alpha })}{I_{i}(\vec {\alpha })} \end{aligned}$$

(2)

and (if $N_{0} \ne 0$, $i=0,\ldots ,M$)

$$\begin{aligned} \frac{N_{i}(\vec {\alpha })}{N_{0}}&= \frac{I_{i}(\vec {\alpha })}{I_{0}} \, . \end{aligned}$$

(3)

Remark 3.2

Assumption 3.1 could be, for the purposes presented in this paper and for the purposes to relate the relative numbers of transmitted photons to the relative beam intensities, replaced by assumption that Eq. (3) holds.

3.2 Example: sequence of linear polarizers

Fig. 2

Example of experimental setup for measurement of relative photon numbers $N_{i}/N_0$ (beam intensities $I_{i}(\vec {\alpha })/I_{0}$) for $i=1,2,3$ in dependence on the rotation angles $\vec {\alpha }=(\alpha _{0},\alpha _{1},\alpha _{2})$ of the linear polarizers when light beam is transmitted through one, two or three polarizers ($M=3$). The measured relative photon numbers correspond to conveniently chosen control surfaces $\Sigma _i.$ (They may not correspond to physical surfaces of the polarizers.) Values of polarization $\theta _{i}$ of photons which passed through surface $\Sigma _i$ form state space $S_{i}$. The values of photon polarization angles must be determined with the help of a probabilistic model on the basis of experimental data

Let us now consider a photon beam passing through $M$ linear polarizers, see Fig. 2 corresponding to $M=3$, as an example of transmission of light through a sequence of polarization-sensitive elements.

3.2.1 Ideal linear polarizers and Malus’s law

In textbooks on optics transmission of light through “ideal” linear polarizers is often discussed (see, e.g., Sect. 12.4 in [12]). If initially unpolarized light beam is sent through a sequence of $M$ ideal linear polarizers, then the transmittance $T_{i}(\vec {\alpha })$ of i-th ideal polarizer ($i=0,\ldots ,M-1$) is

$$\begin{aligned} \quad \qquad \qquad \qquad \qquad \qquad\qquad \qquad \qquad\qquad \qquad T_{i}(\vec {\alpha }) =\left\{ \begin{array}{lll} \frac{1}{2} &{}\quad \text {if }i = 0 &{}\quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \,\,\, \mathrm{(4a)}\\ \cos ^2(\alpha _{i}- \alpha _{i-1}) &{}\quad \text {if }0< i < M \, .&{}\quad \qquad \qquad \qquad \quad \qquad \qquad \qquad \qquad\qquad \qquad \,\,\, \mathrm{(4b)} \end{array}\right. \end{aligned}$$

i.e., the initial number of photons $N_0$ after transmission through the first ideal polarizer drops to one half independently of the orientation of the polarization axis $\alpha _{0}$. The cosine-squared function expresses the known Malus’s law. It represents the first quantitative relationship for treating polarized light intensities describing measurements performed by É.-L. Malus [13] (see also summary of the life of Malus, the historical context and further comments concerning his law in [14]). The transmittances $T_{i}$ given by Eq. (4) do not depend on the value of the initial number of particles $N_{0}$.

Equations (1) and (4) imply that numbers of transmitted photons corresponding to the surface $\Sigma _i$ divided by initial number of photons is

$$\begin{aligned} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \frac{N_{i}(\vec {\alpha })}{N_0}=\left\{ \begin{array}{llll} 1 &{}\quad \text {if }i = 0&{}\qquad \qquad \qquad \qquad \qquad\qquad \qquad \quad \qquad \qquad \qquad \quad \,\,\, \mathrm{(5a)} \\ \\ \frac{1}{2} &{}\quad \text {if } i = 1 &{}\qquad \qquad \qquad \qquad \qquad \quad \qquad \qquad \qquad \qquad \qquad \quad\,\,\, \mathrm{(5b)}\\ \\ \frac{1}{2} \Pi ^{i-1}_{j=1}\cos ^2(\alpha _{j}- \alpha _{j-1}) &{}\quad \text {if }1< i < M \, .&{}\qquad \qquad \qquad \quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad\,\,\, \mathrm{(5c)} \end{array}\right. \end{aligned}$$

3.2.2 Real linear polarizers

The Number of photons transmitted through a sequence consisting of “real” linear polarizers may be very similar to the number of photons transmitted through sequence of ideal polarizers, see Sect. 3.2.1. However, it is known that some other real polarizers differ significantly from the ideal polarizers. Every two real linear polarizers transmit some light, even if they are crossed, and absorb some light if their axes are parallel. This is often expressed by modifying the Malus’s law, as discussed in Sect. 12.4 in [12]. In [15] it has been measured that two crossed real linear polarizers produce unpolarized light. (Value of $N_{3}(\vec {\alpha })/N_0$ behind the third, testing, linear polarizer is independent of its rotation angle.) It has been measured in [16] that transmittance of a pair of real linear polarizers may have local maximum if the polarizer axes are mutually crossed, and around this point the transmittance is not fully symmetrical. (The effect can be measured with sensitive measurement devices.) Some smaller or bigger deviations of measured beam intensity from Malus’s law are visible in experiments conducted in [17,18,19], too. The deviations depend on properties of the polarizers. Some other differences of real linear polarizers from ideal ones are summarized in Sect. 15.27 in [12]. It has not been possible to observe many of these differences by Malus at the beginning of nineteenth century, i.e., before the advent of photomultiplier tubes (PMTs), charge-coupled devices (CCDs) and other devices nowadays commonly used for measurement of numbers of photons (beam intensities).

4 Contemporary widely used descriptions of polarization of light

The study of interaction of light with matter has a very long tradition. Several distinct theoretical approaches have been developed to better understand huge amount of observed phenomena [5, 12, 20,21,22,23,24,25,26,27,28].

The measurement and description of effects related only to polarization of light represent a very broad topic, see [29,30,31,32,33,34] and Part 3 in [12] devoted to optical polarization. A wide range of polarization phenomena, their applications and the history of the polarization of light are discussed in [35].

Theoretical descriptions of polarization of light can be divided into two groups, depending on whether or not they use statistics. Theoretical approaches which use statistics only partially or not at all are discussed in Sect. 4.1, and we will not pay much attention to them. Section 4.2 deals with theoretical methods, which are much more related to statistics. Summary of the theoretical approaches and their possibilities to determine probability density functions $\rho _{S,i}$, $P_{T,i}$ and $\rho _{C,i}$ is in Sect. 4.3.

4.1 Non-statistical descriptions

Various matrix calculi have been developed to describe the effect of polarization-sensitive elements on the state of polarization of a light beam, see Sect. 12.8 in [12] and a historical revision of the development of the differential Stokes–Mueller matrix formalism [36]. The most widely known matrix calculi are the Mueller calculus and the Jones calculus. The distinct matrix formalisms have been developed with different aims and purposes in order to better understand observed polarization phenomena. However, none of the matrix calculi has been designed to describe probabilistic character of transmission of individual photons through a sequence of polarization-sensitive elements, such as the mentioned sequence of three linear polarizers. They describe some average transmission of the whole beam (a state of the beam being represented by a vector). This is common feature of many matrix descriptions used in optics [37]. For example, the description of transmission of light through sequence of three polarizers by Jones [38] does not mention probability at all. Similarly, description of three linear polarizers in the context of astrophysics discussed in [39] is also not focused on probabilistic description of transmission of individual photons. This is also the case of the Maxwell’s equations. None of these non-statistical approaches of description of polarization of light is suitable for determination of the probability (density) functions $\rho _{S,i}$, $P_{T,i}$ and $\rho _{C,i}$.

4.2 Statistical descriptions

Several diverse theoretical approaches more strongly associated with statistics (the probability theory) have been developed in optics, too. Statistical optics [21] deals mainly with a scalar theory of light waves. The scalar quantities are regarded as representing one polarization component of the electric or magnetic field, with the assumption that all such components can be treated independently. Its application to data concerning propagation of light phenomena is based on several other assumptions (such as assumptions accompanied by introduction of complex amplitudes and their properties).

This theoretical approach does not fully explore the concept of quantum of light in which we are interested in. This is also the case of other contemporary theoretical attempts [32] trying to describe various polarization (optical) phenomena with the help of statistics. Quantum mechanics in general can take into account individual photons. However, description of polarization of light in quantum optics discussed, e.g., in Sect. 6 in [40], does not provide straightforward way to determine the functions $\rho _{S,i}$, $P_{T,i}$ and $\rho _{C,i}$ characterizing transmission of individual photons through polarization-sensitive elements (or other optical elements). Description of polarization of light based on quantum optics and discussed in [41] neglects polarization-dependent losses during transmission of photons through a medium and is based on several other limiting assumptions. It is, therefore, not suitable for description of experiment with three linear polarizers where it is necessary to take into account that a photon can be absorbed by one of the linear polarizers. There is also quantum state tomography (QST) which refers to any method that allows one to reconstruct the accurate representation of a quantum system based on data obtainable from an experiment. This is the kind of method we are looking for. However, QST introduces complex probability amplitudes (wave functions), as in quantum mechanics, and also in this case, it seems that the probability (density) functions $\rho _{S,i}$, $P_{T,i}$ and $\rho _{C,i}$ have never been determined in the case of three-polarizer experiment (or similar one) [42, 43].

Monte Carlo (MC) methods represent another broad group of theoretical statistical methods used for description of particle transport phenomena. They are summarized in [44]. They use random or pseudorandom numbers to simulate a random process. To make a MC simulation of transmission of particles through a medium, it is necessary to know parameters of various microscopic processes. Determination of the probability density functions characterizing the processes typically requires detailed knowledge of cross sections of various interaction types, mean free paths, spatial distribution and types of scattering centers, etc. The parameters of the microscopic processes and the dependence of the processes on various random variables are often now known. This kind of information need to be determine first on the basis of experimental data. It makes MC methods hardly usable for determination of the probability (density) functions $\rho _{S,i}$, $P_{T,i}$ and $\rho _{C,i}$ characterizing overall effect of transmission of individual photons through i-th optical element.

4.3 Summary

One can look similarly over other theoretical methods used in optics. It is possible to make several observations concerning the contemporary statistical descriptions widely used in optics:

• 1. States of individual photons are not always taken into account (some methods are more focused on average properties).

• 2. Evolution (transition) operator is standardly assumed to be unitary in quantum mechanics. This makes it delicate to describe phenomena corresponding to probability $P_{T,i}$ which is not identically equal to 1 and depends on random variables characterizing given initial state (for further details see the open problem 4 in Sect. 6 in [45] concerning contemporary descriptions of particle collisions).

• 3. Some theoretical methods require information about a system (such as the Hamiltonian or parameters of various microscopic processes) which may not be known and need to be determined first on the basis of experimental data.

• 4. Introduction of complex probability amplitudes or wave functions is often accompanied by introduction of additional assumptions concerning their properties. We are, roughly speaking, interested mainly in the probabilities given by the square of the absolute value of the amplitudes. Introduction of the amplitudes may, therefore, bring complications rather than benefits.

• 5. It is often difficult to follow under which assumptions various statements are made.

• 6. Only one transition of a photon state to another photon state is often discussed. Sequences (or even nets) of transitions are rarely mentioned.

• 7. The probability of transition of a given initial state to a given final state is not factorized into the three probabilistic effects mentioned in Sect. 2.

We may conclude that the probability (density) functions $\rho _{S,i}$, $P_{T,i}$ and $\rho _{C,i}$ have never be determined on the basis of experimental data. The contemporary theoretical approaches used in optics do not provide straightforward way to do it.

5 Theory of probability and theory of stochastic processes

A branch of mathematics concerned with the analysis of random phenomena is the probability theory. Stochastic or random process is a mathematical object usually defined as a family of random variables. The theory of stochastic processes provides very general and abstract framework to study random processes, experiments having random outputs. It may be seen as extension of the probability theory. Properties of many stochastic processes are intensively studied in the literature, and they have numerous applications in basically all field of research. Optics is, however, one of the fields where it is used only partially or not at all (see Sect. 4).

Section 5.1 is devoted to definitions concerning general stochastic process. Definitions and statements related to stochastic IM process are in Sect. 5.2. Advantages of descriptions based on a stochastic process like IM process are summarized in Sect. 5.3.

5.1 General stochastic process

Definition 5.1

(Stochastic process) A stochastic process is a family $X = \{X_{i} : i \in I\}$ of random variables defined on the same probability space $(\Omega , \mathscr {F},P)$ and, for fixed index i in an index set $I$, taking their values $X_{i}$ in given space $S_{i}$ which must be measurable with respect to some $\sigma$-algebra $\mathscr {S}_{i}$ of admissible subsets. [This is definition 3.13 in [9].]

The probability density functions $\rho _{S,i}(X_{i})$ are given by Eq. (43) in [9]. It is functions of random variables $X_{i}$ and non-random variables $X^{NR}_{{i}}.$ (The dependence on non-random variables will not be written explicitly in this Sect. 5.) The number of states corresponding to given state space $S_{i}$ is denoted as $N_{i}$.

Theorem 5.1

Let $\{X_{i} : i \in I\}$ be stochastic process given by definition 5.1. If $S_{i} \ne \emptyset,$ then

$$\begin{aligned}&\int _{X_{i}} \rho _{S,i}(X_{i})\text {d}X_{i}=1 \end{aligned}$$

(6)

$$\begin{aligned}&{{\,\textrm{dos}\,}}_{i}(X_{i}) = N_{i} \rho _{S,i}(X_{i}) \end{aligned}$$

(7)

$$\begin{aligned}&N_{i} = \int _{X_{i}} {{\,\textrm{dos}\,}}_{i}(X_{i}) \text {d}X_{i} \end{aligned}$$

(8)

for all $i \in I$. [This is proposition 3.12 in [9].]

Remark 5.1

(Density of states) Quantity ${{\,\textrm{dos}\,}}_{i}(X_{i})$ is called density of states corresponding to given state space $S_{i}$ (it is defined in Sect. 3.1.7 in [9]).

5.2 Stochastic IM process

The probability functions $P_{T,i}(X_{i})$ and probability density function $\rho _{C,i}(X_{i}, X_{i+1})$ mentioned in Sect. 2 are given by definition 3.22 in [9].

Assumption 5.1

(Possibility of no transition) Probability $P_{T,i}(X_{i})$ may have values in the interval from 0 to 1, not necessarily in the whole interval. [This is assumption 3.5 in [9].]

Assumption 5.2

(Different state spaces) State spaces $S_{i}$ may or may not be the same for all $i \in I$. [This is assumption 3.3 in [9].]

Assumption 5.3

(Independence of realizations) Realizations (outcomes) of an experiment are mutually independent. [This is assumption 3.6 in [9].]

Assumption 5.4

(memoryless (Markov) property) Let $\{X_{i} : i \in I\}$ be stochastic process given by definition 5.1. Let it have Markov property, i.e., roughly speaking, it means that probability of a future state of a system depends on its present state, but not on the past states in which the system was. [This is assumption 3.7 in [9].]

Definition 5.2

(IM process) Let $\{X_{i} : i \in I\}$ be stochastic process given by definition 5.1. Let it satisfy assumptions 5.1 to 5.4. This process is called independent and memoryless (IM) process. [This is definition 3.19 in [9].]

Remark 5.2

IM process is called state-transition-change (STC) process in [9].

Definition 5.3

(IM sequence) Let $\{X_{i} : i \in I\}$ be stochastic process given by definition 5.2. Let the index set $I$ be totally ordered sequence $(0, \dotsc , M)$ and $N_{0} \ne 0$. This process satisfies assumptions 5.1 to 5.4. [This is definition 3.21 in [9].]

Theorem 5.2

(Transformation of density of states) Let $\{X_{i} : i \in I\}$ be stochastic process given by definition 5.3. It holds

$$\begin{aligned} {{\,\textrm{dos}\,}}_{i+1}(X_{i+1}) = \int _{X_{i}} {{\,\textrm{dos}\,}}_{i}(X_{i}) P_{T,i}(X_{i})\rho _{C,i}(X_{i}, X_{i+1})\text {d}X_{i}\, . \end{aligned}$$

(9)

[This is theorem 3.1 in [9].]

Remark 5.3

Stochastic processes discussed in the literature typically assume $P_{T,i}(X_{i})= 1$. Some phenomena, such as absorption of a photon which may or may not be absorbed by an optical element, cannot be described under this assumption. Assumption 5.1 represents generalization of this assumption.

It is also often assumed that states spaces $S_{i}$ are the same for two different indexes. This is also limiting assumptions in many cases. Assumption 5.2 removes this limitation.

The main assumptions under which one can derive theorem 5.2 are assumptions 5.3 and 5.4. These two assumptions are often used and discussed in the context of stochastic processes. Not all stochastic processes, of course, are based on these two assumptions. However, relatively large and important set of phenomena corresponds to these two assumptions and they allow to derive the very general Eq. (9).

Theorem 5.3

Let $\{X_{i} : i \in I\}$ be stochastic process given by definition 5.3. It holds ($N_0 \ne 0$)

$$\begin{aligned}&\frac{N_{i+1}}{N_0} \rho _{S,i+1}(X_{i+1})= \int _{X_{i}} \frac{N_i}{N_0} \rho _{S,i}(X_{i})P_{T,i}(X_{i})\rho _{C,i}(X_{i}, X_{i+1})\text {d}X_{i}\, . \end{aligned}$$

(10)

for $i \in (0,\ldots ,M-1)$.

Proof

Equation (9) divided by $N_0$ and Eq. (7) imply Eq. (10). $\hfill\square$

Theorem 5.4

Let $\{X_{i} : i \in I\}$ be stochastic process given by definition 5.3. It holds ($N_0 \ne 0$)

$$\begin{aligned} \frac{N_{i+1}}{N_{0}}&= \int _{X_{i}} \frac{N_{i}}{N_0} \rho _{S,i}(X_{i})P_{T,i}(X_{i})\text {d}X_{i} \end{aligned}$$

(11)

for $i \in (0,\ldots ,M-1)$.

Proof

Equations (74) in [9] and  (7) imply Eq. (11). $\hfill\square$

Remark 5.4

Let us assume that relative numbers of particles $\frac{N_{i+1}}{N_{0}}$ are measured as functions of some non-random variables (see, e.g., Sect. 3). In this case, Eqs. (11) and (10) represent system of integral equations. It may or may not be possible to determine uniquely the unknown functions $\rho _{S,i}(X_{i})$, $P_{T,i}(X_{i})$ and $\rho _{C,i}(X_{i}, X_{i+1})$. It depends on dependence of $\frac{N_{i+1}}{N_{0}}$ on the non-random variables and involved assumptions used for description of given system. If some other quantities than $\frac{N_{i+1}}{N_{0}}$ are measured, then similar set of equations may be derived. The more experimental information is available, the better.

5.3 Advantages of descriptions based on stochastic process like IM process

Analysis of data based on IM process has several advantages (other stochastic processes have very similar or the same advantages):

• 1. It allows to take into account and introduce only suitable random and non-random variables characterizing given phenomenon.

• 2. It unifies description of many phenomena from different fields of physics with the help of the theory of stochastic processes (the theory of probability), see [9] in the case of IM process.

• 3. Analysis of data can be separated into two stages (see Sect. 4.5 in [9]). In the first stage, one can try to determine the probability (density) functions on the basis of experimental data. In the second stage, one can go into greater detail and try to explain (interpret) the probabilities in terms of some microscopic processes or some underlying functions (such as the Hamiltonian of the system, complex probability amplitude and wave function). The first stage does not require to introduce the underlying functions or to know various parameters of the microscopic processes. (This information is often not a priori known and must be first determined on the basis of experimental data.) This kind of additional more detailed knowledge about given system can be studied in the second stage. This separation of data analysis into the two stages in many cases allows to study and test various assumptions more effectively than trying to determine too many characteristics of given phenomena at once.

• 4. One can leverage terminology, techniques and results already known in the theory of stochastic processes. (Numerous stochastic processes are successfully used in many fields of research.)

Using IM process, or other suitable stochastic process formulated with the help of the theory of stochastic processes, it is possible to overcome the difficulties and limitations existing in the contemporary statistical methods used in optics, see Sect. 4.3.

Remark 5.5

(Mendelian genetics) As far as we know, similar separation of data analysis has not been systematically done in the context of polarization of light and optics in general. The separation has already helped enormously, e.g., in the field of genetics (biophysics) pioneered by Mendel in 1865 [46]. He performed many plant hybridization experiments. He was able to explain variability of observed properties of organisms in different generations by introducing dominant and recessive traits (characteristics). By introducing additional assumptions how these traits are inherited he was able to calculate probabilities of occurrences of organisms of given properties and in given generation (i.e., transition). The assumptions are now called laws of Mendelian genetics. The calculated probabilities agreed with the observed (measured) numbers. His analysis of data corresponded to the first stage of data analysis (i.e., determination of the probabilities without trying to explain them in greater detail). Genetic research is in many cases nowadays well in the second stage where determined dominant and recessive abstract traits are identified with real biophysical structures in an organism (a gene consisting of two alleles) and related processes are further studied in greater detail. The probabilistic model formulated by Mendel [46] allowed him to easily calculate the probabilities needed for comparison to data. Formulation of the model itself and realization of all the required experiments, however, required surely far more effort and time.

The law of independent assortment in Mendelian genetics is closely related to assumption of independence of outcomes of an experiment, and memoryless (Markov) property is assumed implicitly. Mendelian genetics is essentially described using IM process, even if it is not explicitly mentioned. Mendel used very special case of IM process as neither random nor non-random variable which would be necessary to determine from data was introduced in his probabilistic model. One can find other examples of IM processes implicitly used in the literature. These cases correspond to relatively simple cases. Techniques developed for general IM process can help in some mathematically more complicated cases with the determination of the probabilities (the first stage) which is in general delicate task, see [9] where some more complicated cases are mentioned, too. Explanation of the determined probabilities (the second stage) can be often studied later, and separately, it can be also very demanding. The example from genetics (biophysics) clearly shows that some phenomena can be hardly understood if the corresponding analysis of data is not separated into the two stages and each of them studied separately (or quasi-separately).

We will use the strategy with separation of data analysis (systematically introduced in [9]) into the two stages in the following in the context of optics to describe transmission of light through various optical elements. In the first stage of the analysis (description), it is not necessary to know and understand the microscopic structures of the optical elements or the detailed structure of photon.

6 IM process and polarization of light

Interaction of photons with various optical elements may be considered stochastic process. Therefore, one may ask how the theory of stochastic processes can help to describe this kind of phenomena. Discrete-index IM stochastic process, see Sect. 5.2, is suitable for this purpose. Formulae describing transmission of light through sequence of polarization-sensitive elements are derived in this section.

Measured (relative) numbers of transmitted photons $N_{i}(\vec {\alpha })/N_{0}$ characterize a transmission of light through sequence of optical elements, but they do not explain the phenomenon. If light beam consists of photons then we may ask how a single photon interacts with the optical element, what the probability (density) functions $\rho _{S,i}$, $P_{T,i}$ and $\rho _{C,i}$ characterizing the transmission are.

We will derive main formulae corresponding to three different processes (cases) based on slightly different sets of assumptions. In Sect. 6.1, one can find formulae corresponding to sequence of quite general polarization-sensitive elements (case 1). Sect. 6.2 contains formulae valid for more special case under assumption that $\rho _{C,i}$ does not depend on random variables $X_{i}$ (case 2). In Sect. 6.3, one can find formulae valid in the case of even more special case assuming further that the elements in the sequence are identical (case 3).

6.1 Case 1: sequence of polarization-sensitive elements

Definition 6.1

(Photon polarization angle) Some types of polarization-sensitive elements (such as linear polarizer or Faraday rotator) are sensitive to photon property called polarization and having meaning of an angle $\theta _{i}$. It characterizes the state of a photon when it passed through surface $\Sigma _i$.

Remark 6.1

As to definition 6.1, the photon polarization angle can specify the direction of a vector quantity characterizing the photon (spin, orientation vector, etc.) projected onto the plane perpendicular to the direction of the photon velocity, or projection of direction of some photon oscillations onto the plane. The particular physical meaning of this variable is not important in the presented paper. In the following, it is necessary to only know that it has meaning of an angle.

Assumption 6.1

(Variables) Let random variables $X_{i}$ characterizing i-th state of system (i.e., transmission of a photon through surface $\Sigma _i$) for given index $i \in I$ where index set $I$ is sequence $(0, \dotsc , M)$ be

$$\begin{aligned} X_{i}&= (\theta _{i}) , \end{aligned}$$

(12)

i.e., the photon polarization angles are random variables. Let non-random variables $X^{NR}_{{i}}$ characterizing i-th state of system be

$$\begin{aligned} X^{NR}_{{i}}&= (\alpha _{0}, \dotsc , \alpha _{i-1}) \, , \end{aligned}$$

(13)

i.e., the rotation angles of the polarization-sensitive elements are non-random variables (parameters). $M+1$ random variables and $M$ non-random variables are used in total for description of the whole system.

Assumption 6.2

(State spaces) Let state space $S_{i}$ be a set of states represented by random variables $X_{i}$ and non-random variables $X^{NR}_{{i}},$ i.e., state space $S_{i}$ contains polarization states of photons when they pass through surface $\Sigma _i$ given fixed rotation angles of optical elements preceding i-th optical element.

Remark 6.2

Assumption 6.2 implies that the number of states of a system which were in a state in given state space $S_{i}$ is the same as the number of photons which passed through surface $\Sigma _i$.

Definition 6.2

Functions introduced in definition 3.18 in [9] can be written also as (see assumption 6.1)

$$\begin{aligned} {{\,\textrm{dos}\,}}_{i}(\theta _{i},\vec {\alpha })&= {{\,\textrm{dos}\,}}_{si}(X_{i}) \quad\quad i=0,\dotsc ,M \end{aligned}$$

(14)

$$\begin{aligned} N_{i}(\vec {\alpha })&= N_{i} \quad\quad i=0,\dotsc ,M \end{aligned}$$

(15)

$$\begin{aligned} \rho _{S,i}(\theta _{i},\vec {\alpha })&= \rho _{S,i}(X_{i}) \quad\quad i=0,\dotsc ,M \end{aligned}$$

(16)

where the notation introduced in [9] is on the right-hand sides of the equations. Dependence of the functions on left-hand side on non-random variables (i.e., on the rotation angles of the polarization-sensitive elements) may or may not be written explicitly.

Similarly, the probability of transition $P_{T,i}(X_{i})$ and the probability density function $\rho _{C,i}(X_{i}, X_{i+1})$ introduced by definition 3.22 in [9] can be written as

$$\begin{aligned} P_{T,i}(\theta _{i}, \alpha _{i})&= P_{T,i}(X_{i}){} & {} \quad \quad \quad i=0,\dotsc ,M-1 \end{aligned}$$

(17)

$$\begin{aligned} \rho _{C,i}(\theta _{i}, \theta _{i+1}, \alpha _{i})&= \rho _{C,i}(X_{i}, X_{i+1}){} & {} \quad \quad \quad i=0,\dotsc ,M-1 \, . \end{aligned}$$

(18)

Remark 6.3

In the case of transmission of light through sequence of polarization-sensitive elements, the transmission of a photon through i-th element is described by three functions $\rho _{S,i}$, $P_{T,i}$ and $\rho _{C,i}$ corresponding to the three probabilistic effects mentioned in Sect. 2:

• 1. $\rho _{S,i}(\theta _{i},\vec {\alpha })$ is probability density function characterizing distribution of polarization angles $\theta _{i}$ of incoming photons before an interaction with i-th element (i.e., polarization states of photons which passed through surface $\Sigma _i$). This function is normalized to 1 when integrated over all the possible polarization states $\theta _{i}$. It may be taken as dependent on the rotation angles $\vec {\alpha }$ of the axes of all the elements in the sequence preceding the i-th element. By multiplying it by number of photons $N_{i}(\vec {\alpha }),$ one obtains density of photon polarization states ${{\,\textrm{dos}\,}}_{i}(\theta _{i},\vec {\alpha })$.

• 2. $P_{T,i}(\theta _{i}, \alpha _{i})$ is conditional probability that photon is transmitted through i-th element being rotated by angle $\alpha _{i}$ given that the input photon polarization is $\theta _{i}$. Values of this probability function are in the interval from 0 to 1 (not necessarily in the whole interval).

• 3. $\rho _{C,i}(\theta _{i}, \theta _{i+1}, \alpha _{i})$ is conditional probability density function characterizing change of input photon polarization $\theta _{i}$ to output photon polarization $\theta _{i+1}$ after the photon is transmitted through i-th element rotated by $\alpha _{i}$ given that the photon passed through the element. This function is normalized to 1 when integrated over all the possible values of the outgoing polarization states $\theta _{i+1}$ (independently on the value of the incoming photon polarization $\theta _{i}$ and the rotation $\alpha _{i}$).

The function $\rho _{S,i}$ characterizes property of the photon beam before an interaction with the i-th element, and the functions $P_{T,i}$ and $\rho _{C,i}$ characterize interaction of a photon with the i-th element.

Definition 6.3

(Stochastic process: case 1) Let $\{X_{i} : i \in I\}$ be stochastic IM process given by definition 5.3 which satisfies assumptions 5.1 to 5.4. Let it satisfy also assumptions 6.1 and 6.2.

Remark 6.4

Stochastic process given by definition 6.3 is IM process, i.e., it is based on the two main assumptions 5.3 and 5.4. Assumption 5.3 concerning independence of outcomes of an experiment means independence of transmissions of individual photons through a sequence of optical elements. Assumption 5.4 about memoryless (Markov) property means that transmission of a photon in i-th state through $(i+1)$-th optical elements depends on the i-th state and not on any of the states in which the photon was before.

Several formulae derived in Sect. 5.2 or [9] will be needed for analysis of data corresponding to transmission of light through a sequence of M polarization-sensitive devices using stochastic processes given by definition 6.3. The formulae can be rewritten using the notation introduced above.

Theorem 6.1

Let $\{X_{i} : i \in I\}$ be stochastic process given by definition 6.3. It holds

$$\begin{aligned} 1 \ge&\frac{N_{1}(\vec {\alpha })}{N_{0}} \ge \frac{N_{2}(\vec {\alpha })}{N_{0}} \ge \cdots \ge \frac{N_{M}(\vec {\alpha })}{N_{0}} \ge 0 \, . \end{aligned}$$

(19)

Proof

It follows from Eq. (75) in [9] and assumption 6.1. $\hfill \square$

Theorem 6.2

Let $\{X_{i} : i \in I\}$ be stochastic process given by definition 6.3. It holds

$$\begin{aligned} \int _{\theta _{i}} \rho _{S,i}(\theta _{i},\vec {\alpha })\text {d} \theta _{i}= 1&\qquad i=0,\ldots ,M \, . \end{aligned}$$

(20)

Proof

This normalization condition can be derived using Eq. (46) in [9] and Eq. (16). $\hfill \square$

Theorem 6.3

Let $\{X_{i} : i \in I\}$ be stochastic process given by definition 6.3. It holds

$$\begin{aligned} \int _{\theta _{i+1}} \rho _{C,i}(\theta _{i}, \theta _{i+1}, \alpha _{i})\text {d} \theta _{i+1}= 1&\, . \end{aligned}$$

(21)

Proof

It follows from Eq. (76) in [9] and assumption 6.1. $\hfill \square$

Remark 6.5

The normalization condition (21) holds for arbitrary value of $\theta _{i}$.

Theorem 6.4

Let $\{X_{i} : i \in I\}$ be stochastic process given by definition 6.3. It holds

$$\begin{aligned} {{\,\textrm{dos}\,}}_{i}(\theta _{i},\vec {\alpha })&= N_i(\vec {\alpha }) \rho _{S,i}(\theta _{i},\vec {\alpha }){} & {} \qquad i=0,\ldots ,M \, . \end{aligned}$$

(22)

Proof

It follows from Eqs. (7) and  (14) to (16). $\hfill\square$

Theorem 6.5

Let $\{X_{i} : i \in I\}$ be stochastic process given by definition 6.3. It holds

$$\begin{aligned} N_i(\vec {\alpha })&= \int _{\theta _{i}} {{\,\textrm{dos}\,}}_{i}(\theta _{i},\vec {\alpha }) \text {d} \theta _{i}{} & {} \qquad i=0,\ldots ,M \, . \end{aligned}$$

(23)

Proof

Equations (8), (14) and (15) imply Eq. (23). $\hfill \square$

Remark 6.6

The density of states ${{\,\textrm{dos}\,}}_{i}(\theta _{i},\vec {\alpha })$ essentially defined by Eq. (22) represents spectrum of values of polarization angles of photons which passed through surface $\Sigma _i$; the spectrum is normalized to the number of photons $N_i$ which in total passed through surface $\Sigma _i$, see Eq. (23).

Theorem 6.6

Let $\{X_{i} : i \in I\}$ be stochastic process given by definition 6.3. It holds ($N_0 \ne 0$)

$$\begin{aligned} \frac{{{\,\textrm{dos}\,}}_{i}(\theta _{i},\vec {\alpha })}{N_0}&= \frac{N_i(\vec {\alpha })}{N_0} \rho _{S,i}(\theta _{i},\vec {\alpha }){} & {} \qquad i=0,\ldots ,M \end{aligned}$$

(24)

Proof

Equation (22) can be divided by $N_0$.

Theorem 6.7

Let $\{X_{i} : i \in I\}$ be stochastic process given by definition 6.3. It holds ($N_0 \ne 0$)

$$\begin{aligned} \frac{N_i(\vec {\alpha })}{N_0}&= \int _{\theta _{i}} \frac{{{\,\textrm{dos}\,}}_{i}(\theta _{i},\vec {\alpha })}{N_0} \text {d} \theta _{i}{} & {} \qquad i=0,\ldots ,M \, . \end{aligned}$$

(25)

Proof

Equation (23) can be divided by $N_0$. $\hfill \square$

Theorem 6.8

Let $\{X_{i} : i \in I\}$ be stochastic process given by definition 6.3. It holds

$$\begin{aligned}&\frac{N_{i+1}(\vec {\alpha })}{N_0}\rho _{S,i+1}(\theta _{i+1},\vec {\alpha })= \int _{\theta _{i}} \frac{N_i(\vec {\alpha })}{N_0} \rho _{S,i}(\theta _{i},\vec {\alpha })P_{T,i}(\theta _{i}, \alpha _{i})\rho _{C,i}(\theta _{i}, \theta _{i+1}, \alpha _{i})\text {d} \theta _{i}\, . \end{aligned}$$

(26)

for $i \in (0,\ldots ,M-1)$.

Proof

Equation (10) can be rewritten using Eqs. (16) to (18). $\hfill\square$

Theorem 6.9

Let $\{X_{i} : i \in I\}$ be stochastic process given by definition 6.3. It holds

$$\begin{aligned} \frac{N_{i+1}(\vec {\alpha })}{N_{0}}&= \int _{\theta _{i}} \frac{N_{i}(\vec {\alpha })}{N_0} \rho _{S,i}(\theta _{i},\vec {\alpha })P_{T,i}(\theta _{i}, \alpha _{i})\text {d} \theta _{i} \end{aligned}$$

(27)

for $i \in (0,\ldots ,M-1)$.

Proof

Equation (11) can be rewritten using Eqs. (16) and (17). $\hfill\square$

Remark 6.7

Equation (26) is of key importance. If the initial probability density function $\rho _{S,0}$ and the functions $P_{T,i}$ and $\rho _{C,i}$ are given for all $i=0,\ldots ,M-1,$ then the formula (26) may be used iteratively to calculate $\frac{N_i(\vec {\alpha })}{N_0} \rho _{S,i}(\theta _{i},\vec {\alpha })$ for all $i=1,\ldots ,M$ (i.e., behind each polarization-sensitive element). Let us emphasize that an outgoing photon after being transmitted through a polarization-sensitive element can have different value of polarization, and it becomes incoming photon interacting with the next element in the sequence. The photon polarization angles corresponding to the surfaces $\Sigma _i$ are, therefore, distinguished by index i in Eq. (26).

6.2 Case 2: sequence of polarization-sensitive elements and independence of $\rho _{C,i}(\theta _{i}, \theta _{i+1}, \alpha _{i})$ on $\theta _{i}$

Definition 6.4

If probability density function $\rho _{C,i}(\theta _{i}, \theta _{i+1}, \alpha _{i})$ does not depend on polarization angle of incoming photon, then the function may be denoted as $\rho _{\widetilde{C},i}(\theta _{i+1}, \alpha _{i})$.

Assumption 6.3

(Independence of $\rho _{C,i}(\theta _{i}, \theta _{i+1}, \alpha _{i})$ on ${\theta _{i}}$) Let probability density function $\rho _{C,i}(\theta _{i}, \theta _{i+1}, \alpha _{i})$ do not depend on $\theta _{i}$, i.e., it holds

$$\begin{aligned} \rho _{\widetilde{C},i}(\theta _{i+1}, \alpha _{i})&= \rho _{C,i}(\theta _{i}, \theta _{i+1}, \alpha _{i})&i&=0,\ldots ,M-1 \, . \end{aligned}$$

(28)

Remark 6.8

Assumption 6.3 is equivalent to assumption 3.8 in [9] and assumption 6.1.

Definition 6.5

(Stochastic process: case 2) Let $\{X_{i} : i \in I\}$ be stochastic process given by definition 6.3, and let it satisfy also assumption 6.3.

Remark 6.9

Stochastic process given by definition 6.5 is process given by definition 3.23 in [9] which satisfies assumptions 3.3, 3.5 to 3.7 and 3.8 in [9], and assumptions 6.1 and 6.2. It implies that assumption 6.3 holds (see remark 6.8).

Theorem 6.10

Let $\{X_{i} : i \in I\}$ be stochastic process given by definition 6.5. It holds

$$\begin{aligned} \frac{N_{i+1}(\vec {\alpha })}{N_0}\rho _{S,i+1}(\theta _{i+1},\vec {\alpha })&= \rho _{\widetilde{C},i}(\theta _{i+1}, \alpha _{i})\left[ \int _{\theta _{0}} \rho _{S,0}(\theta _{0},\vec {\alpha })P_{T,{0}}(\theta _{0}, \alpha _{0}) \text {d} \theta _{0}\right] \nonumber \\&\qquad \qquad \prod _{j = 1}^{i} \left[ \int _{\theta _{j}} \rho _{\widetilde{C}, j-1}(\theta _{j}, \alpha _{j-1}) P_{T,{j}}(\theta _{j}, \alpha _{j}) \text {d} \theta _{j}\right] \, . \end{aligned}$$

(29)

where $\prod _{j = 1}^{i} \left[ \ldots \right] =1$ if $i\!=\!0$.

Proof

Equation (97) in [9] can be rewritten using assumptions 6.1 and 6.3 and Eq. (28) to obtain Eq. (29). $\hfill\square$

Remark 6.10

The functions $\frac{N_{i+1}(\vec {\alpha })}{N_0}\rho _{S,i+1}(\theta _{i+1},\vec {\alpha })$ of $\theta _{i+1}$ given by Eq. (29) have the same shapes for all $i=0,\ldots ,M-1$; the functions differ only in normalizations.

Remark 6.11

Equation (29) contains multiplication of integrals, while the integrals in Eq. (26) are calculated iteratively (recursively). Therefore, the functions $\frac{N_{i+1}(\vec {\alpha })}{N_0}\rho _{S,i+1}(\theta _{i+1},\vec {\alpha })$ are easier to calculate numerically using Eq. (29) than using Eq. (26). However, this comes at the cost of loss of generality as assumption 6.3 must be introduced.

6.3 Case 3: sequence of identical polarization-sensitive elements and independence of $\rho _{C,i}(\theta _{i}, \theta _{i+1}, \alpha _{i})$ on $\theta _{i}$

Definition 6.6

(Probability of transmission ${P_{T}(\theta _{\textrm{in}}, \alpha )}$) If the probabilities $P_{T,i}(\theta _{i}, \alpha _{i})$ are the same for all $i=0,1,\ldots ,M-1$, then the probability of transmission of photon through a polarization-sensitive element can be denoted as ${P_{T}(\theta _{\textrm{in}}, \alpha )}$. It depends on the polarization of the incoming photon $\theta _{\textrm{in}}$ and the rotation angle $\alpha$ of the element.

Assumption 6.4

(Identical probabilities $P_{T,i}(\theta _{i}, \alpha _{i})$) The probability functions $P_{T,i}(\theta _{i}, \alpha _{i})$ are the same for all $i=0,1,\ldots ,M-1$, i.e., for all given polarization-sensitive elements, i.e., it holds

$$\begin{aligned} {P_{T}}(\theta _{\textrm{in}}\!=\!\theta _{i}, \alpha \!=\!\alpha _{i})&= P_{T,i}(\theta _{i}, \alpha _{i})&i&=0,\ldots ,M-1 \, . \end{aligned}$$

(30)

Definition 6.7

If the probability density functions $\rho _{C,i}(\theta _{i}, \theta _{i+1}, \alpha _{i})$ are the same for all $i=0,1,\ldots ,M-1$, then the function can be denoted as $\rho _{C}{}(\theta _{\textrm{in}}, \theta _{\textrm{out}}, \alpha )$. It depends on the polarization of the outgoing photon $\theta _{\textrm{out}}$ and the rotation of given optical element $\alpha$.

Assumption 6.5

(Identical functions $\rho _{C,i}(\theta _{i}, \theta _{i+1}, \alpha _{i})$) Let the probability density functions $\rho _{C,i}(\theta _{i}, \theta _{i+1}, \alpha _{i})$ be the same for all polarization-sensitive elements, i.e., it holds

$$\begin{aligned} \rho _{C}{}(\theta _{\textrm{in}}\!=\!\theta _{i}, \theta _{\textrm{out}}\!=\!\theta _{i+1}, \alpha \!=\!\alpha _{i})&= \rho _{C,i}(\theta _{i}, \theta _{i+1}, \alpha _{i})&i&=0,\ldots ,M-1 \, . \end{aligned}$$

(31)

Remark 6.12

Assumption 6.5 is equivalent to assumption 3.9 in [9] and assumption 6.1.

Assumption 6.6

Let all the polarization-sensitive elements in given sequence be identical.

Remark 6.13

Assumption 6.6 implies assumptions 6.4 and 6.5.

Definition 6.8

Let the probability density function corresponding to both assumptions 6.3 and 6.5 be denoted as $\rho _{\widetilde{C}}(\theta _{\textrm{out}}, \alpha )$, i.e.,

$$\begin{aligned} \rho _{\widetilde{C}}(\theta _{\textrm{out}}\!=\!\theta _{i+1}, \alpha \!=\! \alpha _{i})&= \rho _{C,i}(\theta _{i}, \theta _{i+1}, \alpha _{i})&i&=0,\ldots ,M-1 \, . \end{aligned}$$

(32)

Definition 6.9

(Stochastic process: case 3) Let $\{X_{i} : i \in I\}$ be stochastic process given by definition 6.5 and satisfying also assumptions 6.5 and 6.4.

Remark 6.14

Stochastic process given by definition 6.9 is stochastic process given by definition 3.25 in [9] which satisfies assumptions 3.3, 3.5 to 3.7, 3.8 and 3.9 in [9], assumptions 6.1 and 6.2 and assumption 6.4. It implies that assumption 6.5 holds (see remark 6.12) and that assumption 6.6 holds (see remark 6.13).

Let us further assume that all polarization-sensitive elements in given sequence are identical (see assumption 6.6), i.e., let us assume that assumptions 6.4 and 6.5 are satisfied.

Theorem 6.11

Let $\{X_{i} : i \in I\}$ be stochastic process given by definition 6.9. It holds

$$\begin{aligned} \int _{\theta _{\textrm{out}}} \rho _{C}{}(\theta _{\textrm{in}}, \theta _{\textrm{out}}, \alpha )\text {d} \theta _{\textrm{out}}= 1&\, . \end{aligned}$$

(33)

Proof

It follows from Eqs. (21) and (31). $\hfill\square$

Theorem 6.12

Let $\{X_{i} : i \in I\}$ be stochastic process given by definition 6.9. It holds

$$\begin{aligned}&\frac{N_{i+1}(\vec {\alpha })}{N_0}\rho _{S,i+1}(\theta _{i+1},\vec {\alpha })= \int _{\theta _{i}} \frac{N_i(\vec {\alpha })}{N_0} \rho _{S,i}(\theta _{i},\vec {\alpha }){P_{T}}(\theta _{i}, \alpha _{i})\rho _{C}{}(\theta _{i}, \theta _{i+1}, \alpha _{i})\text {d} \theta _{i}\, . \end{aligned}$$

(34)

for $i \in (0,\ldots ,M-1)$.

Proof

Equation (26) can be rewritten using Eqs. (30) and (31). The two functions $P_{T,{}}$ and $\rho _{C}{}$ are the same for all the identical polarization-sensitive elements in the sequence, but it must be correctly integrated over their arguments, see Eq. (34). $\hfill\square$

Theorem 6.13

Let $\{X_{i} : i \in I\}$ be stochastic process given by definition 6.9. It holds

$$\begin{aligned} \frac{N_{i+1}(\vec {\alpha })}{N_{0}}&= \int _{\theta _{i}} \frac{N_{i}(\vec {\alpha })}{N_0} \rho _{S,i}(\theta _{i},\vec {\alpha }){P_{T}}(\theta _{i}, \alpha _{i})\text {d} \theta _{i} \end{aligned}$$

(35)

for $i \in (0,\ldots ,M-1)$.

Proof

Equation (27) can be rewritten using Eq. (30). The function $P_{T,{}}$ is the same for all the identical polarization-sensitive elements in the sequence, but it must be correctly integrated over its arguments, see Eq. (35). $\hfill\square$

Theorem 6.14

Let $\{X_{i} : i \in I\}$ be stochastic process given by definition 6.9. It holds

$$\begin{aligned} \frac{N_{i+1}(\vec {\alpha })}{N_0}\rho _{S,i+1}(\theta _{i+1},\vec {\alpha })&= \rho _{\widetilde{C}}(\theta _{i+1}, \alpha _{i}) \left[ \int _{\theta _{0}} \rho _{S,0}(\theta _{0},\vec {\alpha })P_{T,{}}(\theta _{0}, \alpha _{0}) \text {d} \theta _{0}\right] \nonumber \\&\qquad \qquad \prod _{j = 1}^{i} \left[ \int _{\theta _{j}} \rho _{\widetilde{C}}(\theta _{j}, \alpha _{j-1}) P_{T,{}}(\theta _{j}, \alpha _{j}) \text {d} \theta _{j}\right] \, . \end{aligned}$$

(36)

Proof

Equation (29) can be simplified using Eqs. (30) to (32). $\hfill\square$

Remark 6.15

Numerical calculation of $\frac{N_{i+1}(\vec {\alpha })}{N_0}\rho _{S,i+1}(\theta _{i+1},\vec {\alpha })$ using Eq. (36) is simpler than using Eq. (34), see also remark 6.11.

Remark 6.16

Equations (35) and (34) (or Eq. (36)) allow to calculate the relative number of transmitted photons $N_{i}(\vec {\alpha })/N_{0}$ for any $i=1,\ldots ,M$ according to Eq. (35) if three functions are known: $\rho _{S,0}(\theta _{0},\vec {\alpha })$, ${P_{T}(\theta _{\textrm{in}}, \alpha )}$ and $\rho _{C}{}(\theta _{\textrm{in}}, \theta _{\textrm{out}}, \alpha )$. These unknown functions may be parameterized and determined on the basis of measured (input) values of $N_{i}(\vec {\alpha })/N_{0}$.

7 Definitions of types of optical elements and properties of light (beam)

Different optical elements can transmit light very differently. Standard way of classifying optical elements is based on measured properties of photon diagnostic beam before and after an interaction with given optical element (or using a sequence of elements). In Sect. 7.1, one can find definitions of polarized and unpolarized beam and definitions of two types of polarization-sensitive devices: linear polarizer and Faraday rotator.

Another way how to define various types of optical elements and properties of light beam is using probability (density) functions $\rho _{S}{}$, $P_{T,{}}$ and $\rho _{C}{}$. This method is discussed in Sect. 7.2. The first function characterizes property of photon beam and can be, therefore, used for definition of (un)polarized beam, see Sect. 7.2.1. The other two functions can be used for definitions of various types of optical elements (such as linear polarizer and Faraday rotator), see Sect. 7.2.2. Definitions concerning interaction of light (beam) with polarization-sensitive elements are in Sect. 7.2.3.

A type of an optical element identified with the help of the first approach should be equivalently and consistently identified using the second approach. Determination of functions $P_{T,{}}$ and $\rho _{C}{}$ on the basis of experimental data basically requires experimental data needed to identify the element using the first approach. Some of the definitions below can be specified more precisely, if needed. For our purposes, it is sufficient to show mainly the basic idea behind each definition.

7.1 Definitions based on measured properties of transmitted beam

7.1.1 Definitions of properties of light (beam)

Definition 7.1

(Unpolarized light (beam)) If a photon beam is transmitted through a polarization-sensitive element and the number of transmitted photons does not depend on the rotation of the element, then the light is unpolarized.

Definition 7.2

(Polarized light (beam)) If light is not unpolarized, see definition 7.1, then it is polarized.

7.1.2 Definitions of different types of optical elements

Definition 7.3

(Ideal identical linear polarizers) Ideal identical linear polarizers are polarization-sensitive elements which transmit photons according to Eq. (5) when initial photon beam is unpolarized (i.e., they behave according to Malus’s law).

Definition 7.4

((Real) linear polarizer) An optical element transmits light similarly as ideal linear polarizer (see also Sect. 3.2.2).

Definition 7.5

(Faraday rotator) Let us consider unpolarized beam passing through sequence of a linear polarizer, an optical element and another linear polarizer. The number of photons transmitted through this sequence, divided by initial number of photons, can be measured as a function of the rotation of the second linear polarizer (the first one having fixed angle of rotation). This quantity can be measured with and without magnetic field applied to the unknown optical element. If the two measured quantities have the same dependence on the angle of the rotation, but are shifted by an angle, and the shift depends on the parallel component of the magnetic field, then the unknown element is Faraday rotator (also called Faraday effect-based device).

Remark 7.1

Measurement of this type is closely related to measurement of the Verdet constant and can be found in [47]. The angular shift is clearly visible in Fig. 3 in [47].

7.2 Definitions based on probability (density) functions

7.2.1 Definitions of properties of light (beam)

Probability density function $\rho _{S}(\theta )$ in dependence on polarization angle $\theta$ characterizes distribution of polarization states of photons when they pass through the corresponding control surface.

Definition 7.6

(Unpolarized light (beam)) If $\rho _{S}(\theta )$ does not depend on $\theta,$ then the photon beam passing through the control surface is unpolarized, i.e., photon polarization angles $\theta$ of the photons are distributed uniformly and the distribution is normalized to 1 (see Eq. (20)), and it implies

$$\begin{aligned} \rho _{S}(\theta ) = \frac{1}{2\pi } \, . \end{aligned}$$

(37)

(full angle in radians is $2\pi$).

Definition 7.7

(Polarized light (beam)) If $\rho _{S}(\theta )$ depends on $\theta,$ then the photon beam passing through the control surface is polarized.

Remark 7.2

Definitions 7.6 and 7.7 imply that if a photon beam passing through a control surface is not polarized, then it is unpolarized (and vice versa).

7.2.2 Definitions of different types of optical elements

An optical element (sample) is characterized by functions $P_{T,{}}$ and $\rho _{C}{}$ (i.e., by ${P_{T}(\theta _{\textrm{in}}, \alpha )}$ and $\rho _{C}{}(\theta _{\textrm{in}}, \theta _{\textrm{out}}, \alpha )$ in the case of polarization-sensitive element). They can be determined on the basis of experimental data. These two functions can be, therefore, used for distinguishing various types of optical elements. Let us define, e.g., linear polarizer and Faraday rotator.

Definition 7.8

(Linear polarizer) A linear polarizer is an optical element which may or may not change polarization angle $\theta _{\textrm{in}}$ of an incoming photon such that the possible directions of polarization of the outgoing photon (specified by the polarization angle $\theta _{\textrm{out}}$) are predominantly parallel to an axis (called axis of the linear polarizer), i.e., probability density function $\rho _{C}{}(\theta _{\textrm{in}}, \theta _{\textrm{out}}, \alpha )$ corresponding to this optical element has a peak at the position of the axis for any fixed value of $\theta _{\textrm{in}}$ and rotation of the axis $\alpha$. If the outgoing photons have all the same value of $\theta _{\textrm{out}},$ then $\rho _{C}{}(\theta _{\textrm{in}}, \theta _{\textrm{out}}, \alpha )$ is given by a delta function. One may also assume that probability function ${P_{T}(\theta _{\textrm{in}}, \alpha )}$ corresponding to this element depends on $\theta _{\textrm{in}}$ and the rotation of the axis $\alpha$.

Definition 7.9

(Faraday rotator) A Faraday rotator (or Faraday effect-based device) is an optical element having probability function ${P_{T}(\theta _{\textrm{in}}, \alpha )}$ which does not depend on polarization state of an incoming photon $\theta _{\textrm{in}}.$ (It depends on parameters such as the length of the medium, its temperature and component of magnetic field applied to it and being parallel to the direction of the beam.) Moreover, the difference of polarization angle of an outgoing photon $\theta _{\textrm{out}}$ and the polarization angle $\theta _{\textrm{in}}$ of the incoming photon is the same (resp. roughly the same) for each transmitted photon independently on the value of $\theta _{\textrm{in}}$, i.e., probability density function $\rho _{C}{}(\theta _{\textrm{in}}, \theta _{\textrm{out}}, \alpha )$ as a function of the difference $\theta _{\textrm{in}}$ - $\theta _{\textrm{out}}$ is given by a delta function (resp. by a function having a significant peak).

7.2.3 Definitions of interaction of light with optical elements

Definition 7.10

(Polarizing and depolarizing transmission) If $\rho _{S,i+1}$ differs from uniform distribution more than $\rho _{S,i},$ then the transmission is polarizing. If $\rho _{S,i+1}(\theta _{i+1},\vec {\alpha })$ differs from uniform distribution less than $\rho _{S,i}(\theta _{i},\vec {\alpha }),$ then the transmission is depolarizing.

Remark 7.3

The difference of the two distributions in definition 7.10 can be quantified, e.g., with the help of the second moments of the distributions. If the second moment of $\rho _{S,i+1}(\theta _{i+1},\vec {\alpha })$ is lower (resp. higher) than the second moment of $\rho _{S,i}(\theta _{i},\vec {\alpha })$, then the transmission is polarizing (resp. depolarizing). Or another rule can be used, if one of the moments is not finite. The second moments of the distributions may not be convenient characterization of “(de)polarization” if one of the distributions has more than one significant peak.

Remark 7.4

$\frac{N_{i+1}(\vec {\alpha })}{N_0}\rho _{S,i+1}(\theta _{i+1},\vec {\alpha })$ characterizing density of photon polarization states behind i-th polarization-sensitive element can be calculated on the basis of three functions $\frac{N_i(\vec {\alpha })}{N_0}\rho _{S,i}(\theta _{i},\vec {\alpha })$, $P_{T,i}(\theta _{i}, \alpha _{i})$ and $\rho _{C,i}(\theta _{i}, \theta _{i+1}, \alpha _{i})$, see Eq. (26). The output density of states depends on the input density of states and the properties of the element. Therefore, it is possible that an element can have polarizing or depolarizing effect on the beam in dependence on the input probability density function $\rho _{S,i}(\theta _{i},\vec {\alpha })$. These two effects can be distinguished by compering functions $\rho _{S,i}(\theta _{i},\vec {\alpha })$ and $\rho _{S,i+1}(\theta _{i+1},\vec {\alpha })$.

8 Example of analysis of data using IM process: three-polarizer experiment

In the previous sections, the probabilistic and corpuscular theoretical description suitable for describing optical phenomena, especially those related to polarization, has been explained. In the following, an example data of relative photon numbers $N_{i}(\vec {\alpha })/N_{0}$ ($i=1,2,3$) measured as explained in Sect. 3.1 will be analyzed with the help of stochastic IM process adapted for description of polarization of light, see Sect. 6.

Choice of data sample is discussed in Sect. 8.1. The analysis will be done with the help of the general guidelines summarized in Sect. 4 in [9]. Similar analysis has never been done in the context of optics until now. We will be, therefore, interested mainly in the concept and possibilities of this new kind of analysis. We will not focus too much on numerical details. The analysis has helped to identify several open questions related to determination of functions $P_{T,{}}$ and $\rho _{C}{}$ characterizing linear polarizers, and they are discussed in Sect. 8.7.

8.1 Example data: three ideal identical linear polarizers

Relative photon numbers $N_{i}(\vec {\alpha })/N_{0}$ corresponding to transmission of light through one and two linear polarizers in dependence on their rotation angles are commonly measured (e.g., to compare the measured intensities with the Malus’s law). However, it seems that there are no publicly available experimental data of the relative beam intensities (number of transmitted photons) measured behind one, two and three linear polarizers in dependence on the rotations of the polarizers. In [15], one can find interesting experimental results concerning transmission of light through three linear polarizers, but not the measured values of all the relative numbers of transmitted photons $N_{i}(\vec {\alpha })/N_{0}$ needed for our analysis.

Therefore, let us take (for the sake of simplicity) the dependences given by Eq. (5) for $M=3$ corresponding to sequence of three ideal identical linear polarizers as an input for our analysis of data. It will be assumed that photons in the beam have the same energy, i.e., that Eq. (3) holds. These dependences will be used in our further considerations as an example of measured data. In the following, we will not focus on possible differences existing between the relative photon numbers corresponding to real and ideal polarizers (see Sect. 3.2.2). This well-defined example of measured data will be analyzed with the aim to determine probabilistic (statistical) characteristics of a photon transmitted through the sequence of linear polarizers.

8.2 Initial exploration of data

The data in Sect. 8.1 correspond to a sequence of identical polarization-sensitive elements. (One can assume that the ideal polarizers are identical.) Therefore, one can try to describe them with the help of stochastic process given by definition 6.9, see also Sect. 6.3 for key formulae corresponding to this process.

Experimental data often reveal symmetries. This is also the case of the relative photon numbers $N_{i}(\vec {\alpha })/N_{0}$ ($i=1,..,M$) in dependence on the orientation of the axes of the linear polarizers given by Eq. (5). The ratio $N_{1}(\vec {\alpha })/N_{0}$ does not depend on the orientation of the axis of the first polarizer, and the ratios $N_{2}(\vec {\alpha })/N_{0}$ and $N_{3}(\vec {\alpha })/N_{0}$ are periodic due to the cosine-squared function in Eq. (5c). One may define the following function

$$\begin{aligned} {{\,\textrm{sym}\,}}(x) =\left\{ \begin{array}{llr} \pi - y &{}\text {if }\frac{\pi }{2}< y <= \pi &{} \qquad \qquad \qquad \qquad \mathrm{(38a)}\\ y &{}\text {otherwise}&{} \qquad \qquad \qquad \qquad \mathrm{(38a)} \end{array}\right. \end{aligned}$$

where $y = (|{x} | \mod \pi )$. This function will help to reflect the symmetries. The function is even (${{\,\textrm{sym}\,}}(x)={{\,\textrm{sym}\,}}(-x)$) and is plotted in Fig. 3.

Fig. 3

The function ${{\,\textrm{sym}\,}}(x)$ given by Eq. (38) in the interval from -$\pi$ rad to $\pi$ rad and having period $\pi$ rad

8.3 Check of applicability of the probabilistic model to input data (consistency)

Following the guidelines in Sect. 4.2 in [9], one can check that the input data of photon numbers discussed in Sect. 8.1 satisfy the basic inequalities given by Eq. (19), and the relative photon numbers $N_{i}(\vec {\alpha })/N_{0}$ do not depend on $N_{0}$. For example, if the relative number of photons is determined from the relative beam intensity, see Eq. (3), then assumption 3.1 can be tested experimentally for a photon beam passing through surface $\Sigma _i$ ($i=0,\ldots ,3$). Assumption 3.1 is assumed to be satisfied in our case (see Sect. 8.1). The other assumptions (see Sect. 6) and consequences of the probabilistic model must be tested indirectly.

8.4 Parameterization of unknown functions

According to remark 6.16, the used probabilistic model contains three unknown functions $\rho _{S,0}(\theta _{0},\vec {\alpha }), {P_{T}(\theta _{\textrm{in}}, \alpha )}\hbox { and }\rho _{C}{}(\theta _{\textrm{in}}, \theta _{\textrm{out}}, \alpha )$ which can be parameterized and determined on the basis of experimental data. Parameterizations of some a priory unknown functions in any model are typically accompanied by several additional implicit assumptions. Let us, therefore, formulate for completeness the following assumption:

Assumption 8.1

(Parameterization of unknown functions) Let functions $\rho _{S,0}(\theta _{0},\vec {\alpha }), {P_{T}(\theta _{\textrm{in}}, \alpha )}\hbox { and }\rho _{C}{}(\theta _{\textrm{in}}, \theta _{\textrm{out}}, \alpha )$ be parameterized.

8.4.1 Parameterization of probability density $\rho _{S,0}$

If decrease in measured photon numbers behind the first polarizer $N_{1}(\vec {\alpha })/N_{0}$ does not depend on the orientation of its polarization axis, see Eq. (5b), then one may assume that the probability density function $\rho _{S,0}(\theta _{0},\vec {\alpha })$ does not depend on the photon polarization, i.e., the initial photon polarization states are distributed uniformly (the initial photon beam is unpolarized, see Eq. (37))

$$\begin{aligned} \rho _{S,0}(\theta _{0},\vec {\alpha })= \frac{1}{2\pi } \, . \end{aligned}$$

(39)

In more general case, it would be necessary to introduce a parameterization depending on $\theta _{0}$ and some free parameters; it would correspond to polarized light.

8.4.2 Parameterization of probability $P_{T,{}}$

We may assume that the probability of photon transmission through one polarizer depends only on the difference of the polarizer rotation and the photon polarization (${P_{T}(\theta _{\textrm{in}}, \alpha )}= P_{T,{}}(\theta _{\textrm{in}}- \alpha )$). The following parameterization may be chosen

$$\begin{aligned} {P_{T}(\theta _{\textrm{in}}, \alpha )}= 1 - \frac{1 - g(\theta _{\textrm{in}},\alpha )}{1+a_{2}g(\theta _{\textrm{in}},\alpha )} \end{aligned}$$

(40)

where

$$\begin{aligned} g(\theta _{\textrm{in}},\alpha ) = \textbf{e}^{-a_0 \left( \frac{{{\,\textrm{sym}\,}}(\theta _{\textrm{in}}- \alpha )}{u_0}\right) ^{a_1}} \end{aligned}$$

(41)

and $a_0$, $a_1$ and $a_2$ are free parameters ($u_0 = 1 \text {rad}$). The function $P_{T,{}}$ has meaning of probability; its values should be, therefore, in the interval from 0 to 1 (not necessary in the full interval) for given values of the free parameters.

8.4.3 Parameterization of probability density $\rho _{C}{}$

Parameterization of the function $\rho _{C}{}(\theta _{\textrm{in}}, \theta _{\textrm{out}}, \alpha )$ may be chosen in the form of Gaussian function

$$\begin{aligned} \rho _{C}{}(\theta _{\textrm{in}}, \theta _{\textrm{out}}, \alpha )= \frac{1}{2\sigma \sqrt{\pi }{{\,\textrm{erf}\,}}\left( \frac{\pi }{2\sigma }\right) } \textbf{e}^{-\left( \frac{{{\,\textrm{sym}\,}}(\theta _{\textrm{out}}- \alpha )}{\sigma }\right) ^2} \end{aligned}$$

(42)

where $\sigma$ is a free parameter and ${{\,\textrm{erf}\,}}(x)$ is the error function defined as

$$\begin{aligned} {{\,\textrm{erf}\,}}(x) = \frac{2}{\sqrt{\pi }} \int _0^{x} \textbf{e}^{-t^2} \text {d} t \, . \end{aligned}$$

(43)

The probability density function $\rho _{C}{}$ given by Eq. (42) is normalized to 1 when integrated over $\theta _{\textrm{out}}$(required by the normalization condition given by Eq. (33)). It does not depend on the value of $\theta _{\textrm{in}}$which is consistent with assumption 6.3. It is assumed that it depends only on the difference $\theta _{\textrm{out}}- \alpha$ (similarly as in the case of the parameterization of the $P_{T,{}}$ function, see Eq. (40)).

The parameterization of $\rho _{C}{}$ given by Eq. (42) corresponds to continuous spectrum of polarization angle values $\theta _{\textrm{out}}$ centered around the value given by the rotation of the polarizer $\alpha$. If only a single discrete value $\theta _{\textrm{out}}$ (equal to $\alpha$) was admitted, then the probability density function would be represented by corresponding delta function. The smaller the value of the free parameter $\sigma$ (i.e., the width of the corresponding peak), the closer the continuous spectrum is to the delta function.

Remark 8.1

The parameterizations of $P_{T,{}}$ and $\rho _{C}{}$ given by Eqs. (40) and (42) correspond to the definition of linear polarizer, see definition 7.8. The parameterizations of the three functions $\rho _{S,0}$, $P_{T,{}}$ and $\rho _{C}{}$ a priory restrict set of possible solutions which could describe measured data. The parameterizations represent additional assumptions in the probabilistic model, see assumption 8.1. The parameterizations are consistent with the assumptions of stochastic process given by definition 6.9.

Remark 8.2

Four free parameters have been introduced ($\sigma$, $a_0$, $a_1$ and $a_2$). Their physical meaning is not important, if we stay in the first stage of data analysis, see Sect. 5.3.

8.5 Fitting of the probabilistic model to data

One can try to determine the parameterized functions (values of all the free parameters) in stochastic process given by definition 6.9 on the basis of experimental data by means of optimization techniques. The relative photon numbers $\frac{N_{i+1}(\vec {\alpha })}{N_{0}}$ can be calculated with the help of Eq. (35). The quantity $\frac{N_{i+1}(\vec {\alpha })}{N_0}\rho _{S,i+1}(\theta _{i+1},\vec {\alpha })$ can be calculated using Eq. (34) or Eq. (36) which is less computationally intensive task, see also remark 6.15 in this paper and Sect. 4.6 in [9] for further comments related to computational complexity. Calculation of the relative photon numbers $N_{3}(\vec {\alpha })/N_{0}$ behind the third (i.e., the last) linear polarizer in the sequence according to Eq. (35), and needed for comparison to data, is computationally the most intensive task.

Calculated relative photon numbers $N_{i}(\vec {\alpha })/N_{0}$ ($i=1,\ldots ,M$) behind $(i-1)$-th linear polarizer can be calculated for any value of the rotation angle of the polarizer and the rotations of all the preceding polarizers, i.e., i continuous parameters represented by $\vec {\alpha }$. The rotation angles are typically measured in discrete steps, and the example data, see Sect. 8.1, can be considered only in discrete steps, too. However, even if some discrete angular steps are considered, it may still represent far too many data points (depending on the width of the steps) already in the case of $M=3$. It is, therefore, useful to simplify fitting of the model to data as much as possible.

It is not necessary to take into account all values of $\alpha _{0}$ due to the symmetric dependence of the relative number of transmitted particles on the value of $\alpha _{0}$ (see Sect. 8.2). It is sufficient to consider only one value, e.g., $\alpha _{0}=0\text { deg}$. This is closely related to the fact that the initial light is taken as unpolarized, see Eq. (39).

8.6 Numerical results

It is possible to find dependence of the parameterized functions $P_{T,{}}$ and $\rho _{C}{}$ (i.e., values of the free parameters discussed in Sects. 8.4.2 and 8.4.3) which can describe the input data, under the given set of assumptions on which the stochastic process given by definition 6.9 is based. The solution is, however, not unique. Therefore, only one solution corresponding to the values of the free parameters in Table 1 is discussed in the following. This solution corresponds to the highest value of the parameter $\sigma$ for which one can still fit well the input data. We will come back to the problem of ambiguity in Sect. 8.7.

Table 1 The values of the free parameters of the probabilistic model determined on the basis of data using optimization techniques

The comparison of the measured number of transmitted photons to calculated (simulated) number of transmitted photons with the help of the probabilistic model is in Fig. 6. The model agrees well with the input data and can be further improved by using more flexible parameterizations of the functions $P_{T,{}}$ and $\rho _{C}{}$. Our focus is, however, on conceptually important points and questions, as it has been already mentioned.

The probability $P_{T,{}}$ given by Eq. (40) as a function of $\theta _{\textrm{in}}$ is plotted in Fig. 4 for $\alpha _{0}=0\text { deg}$. The values of this function representing probability are in the interval from 0 to 1.

Fig. 4

The probability of transmission of a photon through one polarizer having the orientation of its axis $\alpha =0\text { deg}$ as a function of the polarization of the incoming photon

Fig. 5

The probability density function of change of photon polarization of a photon transmitted through one polarizer having the orientation of its axis $\alpha =0\text { deg}$ as a function of the polarization of the outgoing photon

Fig. 6

The comparison of the input data of relative photon numbers $N_{i}(\vec {\alpha })/N_0$ ($i=1, 2$) corresponding to transmission of unpolarized light through three ideal identical linear polarizers (Malus’s law) with the probabilistic model (see Sect. 6.3) fitted to the data for several combinations of rotations of the polarizes $\vec {\alpha }$ (in degrees). Blue lines—the input data given by Eq. (5). Orange lines—the probabilistic model, see Eqs. (35) and (34)

Fig. 7

The probability density functions $\rho _{S,i}(\theta _{i},\vec {\alpha })$ as a function of photon polarization angle $\theta _{i}$ if the three ideal identical linear polarizers are rotated by $\vec {\alpha } = (0.0, 45.0 \text { deg}, 90.0 \text { deg})$. In the case of $i=0,$ the function represents initial uniform polarization of photons in the beam given by Eq. (39); in the case of $i=1,2,3$ it describes polarization of the beam transmitted through the one, two or three polarizers. The orientations of the axes of the polarizers are specified by $\vec {\alpha }$, see the vertical dashed lines. The position of the peaks corresponds to the rotation of the axis of the polarizers

Fig. 8

Similar picture to Fig. 7 but the probability density functions $\rho _{S,i}(\theta _{i},\vec {\alpha })$ multiplied by $N_{i}(\vec {\alpha })/N_0$ are plotted, and these expressions are equal to relative densities of states ${{\,\textrm{dos}\,}}_{i}(\theta _{i},\vec {\alpha }) / N_0$, see Eq. (24)

The probability density function $\rho _{C}{}$ given by Eq. (42) as a function of $\theta _{\textrm{out}}$ is plotted in Fig. 5 for $\alpha _{0}=0\text { deg}$. This function is normalized to unity, see Eq. (33), and the values are not in the interval from 0 to 1. (Values of a probability density functions may or may not be in the interval from 0 to 1.)

The probability density functions $\rho _{S,i}(\theta _{i},\vec {\alpha })$ as a function of photon polarization angle $\theta _{i}$ ($i=0,\ldots ,3$) are plotted in Fig. 7 for one fixed combination of rotations of the polarizes: $\vec {\alpha } = (0.0, 45.0 \text { deg}, 90.0 \text { deg})$. The blue line in Fig. 7 corresponds to Eq. (39), i.e., to the assumed initial uniform distribution of photon polarization (unpolarized light). Unpolarized light transmitted through one or more linear polarizers is not uniform, see Fig. 7 (and Eq. (42)). The probability density functions $\rho _{S,i}(\theta _{i},\vec {\alpha })$ ($i=1,\ldots ,M$) have the same shapes (differing only in position). It is the consequence of the used assumptions, see remark 6.10.

The probability density function $\rho _{S,i}(\theta _{i},\vec {\alpha })$ multiplied by $N_{i}(\vec {\alpha })/N_0$ ($i=0,..,3$) for the same rotations of the polarizes $\vec {\alpha } = (0.0, 45.0 \text { deg}, 90.0 \text { deg})$ is plotted in Fig. 8 similarly as the functions $\rho _{S,i}(\theta _{i},\vec {\alpha })$ are plotted in Fig. 7. The function $(N_{i}(\vec {\alpha })/N_0) \; \rho _{S,i}(\theta _{i},\vec {\alpha })$ is equal to ${{\,\textrm{dos}\,}}_{i}(\theta _{i},\vec {\alpha })/N_0$ (see Eq. (24)). This function expresses not only the density of polarization angles but also the decrease in the photon numbers behind $(i-1)$-th polarizer (if $i=1,2,3$), see Eqs. (24) and (25). For the value of $\vec {\alpha },$ it holds $N_1(\vec {\alpha })/N_0=0.5$, $N_2(\vec {\alpha })/N_0=0.25$, $N_3(\vec {\alpha })/N_0=0.12$ (according to both the input data and the probabilistic model fitted to the data). This decrease in number of transmitted photons with increasing number of polarizers is visible in Fig. 8, too.

The numerical results clearly show that it is possible to explain the famous three-polarizer experiment using the idea of quanta of light (photons) and the theory of stochastic processes. It is possible to determine, with the help of the stochastic process given by definition 6.9, the probability (density) functions characterizing the transmission of light through the polarizers mentioned in Sect. 1. The open questions related to the ambiguity of the determination of the parameterized functions will be discussed in Sect. 8.7.

The numerical results presented in this section have been obtained with the help of ROOT [48] and Matplotlib [49].

8.7 Open questions

As it has been mentioned, the determined dependence of the parameterized functions $P_{T,{}}$ and $\rho _{C}{}$ shown in Sect. 8.6 can explain the measured (input) number of transmitted photons through three ideal identical linear polarizers, but it is not unique. There are several open questions concerning determination of the functions:

• 1. Dependence of $\rho _{C}{}$ on $\theta _{\textrm{out}}$

  There is strong ambiguity on determination of the value of the parameter $\sigma$, i.e., the width of the peaks characterizing the function $\rho _{C}{}$, see Fig. 5. The input data could be fitted similarly well with value of $\sigma$ being in the interval from zero to the value specified in Table 1.

• 2. $\alpha$ and ($\alpha - \pi$) (a)symmetry of $\rho _{C}{}$ as a function of $\theta _{\textrm{out}}$

  The parameterization of probability density function $\rho _{C}{}$ given by Eq. (42) (see also Fig. 5) implies that $\rho _{C}{}$ as a function of polarization angle of outgoing photon $\theta _{\textrm{out}}$ has two peaks located at $\alpha$ or $\alpha - \pi$. Moreover, the peaks have the same shape. There is, therefore, the same probability that an outgoing photon has value of polarization angle $\theta _{\textrm{out}}$ closer to either $\alpha$ or $\alpha - \pi$. This symmetry is clearly visible in Fig. 8, too.

  The input data analyzed in Sect. 8 can be, however, fitted equally well with similar parameterizations of $\rho _{C}{}$ normalized to unity and having only one peak or two peaks of different heights and widths. The given input data do not allow distinguishing between these possibilities.

• 3. $\alpha + \pi /2$ and $\alpha - \pi /2$ (a)symmetry of $\rho _{C}{}$ as a function of $\theta _{\textrm{out}}$

  There is one more source of ambiguity. The one or two peaks in the spectrum of values of polarization angle $\theta _{\textrm{out}}$ of outgoing photon may be located at different positions, at $\alpha + \pi /2$ and $\alpha + \pi /2 - \pi = \alpha - \pi /2$. This situation corresponds to the axis of polarizer specified by angle $\alpha + \pi /2$ which is perpendicular to the axis specified by angle $\alpha$. The number of transmitted photons through three ideal identical polarizers (see Sect. 8.1) excludes the possibility that the spectrum could have two, three or four peaks corresponding to both the axes. The peaks must correspond to only one axes. It is, however, not possible to determine on the basis of the input data toward which of the axes the polarization of an outgoing photon is predominantly aligned.

• 4. Dependence of $\rho _{C}{}$ on $\theta _{\textrm{in}}$

  It has been assumed that the parameterization of $\rho _{C}{}$ given by Eq. (42) does not depend on $\theta _{\textrm{in}}$ and depends only on the difference $\theta _{\textrm{out}}- \alpha$. The possibility of more complex dependence of the function $\rho _{C}{}$ characterizing given polarizer on $\theta _{\textrm{in}}$, $\theta _{\textrm{out}}$ and $\alpha$ should be tested.

• 5. Probability function $P_{T,{}}$

  The parameterization of $P_{T,{}}$ given by Eq. (40) depends only on the difference $\theta _{\textrm{in}}- \alpha$. The possibility of more complex dependence of $P_{T,{}}$ on $\theta _{\textrm{in}}$ and $\alpha$ should be tested, too.

Remark 8.3

Some structures (e.g., needle-like crystals or molecules) in a linear polarizer can be aligned predominantly in parallel to one axis of the polarizer. This corresponds to one axis of symmetry. The second axis of symmetry is perpendicular to it. One may ask how the quality of alignment of these microscopical structures is related to the width of the peaks in the spectrum of values of polarization angles $\theta _{\textrm{out}}$ of outgoing photons characterized by the parameter $\sigma$. This is, however, an example of a question which goes beyond the scope of the first stage of analysis of data.

9 Possibilities of unique determination of parameterized probability (density) functions

If a probabilistic model does not allow to describe measured data under given set of assumptions then it means that there is a contradiction which must be removed (it may be necessary to modifying one or more assumptions of the model). The analysis of data performed in Sect. 8 is an example of analysis where input data partially constrain dependences of the parameterized functions, but unique determination of the free parameters (the parameterized functions) is not obtained. This may be in many cases very useful and valuable information. However, there are situations when one would like to determine uniquely the parameterized functions, or at least better constrain them on the basis of experimental data.

Possibilities of obtaining more experimental data are discussed in Sect. 9.1. In some cases, these data can be analyzed with the help of stochastic IM process, this is covered in Sect. 9.2. The other cases are discussed in Sect. 9.3.

9.1 Suitable experimental data

As to more experimental data needed to determine uniquely the parameterized function in Sect. 8, measuring transmission of light through various sequences of polarization-sensitive elements (see Sect. 2 in [9]) can provide essential experiment data which can be further analyzed with the help of the probabilistic approach. This should help to uniquely determine the functions $P_{T,{}}$ and $\rho _{C}{}$ characterizing individual polarization-sensitive elements. Similar measurements are needed for determination of polarization states of photons emitted by a source, i.e., for unique determination of the function $\rho _{S,0}$.

For example, the open questions 2 and 3 could be answered with the help of optical analog of Stern–Gerlach experiment separating photons by their spins (polarization states). One possible experimental method of A. Fresnel is discussed in detail in [50]. (It uses quartz polyprism.)

In the case of optical phenomena not related to polarization of light, one can use very similar approach to obtain required experimental information.

9.2 Probabilistic models based on IM process

In some cases an analysis of measured number of transmitted photons through sequences of various polarization-sensitive elements can be performed with the help of one of the stochastic processes introduced in Sect. 6 with very little or no modification. For example, transmission of photons through a Faraday effect-based device can be analyzed very similarly as transmission of photons through a linear polarizer. Instead of rotation angle of a linear polarizer, one can introduce magnetic field B in the direction of propagation of the beam and the length d of the path where the beam and magnetic field interact in the device (to explain measured number of transmitted photons in dependence on these two parameters, or one of them may be taken as fixed). The functions $P_{T,i}$ and $\rho _{C,i}$ for given index i corresponding to a Faraday effect-based device must be parameterized differently than in the case corresponding to a linear polarizer, see Eqs. (40) and (42), i.e., mainly assumptions 6.1, 6.3 and 8.1 (and also assumptions 6.5 and 6.4, if the beam is not transmitted through sequence of identical Faraday rotators) must be modified. In the case of a Faraday effect-based device, one can try to parameterize the corresponding functions $P_{T,{}}$ and $\rho _{C}{}$ according to definition 7.9 and determine them on the basis of experimental data. The determined position of the peak of $\rho _{C}{}$, mentioned in definition 7.9, as a function of the difference $\theta _{\textrm{in}}$ - $\theta _{\textrm{out}}$ divided by Bd may be then compared to the Verdet constant standardly determined (using a different method without determining the probability (density) functions) to characterize a Faraday effect-based device.

To describe probabilistic character of transmission of light through a linear polarizer only one variable characterizing properties of photon, the photon polarization angle, has been taken into account in Sect. 8. Probabilistic description of transmission of light through other polarization-sensitive elements where photon polarization states denoted as circular and elliptical are involved may require introduction of one or two additional random variables. In this case or in the case of adding some other random or non-random variables (parameters) characterizing photon states and optical elements, the formulae in Sect. 6 can be modified in very straightforward way and then used for analysis of experimental data. The formulae remain essentially the same, only photon polarization angle is replaced by other random variables characterizing states of given system, and rotations of elements are replaced by parameters characterizing the states of the system.

Stochastic IM process can be helpful in description of other optical phenomena not related to polarization at all or related to polarization and some other properties of light, if the two main assumption of IM process (concerning memoryless (Markov) property and independence of outcomes of an experiment) are satisfied. Several types of optical experiments are mentioned in Sect. 2 in [9] where it is discussed that these experiments often include phenomena involving splitting of a photon beam or merging of photon beams (very common in optical systems). In this case, there is in general a net of transitions instead of a sequence of transitions, see Sect. 3.3 in [9]. Coincidence and anti-coincidence experiments common in optics can be also described using IM process. Description of reflection of light on a surface is essentially the same (from the mathematical point of view) as description of transmission of light through an optical element. Experiments with “interference” of light, such as well-known double-slit experiment, can be described using IM process, if the observed pattern does not depend on the intensity of the source of light (i.e., the interactions of individual photons with the slits, or another optical element, are independent).

It may happen that measured numbers of transmitted photons (states of a system) depend not only on some non-random variables, but also on some random variables, i.e., densities of states are essentially measured. In this case, it is necessary to modify some of the formulae in Sect. 6 (with the help of more general formulae in [9]. It concerns mainly the key system of (integral) equations, see remark 5.4.

9.3 Probabilistic models not based on IM process

Not all optical phenomena may be described with the help of IM process based on the two main assumptions (see remark 6.4). Dynamical effects such as saturation of optical media (typical for experiments with stimulated emission) cannot be described under these assumptions. If the intensity of light transmitted through an optical device is too high, then it can completely change optical properties of the device during the transmission. (Transmission of individual photons may not be independent.) In such cases, the theory of stochastic can be also very useful, but it is necessary to formulate suitable stochastic process differing from IM process.

10 Summary and conclusion

Transmission of a photon through an optical element is standardly denoted as probabilistic process. The probability of transition of an input photon state (before an interaction with an element) to an output photon state (after the interaction with the element) can be factorized into three probabilistic effects represented by the probability (density) functions $\rho _{S,i}$, $P_{T,i}$ and $\rho _{C,i}$ with the help of IM process, see Sect. 2. Statistical theoretical descriptions widely used in optics and in physics in general do not allow to easily determine these three probabilistic effects, Sect. 4.

The theory of stochastic processes provides general abstract framework for description of random processes. IM process proposed in [9] and formulated within the framework of the theory of stochastic processes is suitable for description of various processes in physics, see Sect. 5.

It has been shown that formulae suitable for description of transmission of photons through polarization-sensitive elements including, but not limited to, three linear polarizers can be derived with the help of IM process, see Sect. 6. Probability density function $\rho _{S}$ allows to define polarized and unpolarized light. The probability $P_{T,{}}$ and probability density function $\rho _{C}{}$ can be used for definition of various types of optical elements. These new definitions based on probability (density) functions are discussed in Sect. 7. In similar way photon beam and other optical elements not related to polarization of light may be characterized, too.

Stochastic IM processes suitable for description of polarization of light formulated in Sect. 6 can be used for an analysis of measured relative photon numbers passed through $M$ polarization-sensitive elements in dependence on the rotations of the elements. Basic aspects of the measurement are in Sect. 3. It is worth to note that in these cases some properties of the photon beam are measured (the relative number of transmitted photons) and some other statistical characteristics of the individual photons states and their interactions with matter can be determined with the help of the stochastic process. Example data of three-polarizer experiment are analyzed in Sect. 8. This is novel type of analysis showing how to determine the probability (density) functions $\rho _{S,i}$, $P_{T,i}$ and $\rho _{C,i}$ characterizing transmission of individual photon through various optical elements on the basis of experimental data.

Experiments concerning spin-dependent and polarization-dependent phenomena are essential for better understanding connection between spin and polarization of particles. Better understanding of polarization of light at the level of individual photons can stimulate further (already very broad) usage of various polarization-sensitive devices in optical systems. However, the open questions formulated in Sect. 8.7 should be answered before drawing far-reaching conclusions concerning polarization. Suggestions how to address the questions have been discussed in Sect. 9.

The well-known three-polarizer experiment is an example of an experiment that, although not quantitatively described in the literature using the corpuscular theory, can be described with the help of IM process and the corpuscular idea of light. Using IM process (or another stochastic process based on different set of assumptions), it is possible to analyze many experiments and further explore possibilities of the corpuscular theory. With the help of IM stochastic process, it is possible to unify description of various phenomena which may look very differently at first glance, see Sect. 5 in [9]. For example, IM stochastic process has been successfully used in genetics (biophysics, see remark 5.5) and can be used for description of polarization of light (particle–matter interaction) as well as for description of motion of particles having initial conditions characterized by probability density functions, see [51].

Description of experimental data based on stochastic process like IM process has several advantages over other statistical theoretical approaches used in physics, see Sect. 5.3. One of the advantages of analysis of experimental data using IM process is that it can be divided into two stages. In the first stage, one can try to determine overall probability (density) functions characterizing transitions of states of a system on the basis of measured quantities without the aim to explain in more detail the probabilities. It is not necessary to know the Hamiltonian of the system, introduce complex amplitudes, know all microscopic processes, etc. In the second stage, one may try find more detailed internal mechanism (causes) of the transitions of the system leading to the determined probabilities. The analysis of data performed in Sect. 8 corresponds to the first stage.

In some fields of research, the theory of stochastic processes and the separation of an analysis of data into the two stages have been essential to make further progress. The field of optics (studying propagation of light and interaction of light with matter) is one of the fields where many very diverse theoretical approaches for description of optical phenomena have been studied for centuries. However, the potential of applicability of the theory of stochastic processes has not yet been fully explored in this field. In many analyses of experimental data not even the first stage has been achieved. This is the case of analysis of data concerning polarization of light. It has been shown in the presented paper how the theory of stochastic processes and the corpuscular idea of light can open up new possibilities to make further progress in understanding various phenomena in the field of optics. One can leverage terminology, techniques and results already known in the theory of stochastic processes and successfully used in many areas of research.