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arXiv:1202.1103v1 [physics.optics] 6 Feb 2012

About the Bidimensional Beer-Lambert Law

B. Lacaze

Tesa 14/16 Port St-Etienne 31000 Toulouse France

e-mail address: bernard.lacaze@tesa.prd.fr

February 7, 2012

Abstract

In acoustics, ultrasonics and in electromagnetic wave propagation, the

crossed medium can be often modelled by a linear invariant filter (LIF)

which acts on a wide-sense stationary process. Its complex gain follows

the Beer-Lambert law i.e is in the form exp[−αz] where z is the thickness

of the medium and α depends on the frequency and on the medium prop-

erties. This paper addresses a generalization for electromagnetic waves

when the beam polarization has to be taken into account. In this case,

we have to study the evolution of both components of the electric field

(assumed orthogonal to the trajectory). We assume that each component

at z is a linear function of both components at 0. New results are obtained

modelling each piece of medium by four LIF. They lead to a great choice

of possibilities in the medium modelling. Particular cases can be deduced

from works of R. C. Jones on deterministic monochromatic light.

keywords: linear filtering, polarization, Beer-Lambert law, random

processes.

1 Introduction

1.1The Beer-Lambert law

The Beer-Lambert law (B.L law) states that some positive quantity A(0) at the

input of some medium varies following the equation

A(z) = A(0)e−αz

(1)

where z is the covered distance and α is a parameter defined by the medium

[6]. The equality (1) comes from the approximation

A(z + dz) − A(z) ≈ −αA(z)dz

which postulates that the evolution of A(z) on a small thickness dz of the

medium is proportional to A(z) and to dz, with a coefficient α > 0 which is

defined by the medium. Then the differential equation A′(z) = −αA(z) which

leads to (1). A more general method starts from the equality

A(0)A(z + z′) = A(z)A(z′) (2)

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whatever z,z′. Equivalently the quotient A(z + z′)/A(z) depends only on z′,

and any piece of the medium of length z has the same behavior. If we assume

that A(z) is a continuous function on R+, the only solution of (2) is (1) for

some α ∈ C.

If we take A(0) = eiω0t, (1) becomes (R[α] and I [α] stand for the real and

imaginary parts of α)

A(z) = exp

?

iω0

?

t −

z

ω0I [α]

?

− zR[α]

?

. (3)

This means that the monochromatic wave eiω0tis delayed by

ened by exp[−zR[α]] when crossing a thickness z in the medium.

Now we place ourselves from a signal processing perspective. We assume that

a piece of medium of any thickness z is equivalent to a LIF (Linear Invariant

Filter) Fzwith complex gain Fz(ω) and that any piece of thickness z+z′has the

behavior of two filters in series Fzand Fz′ [15], [9]. This means that whatever

the frequency ω/2π

Fz+z′ (ω) = Fz(ω)Fz′ (ω).

z

ω0I [α] and weak-

(4)

Obviously we take F0 = 1. What preceeds implies that for each ω it exists a

complex α(ω) such that

Fz(ω) = e−zα(ω)

By definition Fz(ω)eiωtis the output of the filter Fz when the input is eiωt.

For such an input the power Pzat the output is

Pz= e−2zR[α(ω)]

(5)

(5) summarizes the Beer-Lambert law for wave propagation through a contin-

uous medium, used in acoustics, ultrasonics and electromagnetics. α(ω) gives

the attenuation of the wave (by its real part) and the celerity of the wave (by

its imaginary part). The Kramers-Kronig relation links the real and imaginary

parts of Fz(ω) which constitute a pair of Hilbert transforms [17], [11]. (5) is

matched to monochromatic waves. More generally when the stationary process

Z0= {Z0(t),t ∈ R} is the input of Fz, the output Zz= {Zz(t),t ∈ R} verifies

(E[..] stands for mathematical expectation or ensemble mean)

Pz= E

?

|Zz(t)|2?

=

?∞

−∞

e−2zR[α(ω)]s0(ω)dω

where s0(ω) is the power spectral density of Z0(s0(ω) = δ (ω − ω0) for a unit

power monochromatic wave at ω0). The fact that measurements are generally

performed by non-monochromatic waves lead to gaps with the Beer-Lambert

law in the form (6) [2]. Also, B.L law can be untrue when powers are too

high (which act on the medium properties) [1] or when the beam expands [10].

However, B.L law has many applications in physics, chemistry and medecine

[5], [8], [4], [13].

1.2A counter-example

The Beer-Lambert law applies for light amplitude or power after crossing contin-

uous permanent media which can be viewed as a set of filters in series. However

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light or radar wave is not only defined by an amplitude [3], [14]. Electric field

Ez=?Ez

Ez

x= {Ez

It is assumed that the beam propagates in the neighbourood of the axis Oz and

that the field is orthogonal to this axis. The beam is polarized in the direction

θ at z when

?

for some (real or complex) process A ={A(t),t ∈ R}. The beam is unpolarized

at z when the cross-spectrum sz

xy(ω) of components verifies

x,Ez

y

?is a two-dimensional vector with respect to axes Ox and Oy at

x(t),t ∈ R},Ez

distance z of origin O and with components

y=?Ez

y(t),t ∈ R?.

Ez

Ez

x(t) = A(t)cosθ

y(t) = A(t)sinθ

sz

xy(ω) = 0

whatever the basis Oxy. This implies the equality of power spectra. The wave

is partially polarized in other cases.

The power Pzat z for a stationary wave is defined by

Pz= E

?

|Ez

x(t)|2?

+ E

???Ez

y(t)??2?

. (6)

We know that media act upon polarization and then can influence measure-

ments, for instance in the case of antennas (generally matched to a particular

polarization state). B.L law is not available for behavior of a given component

of the electric field except particular cases. For instance, take the wave

E0

x(t) = eiω0t, E0

y(t) = 0

which propagates in a medium which rotates the beam by angle proportional to

thickness z. We have (c is the celerity in the medium)

Ez

x(t) = eiω0(t−z/c)cos[zα(ω0)]

If we measure Ez

x(t) the medium is equivalent to a filter of complex gain

Fz(ω) = e−iωz/c(ω)cos[zα(ω)]

when a particular direction is chosen (Fz(ω) = e−iωz/csin[zα(ω)] for the or-

thogonal direction and c can depend on ω). The term α(ω) takes into account

a possible dependency of the rotation angle with the frequency. For an antenna

which selects the component in a given direction at distance z, the B.L law is

not verified.

1.3The two-dimensional case

The B.L law is established for one-component monochromatic waves (i.e for

waves defined by only one quantity depending on the time t and on the space

coordinate z). They are time-functions in the form aeiω0t(a ∈ C depends on z).

Quasi-monochromatic waves can be defined as random processes in the form

Ez(t) = Az(t)eiω0t

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where Az= {Az(t),t ∈ R} is stationary with a baseband spectrum which can-

cels outside (−ω1,ω1) with ω1/ω0≪ 1. The spectral band of Ezis included in

the interval (ω0− ω1,ω0+ ω1).

In the two-components case (i.e for wavesdefined by two quantities Ez

depending on the time t and on the space coordinate z) a quasi-monochromatic

beam is defined by

x(t),Ez

y(t)

Ez

x(t) = eiω0tAz

x(t), Ez

y(t) = eiω0tAz

y(t) (7)

where Az

inside (−ω1,ω1). These properties remain whatever the chosen coordinates axes.

It is reasonable to say that a wave is (purely) monochromatic when Az

a relation in the form (for some constant k and real Bz(t))

x,Az

yare stationary with stationary correlation and power spectrum

x,Az

yverify

Az

x(t) = k cosBz(t), Az

y(t) = k sinBz(t)

Bz(t) represents the orientation and keiω0tthe complex amplitude of the field.

This means that its (complex) amplitude is purely monochromatic in the usual

sense. When Bzis degenerate (Bz(t) does not depend on t), the wave is po-

larized (at z). In other cases, the wave is partially polarized. When Ez

have same power spectrum and are uncorrelated, the wave is unpolarized at z.

Both properties are true in any orthogonal system. In optics, beams are often

quasi-monochromatic and partially polarized but it is not always true, partic-

ulary in astronomy and communications. Polarized and unpolarized beams are

convenient idealizations.

The concept of quasi-monochromatic wave is often bad-fitted even in the

light domain. For instance Wolf-Rayet stars and B.L Lacertae have lines of

relative width larger than few percents. Idem for LEDs (light-emitting diodes)

with width larger than 10%. A more general model has to be taken in these

cases.

We have seen that the B.L law for one-component beams can be proved

using a decomposition of media by LIF in series. The aim of this paper is to

generalize the B.L law in the two-components case, using signal theory and

modelling media as more general circuits. Inputs and outputs of these circuits

are stationary processes which represent the components of the field. In the

following section we consider that each component of the field at a distance z is

the sum of two LIF outputs. The field components at the origin point (z = 0)

are the inputs of these LIF. This model allows to determine the shape of the

LIF characteristics, i.e the B.L law for bi-dimensional beams.

Because two inputs and two outputs, 2x2 matrices of filters complex gains

will be defined. In the years 1940, R. C. Jones had developed a ”New Calculus

for the Treatment of Optical Systems” [7]. It was based on 2x2 matrices which

act in the time domain on purely monochromatic waves. Though formally re-

sults of Jones are very close to formulas of this paper, they do not address the

same objects. Actually Jones papers do not mention Beer neither Lambert or

Bouguer, the pioneer of this topic.

xand Ez

y

2Two-dimensional Beer-Lambert law

We deal with a beam which propagates in direction Oz of the orthogonal tri-

hedron Oxyz. The electric field Ezat time t is defined by its components

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Ez

the medium between u and u+z is defined by a set of 2x2 “scattering matrice”

Hzindependent of u

?

where the Hz

jk(ω) depend on the frequency ω/2π and are complex gains of LIF

Hz

Eu+z

x

(t) = Hz

Eu+z

y

(t) = Hz

Equivalently the electric field Ez+uat z + u is linearly dependent on its value

Euat u [12]. The linearity is expressed by the set of the LIF Hz

only on the medium. It is an obvious generalization of what was explained for

the one-component waves propagation. Filters parameters depend only on the

thickness of the considered medium. However results can be obtained without

this hypothesis. Figure 1 shows equivalent circuits of (8).

Filters in series lead to multiplication of complex gains and filters in parallel

to addition. The figure 2 summarizes the following equality

x(t),Ez

y(t) on axes Ox and Oy at distance z of the origin O. We assume that

Hz=

Hz

Hz

11

Hz

Hz

12

21

22

?

jksuch that

?

11[Eu

21[Eu

x](t) + Hz

x](t) + Hz

12

?Eu

y

?(t)

22

?Eu

y

?(t)

(8)

jkwhich depend

Hz+u= HzHu. (9)

(9) is equivalent to

Hz+u

11

Hz+u

12

Hz+u

21

Hz+u

22

= Hz

= Hz

= Hz

= Hz

11Hu

11Hu

21Hu

21Hu

11+ Hz

12+ Hz

11+ Hz

12+ Hz

12Hu

12Hu

22Hu

22Hu

21

22

21

22

(10)

We assume that the derivatives h0

(10) can be written as

jk=

∂

∂zH0

jkare finite. The first equation of

Hz+u

11

− Hz

u

11

= Hz

11

Hu

11− 1

u

+ Hz

12

Hu

u

21

Obviously we have H0

(similar operation is done in the other equations)

11= H0

22= 1 and H0

12= H0

21= 0. When u → 0, we obtain

hz

hz

hz

hz

11= Hz

12= Hz

21= Hz

22= Hz

11h0

11h0

21h0

21h0

11+ Hz

12+ Hz

11+ Hz

12+ Hz

12h0

12h0

22h0

22h0

21

22

21

22.

(11)

The differential system can be split in two subsystems (equ.1+2 and equ.3+4).

We assume that

lim

z→∞Hz

jk= 0(12)

because any wave is evanescent in a passive medium. Two cases can be high-

lighted following the (complex) eigenvalues λ1,λ2of the matrix

?

h0

h0

11

h0

h0

21

1222

?

.

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Solutions are in the form following that λ1?= λ2or λ1= λ2= λ

Hz

or Hz

jk= cjk1eλ1z+ cjk2eλ2z

jk= (cjk1z + cjk2)eλz.

The eigenvalues cannot cancel (or corresponding coefficients cancel). because

(12). Taking into account the initial conditions

H0

11= H0

22= 1 and H0

12= H0

21= 0(13)

leads to two cases

Case 1: λ1?= λ2

By identification with (11) we obtain

Hz

Hz

Hz

Hz

α =

11= αeλ1z+ (1 − α)eλ2z

12=

λ2−λ1

?eλ1z− eλ2z?

λ2−λ1

?eλ1z− eλ2z?

λ2−h0

11

λ2−λ1

−h0

12

21=

22= (1 − α)eλ1z+ αeλ2z

−h0

21

(14)

where the λj,h0

jkcan depend on frequency ω/2π but are independent of z and

λ1=1

λ2=1

∆ =?h0

2

?h0

11+ h0

?h0

22

22+√ρeiθ/2?

?2+ 4h0

2

11+ h0

22−√ρeiθ/2?

11− h0

12h0

21= ρeiθ.

(15)

The eigenvalues have real parts strictly negative (to fulfill the condition (12)).

Case 2: λ1= λ2

The solutions are given by the equalities

Hz

Hz

a =

11= (az + 1)eλz,

22= (−az + 1)eλz,

h0

22

2

Hz

12= h0

Hz

11+h0

22

2

12eλz

21eλz

21= h0

.

11−h0

,λ =

h0

(16)

The case h0

equalities are verified in any system of coordinates. Components evolve inde-

pendently, with same attenuation and celerity. This corresponds to a medium

with all possible properties of symmetry.

In all cases, the real part of eigenvalues different from 0 have to be negative

for passive media which weaken waves. Moreover, the solutions are only matched

to equations (11),(12),(13).

11= h0

22?= 0,h0

12= h0

21= 0 leads to the usual B.L law. These

3 Examples

In examples, we assume that the parameters h0

supports of inputs E0

y.

jk(ω) are constant on spectral

x,E0

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3.1 Example 1

Let assume that

h0

11?= h0

22and h0

12= h0

21= 0.

We are in the case 1 with

Hz

11= eh0

11z,Hz

22= eh0

22z,Hz

12= Hz

21= 0.

This means that a beam eiωtpolarized on Ox is transmitted with weakening

exp?R?h0

first component disappears before the second component, independently of the

values of I?h0

when

Ez

11

?z?

and delay I?h0

11

?z/ω. If polarized along Oy, the weakening is

exp?R?h0

22

?z?

and the delay is I?h0

?and I?h0

22

?z/ω. When R?h0

11

?/R?h0

22

?

≪ 1, the

11

22

22values fitted to a dichroic material. More generally

?which define the refraction indices of the medium.

Then we can give to h0

11,h0

x(t) = eh0

11zE0

x(t), Ez

y(t) = eh0

22zE0

y(t)

both components evolve independently and the usual B.L law is verified for

each component. The power Pz at z becomes (we have assumed that the h0

are constant with respect to frequency)

jk

Pz= e2zR[h0

11]σ2

x+ e2zR[h0

22]σ2

y

where σ2

R?h0

x=E

?= R?h0

???E0

x(t)??2?

,σ2

y=E

???E0

y(t)??2?

. We retrieve the usual result when

11

22

?. In other cases usual B.L law is no longer valid because the

two terms in Pzhave different behaviors.

3.2 Example 2

Let assume that we are in the case 1 (two distinct eigenvalues λ1,λ2different

from 0) and that we deal with a monochromatic wave polarized along Ox:

E0

x(t) = eiω0t, E0

y(t) = 0.

By (8) we have

Ez

x= Hz

11

?E0

x

?, Ez

y= Hz

21

?E0

x

?.

E0is transformed in Ezdefined by

Ez

x(t) =

Ez

y(t) =

?λ2−h0

λ2−λ1

11

λ2−λ1eλ1z+h0

−h0

21

?eλ1z− eλ2z?eiω0t

11−λ1

λ2−λ1eλ2z?

eiω0t

where parameters can be complex. At z, both components are weakened and

delayed through two terms functions of z (eλ1zand eλ2z) and not one as in the

usual B.L law.

1) When h0

21= 0 we have

Ez

x(t) = eh0

11z+iω0t,Ez

y(t) = 0.

Ezis polarized along Ox with R?h0

11

?and I?h0

11

?as parameters of weakening

and of delay. Pz= e2zR[h0

11]has the shape (5) of the usual B.L law.

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2) Except when h0

Assume that h0

(14)

21= 0, the monochromatic wave Ezis no longer polarized.

21?= 0,h0

12= 0 (which implies h0

11?= 0,h0

22?= 0,h0

11?= h0

22). From

?

Ez

Ez

x(t) = eh0

y(t) =

11z+iω0t

h0

21

h0

11−h0

22

?

eh0

11z− eh0

22z?

eiω0t.

Even when R?h0

polarization, and untrue for other polarizations.

11

?= R?h0

22

?it is impossible to have Pzlike (5). Consequently

we understand using symmetries that usual BL law can be true for a particular

3.3Example 3

1) The case

h0

12= −h0

21, h0

11= h0

22?= 0 (17)

is particularly interesting. We have

λ1= h0

11+ ih0

12, λ2= h0

11− ih0

12

and (14) becomes

?

Hz

12= −Hz

11= Hz

22=1

21= −i

2

?eλ1z+ eλ2z?

Hz

2

?eλ1z− eλ2z?.

Now we assume that the electric field E0is monochromatic and polarized at

angle φ with respect to Ox

?

E0

E0

x(t) = eiω0tcosφ

y(t) = eiω0tsinφ.

From (8) and after elementary algebra

?

Ez

Ez

x(t) = eiω0t+h0

y(t) = eiω0t+h0

11zcos?φ − zh0

12

?

11zsin?φ − zh0

12

?.

(18)

For real h0

a monochromatic wave at the frequency ω0/2π is rotated by the angle −zh0

and attenuated by exp?zR?h0

is equal to (c is the light velocity in vacuum)

12, the electric field Ezat z is polarized at the angle?φ − zh0

11

??. Moreover −z

12

?. Then

12

ω0I?h0

11

?

is the time spent by

the wave between O and z which shows that the medium refraction index n(ω0)

n(ω0) = −c

ω0I?h0

11

?.

The rotation angle is independent of the polarization angle φ. Because Pz =

e2zR[h0

components.

2) When h0

12is not real (18) is developed in

11]usual B.L law is obeyed for the amplitude and the power, but not for

Ez

Ez

x(t) = eiω0t+h0

y(t) = eiω0t+h0

a = φ − zR?h0

11z(cosacoshb + isinasinhb)

11z(sinacoshb − icosasinhb)

12

?, b = zI?h0

8

12

?.

(19)

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Consequently Ezhas two components polarized in orthogonal directions of an-

gles a and?a +π

???Bz

time a second component appears which is orthogonal to the first component.

The ratio of ”heights” increases with z and tends towards 1. The power Pzat

z is given taking into account othogonality of components

2

?with respect to Ox and of ”height” Bz

??? increases from 0 to 1 when z increases from 0 to ∞. The main component

1and Bz

2such that

Bz

Bz

1= eR[h0

2= −ieR[h0

???Bz

11]zcosh?zI?h0

??? = tanh??zI?h0

12

??

12

11]zsinh?zI?h0

??

2

Bz

1

12

???.

(20)

2

Bz

is rotated by −zR?h0

1

12

?and attenuated by exp?zR?h0

11

??coshb. In the same

Pz= e2zR[h0

11]cosh?2zI?h0

12

??

which shows that the usual B.L law is not valid (except in the case of real h0

which leads to Pz= e2zR[h0

follow the usual B.L law. For instance, for the power of the component Ez

12

11]). The power of each component Ez

x,Ez

ydoes not

x

Pz

x= e2zR[h0

11]?cos2?φ − zR?h0

12

??+ sinh2?zI?h0

12

???.

The same remark is true for components in the directions a and?a +π

2

?with

respect to Ox.

3.4Example 4

We assume that E0is a quasi-monochromatic beam (see section 1)

E0

x(t) = eiω0tA0

x(t), E0

y(t) = eiω0tA0

y(t).

The h0

ditions (17) are fulfilled, the components E0

orthogonal parts. With respect to Ox′y′defined by

jk(ω) are constant with respect of ω on the beam spectrum. If the con-

x,E0

yare split by the medium in two

(Ox,Ox′) = (Oy,Oy′) = −zR?h0

12

?

the beam at z verifies using (19) and (20)

?

Ez

Ez

x′ (t) = eR[h0

y′ (t) = eR[h0

11]z?A0

x(t)cosh?zI?h0

12

??+ A0

y(t)sinh?zI?h0

12

???

11]z?A0

y(t)cosh?zI?h0

12

??− A0

x(t)sinh?zI?h0

12

???.

We deduce the power Pzdefined by (6)

?

Pz= e2R[h0

11]z?P0cosh?2zI?h0

12

??+ θsinh?2zI?h0

12

???

θ = 2I?E?A0

x(t)A0∗

y(t)??.

Whatever the polarization state of the beam at z = 0, the usual B.L law is

verified if and only if h0

12is real, i.e when the effect of the medium is a rotation

(added to a weakening). Whatever h0

polarized (E[..] is real) or unpolarized (E0

12, we have θ = 0 for instance when E0is

xand E0

yare uncorrelated).

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3.5 Example 5

We assume that E0is a quasi-monochromatic beam like (7) and

h0

11= h0

22, h0

12= h0

21

i.e transfers between the coordinates are symmetric. From (8) we have

?

Hz

Hz

11= Hz

12= Hz

22= eh0

21= eh0

11zcoshh0

11zsinhh0

12z

12z.

However we remark that these relations are not maintained in other reference

systems. For instance, in Ox′y′with (Ox,Ox′) = φ we have (the k0

new parameters)

Elementary algebra leads to

ijare the

k0

k0

k0

11= h0

22= h0

12= k0

11+ h0

11− h0

21= h0

12sin2φ

12sin2φ

12cos2φ.

?

Pz= e2R[h0

11]z?P0cosh?2zR?h0

12

??+ θ′sinh?2zR?h0

12

???

θ′= 2R?E?A0

x(t)A0∗

y(t)??

which proves that the usual B.L law is not true apart from particular cases.

4Conclusion

The Beer-Lambert law (actually due to P. Bouguer around 1729) was firstly

used to measure the concentration of solutions . It addresses the problem of

concentration measurement of some kind of molecules in a liquid. In equation

1, we have

α = k (ω)a (21)

where a is the concentration and k (ω) is a wavelength-dependent absorptivity

coefficient. k is deduced from a measurement of the attenuation for a known

value of a. The property of linearity with respect to the distance is due to Lam-

bert and the linearity with respect to the concentration of absorbing species in

the material was highlighted by Beer. The BL law intervenes in wave propaga-

tion to explain together the attenuation and the dispersion whatever the crossed

medium.

A version of the Beer-Lambert law addresses the power as a function of the

medium thickness, whatever the nature of the wave, acoustic or electromagnetic.

The result has the form A(z) = A(0)e−αzwhere α is a function of the medium

and of the frequency ω/2π. Very often α is a power function of ω. Its estima-

tion has numerous applications in medecine [13], [16]. Also, chemistry uses the

measurements of α because it is a linear function of the number of particles

imbedded in the medium. The B.L law can be wrong for instance when multi-

paths in fibres or in case of too strong transmitted powers [1], [2]. Moreover the

medium and the devices can be sensitive to polarization state. For instance the

B.L law can be untrue in the case of birefringence for the medium and when

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devices select only one component of the field. However a generalization is pos-

sible studying separately both components of the electric field which defines an

electromagnetic beam.

The B.L law is easily proved modelling a medium thickness as a linear in-

variant filter (LIF) where input and output show the evolution of the quantity

of interest (for instance an amplitude or a power). In this paper we study the

evolution of two quantities, the components of the electric field of an electromag-

netic beam. To take into account interactions between components, a piece of

medium is modelled by four LIF. We assume that the crossings of two successive

medium pieces are independent events. This hypothesis suffices to determine

the shape of the LIF complex gains. They are defined by four parameters h0

h0

22which depend on the medium and which may depend on the fre-

quency. This model generalizes the Jones matrices used in the deterministic

monochromatic beam to stationary random beams. As soon explained, Jones

papers do not contain comments about the BL law. Examples of section 3 show

that the set of parameters can be fitted to realistic situations.

11,

12,h0

21,h0

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