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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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