2. THE THEORY OF ANOMALOUS DISPERSION

In 1875 HELMHOLTZ presented a theory which gave a relation between the absorptions bands and the anomalous dispersion of the refractive index. His theory was based on a mechanical analogy. For the free "ether¹¹ he assumed that the equation of motion could be written

a2t. a2~

p ^, = a

at 2 ay2

p is the ¹¹density¹¹ of "ether¹¹ 3 and a is an elasticity constant per volume unit of the ether, ^~ is the ^dis~

placement along the y axis of a parcel of ether from its mean position. If the motion has the character of a free]

progressive harmonic wave along the y axis, it may be ^de~

scribed by

~

=

^i;o e

i ²⁷~( y-et)

A

1;0 is then the amplitude of the wave, A is the

wave-length and c the phase velocity. Incerting eq. 41 in 1, we obtain the relation

c2

=

^a/p

When molecules are present, HELMHOLTZ thought that there are a Hook type of elastic force between each atom of the molecule and the ether, that is of the form

where S is the Hook constant, and x is the displace-ment of the atoms from their mean position. The equation of motion for the ether then becomeq

(40)

(41)

(42)

-44

-= (43)

If there are j different types of atoms present, the equation should read

=

⁽⁴⁴⁾

For the j-th type atbms with mass r.

J the equation of motion is

a

2 x. ax.

r. J ~at

2

^=-f.x. J J ^{+ B·(~-x.)} J J ^{- g .} J ^~tJ o (45)

The first term on the right side of the equation is the intermolecular Hook force which results from the dis-placement of the atom within the molecule. The molecule as a whole is supposed to be at rest. The second term is the reactio of the actio in eq. 44, and the third term describes the dissipation of energy, which is supposed to be proportional with the velocity of the oscillating atoms.

A light wave passing through this medium must then have a t~rm which accounts for the dissipation or absorption effect. Instead of eq.

41

we should write

i ²9-TI ( y-vt ) ~ ~ ^y

~o e

and x should be of the form

X.

=

J

i 2

9-TI (y-vt )- ~ y x. e

JO

1 and v are the wavelength and velocity of the wave in this medium~ and a/2 is the absorption coefficient of the amplitude. (Since the energy is proportional with

(46)

(47)

-45

-the squax·e of the amplitude, the absorption coefficient of the energy is then a).

The exponents of the eqs. 46-47 may be written

· 27T((l • aR-) - v·t).

l T +l1f7fY

By introducing the refractive index of the medium relative to ether

n

=

^c _v ^:::

and the absorption index

K ::: at.

1f7f

the exponent becomes

i ~7T ⁽⁽ⁿ + iK)y- ct).

The complex refractive index m may in this context be defined as

m

=

n + iK

(The phase in eq. 46 may just as well be expressed by vt - y instead of y - vt. m will then be defined as

n - iK.

.

It should also be noted that both n and K are indices relative to the refractive index of the medium in which t. is the wavelength). The eqs. 46-47 now become

. 27T ( )

1

T

my - ct

X.::: X· e J JO

Incerting these functions in eqs. 44-45, we obtain

(48)

(49)

(50)

(51)

(52)

(53)

-

46-~rj(2>.'!!_ '--I ^{~-1 2} x. = -fjxjo + ej<~o-xjo) + ig. T ^21T c X·

JO J JO

The last equation may be solved for _{·X. '}

JO and substituting this into eq. _53~ the result becomes after some ordering

The first term on the right side of eq. 55 is 1, according to eq. 42.

The second term is a constant,

multiplied with >.2 . It will make m2 decrease mono-tonously as ^>. increases~ and consequently does not de-scribe the phenomenon of anomalous dispersion. Probably the term is very small or is compensated for by the third term, otherwise m2

would soon aquire very unreasonable values outside the absorption bands, due to the ^>.2 in-fluence. We shall then omit this term here.

The third term gives rise to the anomalous be-haviour of m2 . If we assume that the influence of each j-contribution is restricted to a very small wavelength interval, so that the factor >.2 before the E sign may be regarded as a constant for each j-terms we may as a first approximation write

A. J

=

It will also be convenient to write

A. 2 2 2

= 41T c r./(13 .+f.)

J J J J

and

G.

=

21TC g./(f3.+f.)

J J J J

(54)

(55)

(56)

(57)

(58)

Eq. 55 then becomes

2 2 .

A. -;\. ~ 1.A.G.

J J

This equation may be separated into its real and imaginary parts:

2 2

n - K = l + 2:

2nK

=

These are the equations presented by JENKINS and WHITE (1957, p. 476) as resulting from the theory of HELMHOLTZ.

An interesting point is that a later model by LORENTZ (1915), based on electromagnetic theory, gives a formula of the type

= 2 ?

I. -A."'~ iA.G.

J J

(Lor..:;;:\!'l'Z gives the formula explicit for a non~-absorbing

mix::·cre (pae;e 311, note 58)). If the relative variation o+~ .11²~--1 is much greater than the relative variation of

.-.

c: . • t:h t .

m ~.~ ~ v a J..s) m is close to 1 , tl 1en m2 +2 ::: 3 and eqs. 59 and 62 become of identical form.

(59)

(60)

(61)

(62)

The indices n and ^K may be found from eqs. 60-61, but t.;he resulting expressions are complicated. To simplify

tJ-;.:- '1iscussion we shall just look at the behaviour around

on~ of the absorption bands. We will further ass~me that the ^D i.~her bands contrib-ute to a rather constant refractive index N outside the actual bandl and that their absorption

- 48

-is pl'a·.;\.;ically zero at these wavelengths. The eqs. 60-61 then become

(63)

2nK

=

⁽⁶⁴⁾

From eq. 64 it is seen that if n is fairly constant~ K

will have its greatest values around A1• Wb shall now again make use of the earlier assumption that the influence of the absorption is restricted to a small interval, so that A may be regarded as constant compared with (A-A

1), and write

A2 - A 2

1 = (A - A

1)(A + A

2

^{)~(A - A}1)2X1 (65)

A2 ^~ A 2 (66)

1

A3 ^~ _{), 1} 3 (67)

Further we shall assume that the absorption index K is small compared with the refractive index n, but that its relative variation is greater, that is

K << n ^~ N (68)

I~ ~~I

^>>

I~ ~I

⁽⁶⁹⁾

vJith these assumptions eqs. 63-64 may be written

(70)

2NK ~ (71)

-

49-We mus· ::;:i.mplify eq. 70 further by observing that n 2-N2

=

^(n-N)(n+N)

=

(n-N)(2N)

If we write B = A/2N eqs. 70~71 become

n z N +

K ~ ~

These expressions are more suitable for discussions than eqs. 60-61.

We see from eq. 75 that K will have its peak value Kl for A

=

^A.l:

Kl

= ---a

^A.1B

Solving for B and substituting the expression for B in Rq. 75 gives

K :

4(11.-:\1)2 + G2

It should be noted that K is a symmetric function around J.

1.

There are two wavelengths 11. 2 where K has the value K1/2. From

=

it is obtained that these wavelengths are

G 2

(72)

(73)

( 711)

(75)

(76)

(77)

(78)

(79)

-50

-Thus G is the width of the absorption band where K

has half of its peak value.

With the substitution of B from eq. 76, eq. 74 becomes

n

=

It is seen that n - N is antisymmetric around A = A1. For A

=

_A1, n - N

=

0, or n ~ N.

By differentiating the last equation with regard to A~ one obtains

dn

=

dA

This expression is zero for

±

2

G

that is for A

=

^A₂^• At these wavelengths eq. 74 gives that

K

n

=

^N ± 21

The approximate model described by eqs. 74-75 thus gives that n will have a minimum value at

cross its mean value N at A1, and obtain its maximum

' G

at /\1 + 2' The difference between the maximum and minimum value of the refractive index equals the peak value of the absorption curve.

An interesting point is that if we have just one absorption band, and we are far from it, at the long wave -length side~ then

(80)

(81)

(82)

(83)

-

51-\ >> .Al

and eq. 60 may be written

which is identical with the SELLMEIER equation presented in Chapter

7.

(84) (85)

(86)

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Fig.l. The .refractive index of a two-component system, based on diffe-rent equations, as a func-tion of the partial water volume.

Fig.2. The mean refrac-tive index of different plankton species, based on Table 2, as a function of the partial water vo-lume.

-59-X

w a z

n

1.40

(.J

<(

~ u..

w 1.37

~

1.36

n

0.7

PARTIAL

v.. 0.9

VOLUME

1.4 5 ~--+---+----1---+----1

.5

.6 .7 .8 .9 1.0

Fig.3. Dispersion of the main algal constituents, relative to water.

Fig.4. Absorption in-dex K and refractive index n of an ideali-zed homogeneous alga, due to chlorophyll a

Fig.5.

Absorp-tion and scattering efficiencies for

dif-ferent diameters, when the chlorophyll

a content is 10 mg cm- 3 .

Fig.6. Absorption and scattering effi-ciencies for

diffe-

-61-0.0 0 1 L...-...i-....J~__._..._._-L.._,_--<--J

600 700 nm

100 J.lffi

rent diameters, when 0.11--+---1-+-t--, the chlorophyll a

content is 1 mg

c~- 3.

600

10

0. 1

0.0 1

100 J.lm

- - X ' = X'(~) , n =nO)

· · ... · K = 'r<i( t.) , n = ^N

- "':' - -X= 0 ' n = N

1J.1m

600 700 nm

10 J.lffi 100 jJffi

---

^{1 - -} lJJm

-

---600 700 nm

COMMENTS AND CORRECTIONS TO REPORT N0.46:

THE REFRACTIVE INDEX OF PHYTOPLANKTON

Chapter 4.1, page 22

The waterish type of opal, with p=2.07 g em ^-3 and n=l.43, was chosen for the calculations since the opal of the phytoplankton contains water. However, the relative mass concen-trations of ^s~lica given in Table 2 (page 18) probably represents dry Sio

2 without any water content. More correct values for the calculations are then perhaps p=2.65 and n=l.486.

This change leads to some corrections.

lines in Table 8 (page 29) should now be

Diatoms .1. 538. 1. 4 3 8. ^II 1.536 Mean from Table 2 1.555 1.438 ^II 1. 342

~ine no.9 on page 30 should read:

n

^~ (1.55 ^± 0.01) - (0.21 ^± O.Ol)vw and line no.l5:

Two ot the

1.281 ^II 1.201 ^II

density becomes (1.10 ± 0.05)g em ^.:..3 If only species with

Fig.2 consists of six curves for n as a function of c .

w

The lowest of these curves, for .diatoms, should now be omitted, since the new curve for diatoms will coincide with the one

for blue-green algae.

In document The refractive index of phytoplankton (sider 44-63)