A thin absorbing layer at the center of a Fabry-Pérot interferometer

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A thin absorbing layer at the center of a Fabry-Pérot interferometer

P. Mächtle, C. Müller, C. Helm

To cite this version:

P. Mächtle, C. Müller, C. Helm. A thin absorbing layer at the center of a Fabry-Pérot interferometer.

Journal de Physique II, EDP Sciences, 1994, 4 (3), pp.481-500. �10.1051/jp2:1994139�. �jpa-00247974�

Classification Phj,.vie.,v Ah,vtiac.t.v

33.20K 42.60D 82.65D

A thin absorbing layer at the center of

_a

Fabry-Pdrot

interferometer

P. Michtle, C. MUller and C. A. Helm

(*)

Institut for Physikalische Chemie, Johannes Gutenberg-Universitht, Jakob-Welder Weg II. D-

55099 Mainz,

Germany

(Rec.eiied 7 September ^f993, ac.cepted _in

final

foim ^{9 Decemher} 1993)

Abstract. The influence of _a dye (Rhodamin B) dissolved in solution _or adsorbed at _an interface

at the _center of

a Fabry-Pdrot interferometer _on the transmission _was investigated ^both experimentally and theoretically. We show that (ii _~pectra of extremely thin films _are measurable at sub-monolayer concentration lone monolayer reduces the transmission

by

₌ 50 %), (iij the

transmis~ion is sensitive _to the location of _an extremely thin film within _nm range and (iii) the

absorption

coefficient is determined

quantitatively

by

comparison

between theory and e~periment thus permitting _one to estimate the local concentration. The results _are relevant for the _use of dye probes to monitor changes ^of organic interfaces in the Surface Forces Apparatu~ _as well _as generally to detect changes ^{of interfaces between} adjacent ^surfaces.

1, Introduction,

The Surface Forces

Apparatus

has become _an established method for

accurately measuring

the total interaction force between surfaces immersed in solution

[1, 2]. Very often,

these forces

are due to osmotic pressure

gradients.

Such _an osmotic pressure

gradient

_occurs, if the

density

and/or

composition

at the

solid/liquid

interface deviates from the bulk

liquid.

Osmotic pressure

gradients

contribute to very different forces, such _as the electrostatic, the solvation and

structural and the steric and

hydration

^forces

[3].

Until _now, there _{was no} direct way to monitor the interface while the interaction between the surfaces is measured as a function of their

separation.

This is

especially

inconvenient, if the interfaces

change

due to molecular rearrangements

[4, 5].

To eliminate this latter drawback, we

developed

a more direct method to determine solute concentrations. We _measure the

absorption

of

dye

molecules between the

interacting

surfaces and thus determine the

optical density.

This is _a convenient

method,

since

optically

the Surface

(*) To whom

co«espondence

should be sent.

Forces

Apparatus

^is a

Fabry-Pdrot

interferometer which _was

originally designed

to determine

the distances between the surfaces. The measurement of enhanced and

position

sensitive

absorption requires

_a

quantitative knowledge

^of local fields and the relation between

optical density

and local concentration. In this

work,

_we calculated and measured the effect of _an absorber in the center of such _a

cavity.

There, the

spatially oscillating intensity

_waves exhibit either _crests

(even-order fringes)

_or

troughs (odd-order fringes).

Therefore _an

extremely

thin film

(down

to 0, I fb of _a

wavelength)

attenuates and shifts

only

even

fringes

without

affecting

odd

fringes. Actually,

_a thin

dye layer

(about two

monolayers) trapped

between the surfaces exhibits _an extinction coel'ficient which exceeds the _one in the bulk

by

about _two orders of

magnitude indicating

_a

substantially

increased

dye

concentration at the interface. If the film thickness is increased _to

extremely large

distances

(~Lm)

we observe the Rhodamin B spectrum

as measured in

solution,

_{yet we} still find _an increased

dye

concentration _at the interface.

The paper is

organized

_as follows. The _next section describes the

optics

of _a

Fabry-Pdrot

interferometer with _a thin

absorbing layer

at the center. Since this

technique

has _not been described before, our account is

fairly

detailed. Section 3

gives

the

experimental procedures.

In section 4 _we show _our

optical

results and in section 5 we discuss them in _terms of the

density

distribution of the

dye

at the interface.

2,

Theory.

First, _we shall describe _a

simple Fabry-Pdrot

interferometer with _one

homogeneous layer

between the mirrors

(following [6]),

then _a

symmetric three-layer

interferometer _as it is used in the Surface Forces

Apparatus.

Next _we consider _a

one-layer

interferometer with _a loss, and

eventually

we discuss the _most

interesting

_case, an

absorbing layer

in the Surface Forces

Apparatus.

For the whole section, _{we assume} that _no

dispersion

_occurs, I-e- that all reflection and transmission coefficients _{are constant} for the _resonance

wavelength

of _a

given

order _even when

the _resonance conditions _are affected

by changes

of the _center

layer.

The

Fabry-Pdrot

interferometer, named after its inventors, _can be considered _as the

simplest

type ^of

optical

resonator.

Normally,

such _an instrument consists of two

parallel

dielectric

mirrors.

To

begin with,

_we consider _a

simpler

structure, that consists of _a

plane-parallel plate

of thickness D and refractive index _n immersed in _a medium of _a different index. The reflection and transmission coefficients R and T are,

according

_to standard

optics [6], given by

(I e~~'~),j

R

= ~

(l)

I p e~ '~

~

~-,~

T

= ~

(2)

p e~ '~

where _we used the

symmetry

^{of the} system

(I.e. r(

=

r,)

as well _as the definitions

(cf. Fig.

I, without the center

layer)

where _p and _{T are,}

respectively,

the fraction of the

intensity

reflected and transmitted at each interface and will be referred to in the

following

discussion _as the mirror's reflectance and transmittance,

following

convention. If the incident

intensity (watts

per unit

area)

^{is taken} as

unity,

we obtain the

following expression

for the fraction of the incident

intensity

that is

reflected _:

j~j2 ~

~ _P SIn~ _~b

~~

(l

_{p )~} + 4 _p

sin~

~b

Moreover,

from

equation (2)

~~

^~

(l _p

~

+

~sin~

~b

~~~

for the transmitted fraction. This basic model contains _no loss

mechanism,

so conservation of energy

requires (R

_{(~ +}

(T(~

₌ l, _as is indeed the _case.

Let _us consider the transmission characteristics of the interferometer.

According

to

equation (5)

the transmission is

unity

whenever

By using

A

= c/v, condition

(6)

for maximum transmission _can be written _as

where _c is the

velocity

of

light

in _{a vacuum} and _v is the

optical frequency.

In the literature, the

sharp

transmission

peaks

_are also known _as

Fringes

of

Equal

Chromatic Order

(FECO).

Anyhow,

for _a fixed thickness

D, equation (7)

defines the

unity

transmission

(resonance) frequencies

of the

Fabry-Pdrot

interferometer. On the other

hand,

the minimum transmission

approaches

zero as p

approaches unity.

In the

preceding paragraphs,

_we used _a

plane-parallel plate

^of dielectric medium _as the

simplest

type ^of

Fabry-Pdrot

etalon. In the _case of the Surface Forces

Apparatus

the transparent

thin film of thickness D is sandwiched between _two

(mica)

sheets each of thickness

D~,

and refractive index n~,. The _outer surfaces of the sheets _are silvered and

provide high-

reflectance mirrors (cf.

Fig, I).

Now,

the _« mirror's _»

reflectivity

coefficient

,,'p

is _a function of r, and i~, the Fresnel

reflectivity

coefficients of the silvered

mica/support

and the mica/medium interfaces,

respectively

-?>~~j -2,~~j

'~ ~ _~~e~

~'~~'

~ '

~

~~~

mica

dye

mica

silver silver

~ D~~, n~~ D, D~~,

n~~

J~

n~-iK

ri -ri r~ -r~ -r~ r~ ri

ti t'i t2 t'2 t'2 t2 t'l tl

Fig. I. An interferometer made up of three layers of thickness D~~, D. DGI and refractive index

n~j, n n, ix, n~j with the outer faces ~ilvered. The reflection and tran,mission coefficients _as u~ed in

the text _are indicated. A ray of light incident _on the left face emerges from the right after multiple reflection~.

where _we used the definitions

2<~Gl

~~~

2 grn~j

~/

= r, "2 e

~~

~jj

_~ ^{, ~$~j =} _~

^~~~

^'

Obviously, ,,~

_is

_complex.

_We

may define

,,~

=

,~

_{e'Y and thus}

p may be written _as

P =

(P( ^e~'Y

The « mirror's

» transmittance

(r( depends additionally

_{on t,} and t~, ^{the Fresnel}

transmission coefficients of the

support/silvered

mica and the mica/medium interfaces,

respectively

t, t~ e

'~~'

t( ti

e

'~~'

(t, t( )(tl

t~)

~, ~

~

l r, r~

e~~'~~'~

I r, r~

e~~'~~' N~

^{~ ~~ ~~~~}

Note, that

sandwiching

the thin film of interest between _two sheets did _not introduce any loss

mechanism,

I-e- _we still have

[p

+

jr

= I.

The reflected fraction of the incident

light

is

~j~

~

41pi ^sin2

(~b

y) (i lpi)2+41pi sin2

_{i~b yi} ~~~~

and the transmitted fraction is

~ ii

jp

_)~

'~" j12)

~

(i

lpi

_{)~ +}

4jp sin~

_{(~b Y)}

To find the _resonance condition of this

Fabry-Pdrot,

_we have to consider the additional

pha~e

shift _y due _to the insertion of the

(mica)

sheets into the

Fabry-Pdrot

interferometer. The transmittance is

unity~

whenever _{~b y}

equals

_{zero or} some multitudes of _gr, I-e-

tan ~b

~~~'' _~

Re,~ (13)

This _resonance condition is

unambiguous.

This is _not

trivial,

since both _{p s} absolute value,

p

[,

and its

phase,

2 _y, do

depend

on the

wavelength.

Yet, _on

considering

_p

[,

one finds

Only [N

^~

depends

_on the

wavelength.

On

evaluating equation (12), [N

_[~ cancels. Since _we have for all

wavelengths

_p

^N~

~ 0,

equation (13)

is the

unambiguous

_resonance condition.

It may be rewritten _as

tan ~b =

"~ '

~l

~'~

(2 lbGi)

'2('

+ '

~)

(i +

I'()

_COS

_{(2 ~b~,)} ⁽¹⁵⁾

Assuming

furthermore the reflectance _at the silvered

mica/support

interface to be

unity ii-e-

r, = I, this condition

implies [p

= I,

too),

one finds

(I 11)

_~'~ ~~

~Gl)

~~~ ~

(16)

21'2

(1 ₊

1'()

COS

(2

_~bGl~

the standard

equation

to calculate the distance between the

(mica)

surfaces in the Surface Forces

Apparatus [2].

Note,

that _we

approximated

ideal reflectance of the silvered

mica/support

interface

by

the

approximation

_tj

- 0. Thus in the limit of t, = 0 _{we see} from

equations (2, 12)

that the

transmission remains finite

only

if

[p

= I, _as is indeed the _case.

In _an

experiment,

a

slightly

different form of

equation (16)

is used to minimize _errors

arising

from _some little known

quantities,

such _as

dispersion

and

phase

shifts due _to the silver. We focus _our attention _on the _m th order _resonance

wavelength,

A~,. ^{If the sheets} are in contact

(I,e.

D

=

0) _we know the _resonance

wavelength

from

equation (6)

'~l~ ^~

~ ~ ~Gl

~Gl (17)

On

separating

the sheets

by

a distance D, the _m th order

fringe

is shifted

by AA,,,

to a

longer wavelength

A. Therefore, ~b~, may be rewritten _as

~bGi =

j'n"

_'

j ₍₁₈₎

Using equation (16)

and _some

trigonometric

relations like

cos(gr-6)=-cos6~

cos

(2

_gr 6

= cos 6~ sin (gr 6)

=

sin 6 and sin (? _gr 6

=

sin 6 _{we can} write

equation (16)

in the _more convenient form

(1 _1-j sin (m _gr AA~,/A

tan ~b =

~

(19)

± ^? r~ + + i-i cos (mar AA~,/A

where the ₊

~ign

refers to m odd, and the

sign

to m even.

Finally,

for small gaps between the sheets, _{one may}

approximate

sin (mgr AA ~,/A ) _{= mar} AA

~,/A,

cos (mgr

AA,,,/A

)

=

and tan ~b = 2 _grn DIA. Thus, one obtains

D

= m

AA,,,/2

_n~,, m odd

(20)

and

Dn~

= m AA

~,

n~,/2,

_m even

(21)

The shift of the _even

fringes

is

strongly

influenced

by

the medium between the sheets, whereas the odd

fringes

do not _« see » the medium. For _an intuitive

understanding

of

why

the odd and _even

fringes

shift

by

different amounts we follow

[2]

and refer to the

analogous

case of

an elastic string ^{stretched between} two walls. The normal modes of transverse vibrations _are

considered. The first

(fundamental)

mode has _a

trough

at the center of the

cavity

_; the second

mode has a crest at the center the third mode has

again

a

trough

at the center, and _{so on.} At the

center, the

standing

waves have either _{a crest} (the even

fringes)

_or a

trough (the

odd

fringes) (cf. Fig.

2a). At the center, where the medium is situated, the electrical field of the odd

fringes

is _zero.

Consequently,

the medium and the odd _waves do _not interact with each other. For the

even

fringes,

the

opposite

situation _occurs, the electrical field is _{at a} maximum

just

at the

position

of the medium and

thus,

the interaction is

extremely

strong.

If _we allow for existence of losses in the medium of _a

simple Fabry-Pdrot

interferometer, _we find that the

peak

transmission is les~ than

unity.

The refractive index of the medium become~

complex,

I,e, _n

= n, ix. _x is known _a~ the extinction coefficient.

During

the passage of

light through

the medium not

only

a

phase

^~hift occurs, but also

absorption.

Then

equation (2)

becomes

~

~-'~ ~~'kIJ

T

~ ~ ~ ~

(22)

1-pe~-"

e~-'

with the abbreviations

2 grn~ 2 _~r~

~b =

D and _a

=

(23)

(For

visible

light,

I.e. A

=

550 _{nm we} have _K

> 100

a).

The maximum transmission

drops

from

unity

to

T ^~

=

~~ ~~~

~j~ (24)

(1

_{P e~} ~

Yet, the condition for maximum transmission _as stated in

equation (6)

remains valid.

However,

the situation gets more

complicated

and _more

informative,

if _we introduce _an

absorbing

medium between the

(mica)

sheets in the SFA. Then

equation (2)

reads

~

~-,~

_~- aD

~

l

[p ^{e~'~Y~~~ e~~~~}

^~~~~

and for the

reflectivity

coefficient r~ ^at the mica/medium interface and the

corresponding

transmission coefficients t~ ^and

t(

_we find

n~, n~ + I _K

~ 2 n~, 2(n~ I _K

r~ = =

(r~

^e'

,

t~ = and

t(

=

(26)

n~, + n~ i K n~, + n~ i K ncj + n~ i K

However,

the definition of the _« mirror's _» transmission coefficients _as

given

in

equation (12)

still holds.

Thus,

the transmitted fraction of the incident

light

is

~

~j2 ^~-2aD

~~~

ii (p e~~~~)~

₊

4(p ^{e~~~~ sin~}

_(~b

y)

~~~~

with the abbreviations

[p

= (

[r~

_[~ +

r( _{21-j [r~(}

_cos

₍₂

_{~b~j + 6}

_{)) (28a)}

(N~(

and

(N~

_{= r~} ^~

r(

_{+ 2 rj r~ cos}

(2

_~b~, 6

(28b)

It is very inconvenient _to find that

(N (~(l

_p

^e~~ "~)

_does

_depend

on the

wavelength

A.

Therefore, _no

simple

_way exists _to find _{a resonance} condition similar to

equation (13).

Since the

equations

do _not have obvious solutions, _we will describe first the results from calculations _on the base of the matrix formulation

[6] together

with our intuitive

picture.

These

calculations _were

performed assuming

a

layered

system as shown in

figure

I. Let

I~ be the incident

intensity

and

lo

the transmitted

intensity.

Inside the

Fabry-Pdrot

interferome- ter, we may

split

the electric field _as the sum of _a

right-traveling

wave and _a

left-traveling

wave. Let

Ii

_,

and

I~_,

be the intensities of these _waves,

respectively.

These intensities _are related

by

lo

=

(T(

² _I~

(29)

lo

= t,,

t(.I,_,

=

(I r().1,_~ (30)

'<-1

"

~~'ii

<

(31)

According

to these

equations,

the

intracavity

intensities _are

given by

1,

_, ⁼

~~~

I~

(32)

rj

r([T[2

1,_j

= /~

(33)

r(

rj is assumed to be close _to

unity.

If maximum transmission _occurs, the

intracavity intensity

in such _a

Fabry-Pdrot

interferometer is much

larger

than the incident

intensity (cf. Fig. 2).

Now, let _us

again

consider _an

extremely

thin

layer

of

absorbing

material between the

(mica)

sheets and its effect _on both the _resonance

wavelength

and the transmittance. It _was shown

before,

that the odd

fringes

do _not interact with _a thin

layer

at the _center of the

cavity,

since there, their electrical field is _zero and their

intensity

_waves have

troughs (cf. Fig. 2). Thus,

_we may expect the shift of the

wavelength

AA to be

independent

of the index of refraction of the thin film and

equation (20)

to be valid. Also, _{as can} be _seen in

figure

2a

(or

rather, cannot be

seen, the _waves with and without _a thin film _are

indistinguishable

within the resolution of the

figure),

the

intracavity intensity

is the _same. Therefore the

intensity

of the odd

fringes

_as

observed after

leaving

the

Fabry-Pdrot

interferometer is

expected

to be the _same.

To understand the behaviour of the _even

fringes,

let _us first consider the electrical field

waves within the

cavity.

Here, a thin

absorbing

film is

positioned exactly

at an extremum of

the electrical field

(in Fig.

2b the thin film is

positioned

at _a minimum of the electrical field.

The

intensity depends

on the square of the electrical field, therefore both minima and maxima of the electric field _wave

correspond

to maxima of the

intensity wave).

A

right-traveling

wave

will

undergo

_a

phase

shift

entering

the film

(due

_to the

complex

transmission coefficient t~. ^{Note that the}

imaginary _parts

of

t(

and t~ are

equal).

^{In the}

absorbing film,

the electrical field

will decrease

exponentially,

then, on

leaving

the film the _wave

experiences

the _reverse

phase

shift. Due _to the

absorption

at the crest, the _wave lost its symmetry, ^{the maximum of the} electric field is

slightly

shifted to the left where the _wave comes from.

Similar,

a wave,

moving

in the

opposite

direction

(from right

to

left) experiences absorption

at the center crest and the

opposite

_symmetry break. The electrical field in _a normal

Fabry-Pdrot

at a

given wavelength

is the sum of the electrical fields

generated by

the _waves

traveling

in both directions. It is

symmetric. However,

in _case of very strong ^absorbers

(large

K), the symmetry ^{break of the} center crest may be

quite pronounced.

^{This leads} to a

broadening

of the center node, in extreme cases the

« crest _» even exhibits two extrema

(minimia

in the _case of

Fig. 2b).

The

broadening

of the crest in _one

period

of the _{wave can} be _{seen as an} effective decrease of the

optical path,

therefore the _resonance

wavelength

is decreased. Also, as is demonstrated in

figure

2a, the

intracavity intensity

is much lower than without _an absorber.

To find _some

approximate

formulas for the _resonance

condition,

the

following approach

_was

chosen: to allow _an

unambiguous

_power series

expansion

to the denominator of the

transmittance,

equation (27j,

the

complex

numbers have to be broken into

explicit

real and

imaginary

parts ^{instead of absolute values and their}

phase.

The power series

approximation

was calculated for AA~ and D, up to terms of order AA ^~ and D. Then, to find the maximum of the transmission, the derivative with respect to AA _was put to zero furthermore ri was taken to be

unity.

As

expected,

we find the _resonance condition for the odd

fringes unchanged (cf. Eq. (20))

D

= m AA

~/2

_{n~, m} odd

(34)

However, for _even

fringes

we obtain

(n)

K ~) D ₌ m AA

~,

n~j/2

,

m even

(35)

even, D

= 0

20 odd, D=)D=2nm

even,D=2nm

15

j

^[

:_ ,I "; ;' [ _{."_} / '.

-

I _I

I

_{I I}

_/

f

l _I ; ^I I

- I II j I i ;I

2~ 10 _I; il ;÷ ^{1. .~}

i

_{~) l} II _).

Ii

I

I

it

/; I( II

^~

ii

5

jl );

l Ii ; Ii

ij _j Ii _j II

0

~'~

«300 «200 «100 0 100 200 300

Distance from Center [nm]

a)

E, 1~ _r Thickness: 2 nm

E~r~l n=1.33-5i

Z E

_

1.5

$

_1.0

il

0.5 ~"'

[. ,,

~

o-o

,

.."'

' "., _,'

;""

^',

.._

.(

«0.5

"ji--I)

I

-1.0 LU

-1.5

-300 -200 -100 0 100 200 300

Distance from Center

[nm]

b)

Fig. 2. a) The ~patial

inten~ity

di~tribution at the center of the cavity without and with _an ab;orber at

the _{center, as} obtained from calculations using the matrix formalism. Without _an ab,orber, at the _center

of _a simple cavity the

intensity

of the odd iringe, _{i; zero,} while the inten;ity of the _even fringe, _i, at a

maximum, about fifteen times the incident inten,ity i,. With and without _a 2 _nm thick ab;orber at the

center of the cavity, the odd

fringes

coincide within the re,olution oi the figure. However, _on in,ertion of

the dye, ^the

intensity

of the _even fringe is

profoundly

attenuated, al;o _a

splitting

oi the _{cre~t al} the position of the dye _can be detected. b) _was designed, to explain the ,plitting of the _cre,t, Here, the electrical field of the _even

fringe

aiter the in~ertion of the dye i~ shown. The electrical field (3E is the _sum of the fields of the _wave traveling ^from left

- right and of the _one

traveling

from right

-

left. On

pa~sing

the ab,orber, both _wave~ experience a ~ymmetry ^break a~ well _{a~ a}

broadening

oi the

crest. In the matrix calculation~, the following constants were u~ed the ~ilver layer _wi 40 _nm thick with

an index of refraction of 0.05 2.87 I, the mica _was ?.754515 _~m thick with iicj 1.58 and the dye- layer wa~ 2 _nm thick with _{n, ix} 1.~3 5 _1, Without _an absorber, the wavelength of [he odd fringe

was 550.018 _nm, the _one of )he _even

fringe

586.02 _nm. With the dye, the _re~onance wa~elength~ ~.ere

~hifted by ^0,19~ nm and 1.967 _nm, re~pectively.

As mentioned above, for

large

_K, the _resonance

wavelength

decreases with

increasing

film thickness. This is due _to the symmetry ^{break the} wave suffers when

traveling through

_a thin film. If n, exceeds _K, the _resonance

wavelength

^{increases with}

increasing

film thickness. In the limit of _K

- 0 _we have the _same

equation

_as for _a

non-absorbing

thin film,

equation (21).

Now, let _us consider the maximum transmittance, I,e, the

intensity

losses due to the thin film. It _was calculated

by replacing

the shift of the _resonance

wavelength by

^{the distance} D

(I.e. Eqs.(34, 35))

in the

equation

for the transmittance

equation (27) (the

latter broken into

explicit

real and

imaginary parts).

We find for the odd

fringes

T(D)[

_)~~

=

l D ^{~ "} ~r' K

(I

1",

~~

_~G'

(I

₊

"1) ⁽³⁶⁾

The transmittance is almost

unity,

since both

D/A°

and r, are very small. However, for

even

fringes

we have

~

12

~r n~. K (I + rj ) ²

T(D

_j

= I ₊ D

(37)

~~~~ A^~ ~Gl I 1',

Now, the small D/A ^° is _to be divided

by

the

similarly

small _r,. Therefore, the

intensity

starts to decrease _even if D amounts to 0.I fb of A. The effect is _more

pronounced

the _more the

reflectivity

coefficient of the silvered

mica/support

interface

approaches unity.

The strong ^{influence of} the

reflectivity

coefficient of the silvered

mica/support

interface will be discussed

briefly,

first for _a

simple Fabry-Pdrot

interferometer

consisting only

^of mica

without any film in the center. The

reflectivity

of the mirrors does not influence the

transmittance at resonance. Also, the _resonance conditions

quoted

in

equation (7)

are

independent

of _r,, and _{so are} the _resonance

frequencies,

which are

separated by

^{the free}

spectral

_range, Av cl (2 _n~, D). However, the

reflectivity

is

important

for transmittance at

frequencies

which

slightly

deviate from _resonance, I.e, the width of the transmission

peaks

is finite. This effect is

usually quantified by considering

the Full Width of these

peaks

at their

Half-Maximum values

(FWHM). According

to

equation (5),

half-maximum transmission

(I.e., (T(

^~ ₌

0.5)

_occurs at

~

(l

r))~

sin- ~b~, =

(38)

4

r(

For _most

high

resolution

Fabry-Pdrot

interferometers,

r(

is very close _to I. Thus, the solutions of

equation (38)

are

given approximately

as

1_,-(

~b~, = mm =

139)

~,.,

The FWHM in terms of the

frequency dv,/~

is thus

given by

I

r(

d_v

~~ ⁼ A_v

=

A_v IF

(40)

grr,

The finesse F is the ratio of the

separation

between

peaks

to the width of _a transmission

peak.

Here, it is

given

as a function of the reflectance of the silvered

mica/support

interface. If _a thin

absorbing

film is _now situated _at the _crest of _{a resonance wave,} the symmetry ^{breaks shown in}

figure

2 lead to shifts in the

wavelength,

_away from the almost

symmetric right

and left

traveling

waves in _a usual

cavity.

The

sharper

the transmission

peaks,

the _more decreased is the

intensity

of the thus shifted _wave.

If _{one wants to} compare the

absorption

_spectra of _a very thin

layer

to those from the bulk

medium,

it is convenient to

give

the

absorption

in _a form similar _to the Lambert-Beer law.

Usually

the

absorptance

A is

given,

which is

proportional

_to the

logarithm

of the transmitted

intensity

I divided

by

the incident

intensity

_I~

A _cc 2 grKD/A

=

Injo (I/I~).

Cte.

(41)

Here, _we find with

equation (37)

T(0

) _{)~~~ n~,}(I rj

2 grKD/A

=

C

~

l with C

=

(42)

T(D

~~~~

n~(

^I + r,

)

Let _us

briefly

mention the _case of

large separations,

I,e,

large

^{~ "~} D. For a very

rough

estimate of the transmittance, we may put _{p ~} l and obtain T(D

)[

² _{= r [2} e~ ^{~ ~~}

,

D

large (43)

At

large separations

of the mica

sheets,

the transmittance

decays roughly exponentially

with distance.

To conclude the

theory

section, _we would like _to

emphasize

that _we discussed

only

the

principles

of _an

homogeneous

thin

absorbing

film at the _center of _a

Fabry-Pdrot

interferometer.

We also

neglected dispersion,

which is _no

problem

as

long

as the shifts of the

wavelength

are

very small. If _we intend _to simulate _an

experiment

which involves

large

distances and thus

large wavelength shifts,

_we would have _to take into account both the

dispersion

of the thin film

at the center of the

cavity (that

is

r~)

_as well _as the

pronounced dispersion

of the silver

(that

is

r,).

However, in this _{case we} expect an

inhomogeneous

concentration

profile

at the interface,

and the whole

three-layer picture

_appears to be

grossly inappropriate. Then,

one would

presumably

use a matrix formalism

involving

_{as many}

layers

_{as necessary, or may} be _one

layer

with _a

continuously varying

index of refraction.

Anyway,

in this _{case one} would have _to do numerical calculations. And the aim of this section _{was to}

give

_a

physical understanding

for enhanced

absorption.

3. Materials and Methods,

MATERIALS. -Rhodamin B _was from Merck,

Darmstadt, Germany

and used _as received.

The water was filtered in _a

Milli-Q

unit. The

dye

concentration _was in all

experiments

0.01 M.

SURFACE _FoRcEs APPARAI'US. In these

experiments

the

Fabry-Pdrot

interferometer consists

of _two mica sheets of

equal

thickness, which _are silvered _on their backside (about

5001

silver).

The sheets _were

glued

on crossed

cylindrical

disks

(radius

~ 2 cm), a geometry ^which

corresponds

to a

sphere

on a flat. The sheets _were

quite large

and covered the

glass

^disks

completely.

Then, the disks _were mounted in the apparatus (Mark IV, Anutech,

Australia).

One of the

mountings

is movable, the other is

rigid.

The former is

supported by

_a stiff

spring.

At the

beginning

of each

experiment,

the mica surfaces _were

brought

into contact in

dry

air and their

optical properties

were studied. Then, the surfaces _were moved far apart ^and a

droplet

of Rhodamin B solution (~ 60 _{~Ll) was}

injected

with _a

syringe

between the disks. Next, _a water

dish _was

placed

at the bottom of the apparatus, ^{and the}

atmosphere

was allowed to

equilibrate

for about _two hours.

OPTICS. The

experimental

_set-up is shown in

figure 3,

the

principle

of the

optics

in

figure

4.

Briefly, light

from _a 100W

halogen lamp (MUller Instruments, Grasbrunn, Germany)

is focused _on the

Fabry-Pdrot

interferometer. Behind the interferometer _a

microscope objective (magnification lo,

effective focal

length

13.9

mm)

focuses the transmitted

light

_on the

entrance slit of _a

spectrograph (HR

3205

f/4.2,

Instruments S. A. Riber-Jobin-Yvon, Gras-

brunn, Germany),

which is >500 _{mm away} from the

interferometer,

thus

leading

to a

magnification

factor of >50. Behind the

spectrograph

a camera

(Proxitronic, HLS,

Bensheim,

Germany)

and _a video-recorder allow

recording

and observation of the

fringe

pattern. ^The

spectrograph

is also

equipped

with _an

eye-piece,

^which contains _a small moveable scale. With the

eye-piece

shifts of the

wavelength

down to 0.02 _{nm can} be resolved in intense

fringes.

ID M2 S

Prism Spektrograph

CL Video

Microscope

System

Objective

^LG

Crossed Mica spektrograph

Cylinders

HM

OSMA

~

Lamp

Fig.

3. The experimental set-up. HM _: heat reflecting filter (hot mirror), Ml mi«or, ID iris

diaphragm,

M2 _: moveable mirror, S slit, CL

collimating

lens system, ^LG

light guide.

The

intensity

of the

fringes

_was determined with _an OSMA

(Optical

Simultaneous Multichannel

Analyzer)

_system. A movable mirror is used _to deflect the

light

at the

position

indicated in

figure

3. A lens system ^{collimates the beam} on a

light guide (OFC200,

Spectroscopy Instruments, Gilching, Germany)

with _an actual

light transmitting

_area of

~0.3mm2. Then, the

light

travels

through

a monochromator

(HR320, Spectroscopy

Instruments)

to a Multichannel

Analyzer (IRY-700G, Spectroscopy Instruments).

The

spectral

resolution is determined

by

^the

grating (3001/mm

and

12001/mm),

~160 _nm and

m40nm _can be observed

simultaneously

with _a resolution of

0.23nm/pixel

and

0.057

nm/pixel, respectively.

The _two

gratings

have different reflectivities. The

sensitivity

of the diodes of the Multichannel

Analyzer

varies

by

_up to 10 fb.

Therefore,

whenever intensities

of different spectra were

compared, only

measurements

performed

at identical

grating

positions

were used. Between Is and I min _are necessary ^to measure a spectrum ^with

satisfactory

statistics.

Since mica is

birefringent,

and the orientation of the sheets

against

each other is random, sometimes

peak

doublets _were observed. In this _{case, a}

polarizer

_was used.

To select the center spot ⁱⁿ the interference pattern (cf.

Fig. 4),

_an iris

diaphragm

is situated in the

path

of

light

before the mirror. For

spectral calibration,

_{a mercury pen ray}

lamp

_was

employed.

lo _Xi

e

m

~

QJ

b

g

O

~ u

tll QJ

B

I

QJ

j~

^~

~ tll

~

X

Real

Space Wavelength ~

lo xi

Wavelength ~

Fig.

4. Cartoon of the

Alignment.

Top leit it the cros~ed cylinder~ _are irradiated by monochromatic light. Newton'~ rings _are ob~erved, Top right _: the pattern as observed in the ~pectrograph, if the central

area of Newton's rings is focused _on the _entrance flit. Bottom left when the surfaces _are irradiated by white light, the FECO fringe~ _are observed. Their shape reflects the deformation~ of the surfaces. Bottom right _: at the smallest wavelength ^of a given fringe the suriaces _are close~t together. ^This area i~ ~elected

by

an iri~ diaphragm, ^and there, the intensity of the

iringe

is measured.

CALCULATIONS. To simulate the

optical

_set-up, numerical calculations

relying

_on the

matrix formali~m _were used

[6].

This _was executed with _a Mathematica (Version

2.04)

program on a Macintosh llvx. The sequence of

layers

was as follows silver, mica, medium, mica, silver. Mathematica _was also used _to find the

approximate

formulas for the

wavelength

shift and the intensities

(I,e. Eqs.(34-37))

in _cases where the

absorbing layer

is very thin.

Furthermore, all

analytical

formulas _were cross-checked with the matrix formalism for _some

typical

numerical

examples.

4. Results.

This section is

organized

_as follows first _we will describe how _we

distinguish

odd and _even

fringes

in air, and how _a very thin film of

dye trapped

between the mica sheets affects the transmission of the

Fabry-Pdrot

interferometer. Next, to

quantify

the effect, _we have _to know the

properties

of the

Fabry-Pdrot

interferometer such _a~ its

spectral

resolution, its finesse and

the transmission of the

contacting

mica sheets without

dye.

Then _{we can} determine the

thickness of the thin

dye layer

and its

complex

^{index of} refraction. If the distance between the mica surfaces is increased, ^the spectrum ^resembles more and _more that of the bulk solution.

Yet, _even if the surfaces _{are ~Lm} apart, we still _see the concentration increase of the

dye

^close

to the mica surfaces.

Odd and _even

fringes

_can

already

be

distinguished

with mica in air. This is due to the

large

adhesion of clean mica surfaces in

dry

air

[3].

If the surfaces _are

brought

into _contact~ the

contact zone flattens because of the

large

adhesion. This is shown

by

^the

shape

of the

interference

fringes,

which have _a flat _center and _{are no}

longer

rounded _as sketched in

figure

4.

The diameter of the contact zone is 10-50 _~Lm~

depending

_on the mechanical

properties

of the whole system

[7].

The

large

adhesion induces the surfaces to

bulge

forward the bifurcation _at the

edges

of the _{contact zone} is _very

pronounced.

This bifurcation is detected

differently by

even and odd

fringes.

At the

edges

of the contact _zone, the distances between the surfaces _are

still small and

equations (20, 21) apply.

^{The shift of} the odd

fringes

is

proportional

to

n~, = 1.58, whereas the shift of the _even

fringes

is

proportional

to

n~/ii~j

₌ 0.63

(assuming

n~, =

1.58 and _n

= n~,, ). Therefore~ above and below the flat _zone, the odd

fringes

_appear

to have

sharp edges,

^{while the} even

fringes

seem to be _more rounded.

Figure

5~ top, shows the spectrum ^{of the} resonance

wavelengths

of

contacting

mica surfaces

in air _as measured with the OSMA and the 3001/mm

grating. Sharp

_resonance

peaks

are

observed, ^whose

positions

_are determined

by

^{the thickness of the mica sheets}

(Eq. (17)

_). The maximum

intensity

follows _an

envelope

which is determined

by

the

lamp

spectrum, ^{but also} influenced

by

the

spectral

^{resolution of the}

grating

and the detector. The width of the

peaks

is

resolution-limited.

However, if _one intends to

quantify

the

absorption

coefficient of _a thin

layer,

it is essential _to determine the fines~e and thus the

reflectivity

of the silvered mica/substrate interface. To

achieve this end, the 12001/mm

grating

_was used, but still the

peaks

_were

only slightly

broader than the resolution. Therefore, _we used mica with about half the thickness of the _one whose transmission spectrum ^{is shown in}

figure

^5.

Keeping

the finesse _{more or} less constant

(I,e, using alsways

the _same amount of silver to achieve the _{same i-j}) and

dividing

the thickness

by

a factor of _two increases both the free

spectral

_range and the distance resolution

by

that

factor of two. (Of _{course~ any} measured

absorption

spectrum ^of a very thin

layer

has _now fewer

points

that is the

pay-offj.

One transmi~sion

peak

of _a

Fabry-Pdrot

interferometer

consisting

of very thin mica sheets is shown in

figure

6. To obtain

quantitative

information _{one must}

la)

_assess and subtract the

background

from the total

intensity

to obtain the

signal intensity,

and

(b)

^{model the}

signal intensity by

a convolution of _an assumed, intrinsic line

shape

^with the

experimental

resolution.

The latter is determined

empirically

with the 1?001/mm

grating

and the

yellow

^{lines of} a

mercury pen ray

lamp.

The theoretical

shape

^of a transmission

peak

^is

given

in

equation (5).

Thi~

shape

_may be well described as a Lorentzian, after converting the

wavelengths

to wave numbers. The convolution

450 500 550 600 650

Wavelength [nm]

Fig. 5. The transmission pattern as observed with the OSMA,

using

the 3001/mm

grating.

The

diffraction peaks ^measured at two different

angles

of the

grating

_are

superimposed,

they overlap ^between 540 _nm and 550 _nm. The _mica is rather thick (D~j ₌ ^7.52 ~m ), therefore the distance between the peaks (I,e. the free

spectral

range) ^is

quite

small. Top _: the mica surfaces in air _are

brought

into _contact.

Bottom _a droplet of 0.01 M Rhodamin B solution is placed between the mica surfaces, then the surfaces

are

brought

into contact

again.

It is

impossible

to _remove the last

remaining layer

of Rhodamin B, even

though large compressive _{pressures are} applied. The thickness of the remaining layer is 1.2 ₌ 0.6 _nm.

Only

the electric field of the _even

fringes

does interact with the Rhodamin B, therefore

pronounced absorption

of every second fringe is observed.

559.32nm 557.67nm

8000

~i

6000

2000

17700 17800 17900 18000 18100

Wave Number

[cm

_~]

Fig. 6. The transmission peak of _{an even} fringe (m 32) measured with mica _in air. Shown is _a

spectrum as observed with the OSMA,

using

the 2001/mm

grating.

The _two maxima _are due to the pronounced birefringence ^{of the mica.} The straight ^line gives the resolution _as determined from _{a mercury} pen ray

lamp.

The mica is _on the thin side

(D~j

₌ 2.83 _~m, A

(

= 575.95 _nm and (j = 577.71 nm),

therefore both the distance between the peaks and their width _are quite large. ^{The finesse of this} Fabry- Pdrot interferometer _was 20.5 ₌ 0.5. In this

experiment,

_we had _a slightly thicker than usual silver layer,

leading

to

quite

low transmission.