Multiple beam interference of light in an absorbing wedge

Vani.V.C

Department of Instrumentation

Indian Institute of Science

Bangalore 560 012

INDIA

e-mail [vani@isu.iisc.ernet.in](mailto:Vani@isu.iisc.ernet.in),
<chat@bgl.vsnl.net.in>

# Abstract

Multiple beam interference of light in a wedge is considered when the
wedge is filled with an absorbing medium.The aim is to examine a method
that may give values of both the real and the imaginary parts of the
refractive index of the absorbing medium. We propose here a method to
determine these quantities from simple techniques.like fringe counting
and interferometry, by using as the incident wave either a single
Gaussian beam or two parallel Gaussian beams.

Key words: Multiple beam interference, wedge, fringe counting,
intensity.

PACS: 42.15Dp, 42.25Fx, 42.25Hz

**1. Introduction**

Intensity pattern as produced by multiple reflections in a transparent
plane wedge is a subject of interest in various contexts of
optics.\[1,2\]. The need for considering a wedge is in fact two fold.
Firstly in making large surfaces, errors of the order of 10 ^--3^
radians are difficult to avoid and secondly by using a wedge one can
increase the dispersive properties of the system which improve the
accuracy with which the real and imaginary parts of the refractive index
can be measured. In our present analysis the refractive and absorptive
properties of the dielectric medium are expressed by a complex
refractive index. The calculation of the intensity pattern requires the
proper evaluation of the path lengths of the different rays that emerge
from the wedge such that one can find the electric field, by taking into
account the proper amplitude and phase for every emergent ray \[3, 4,
5\]. The situation, generally encountered is shown in figure1. The
intensity pattern is very sensitive to the phases of the different rays
and therefore a proper calculation of the path lengths is necessary.
Calculation of this path length also helps to find the amplitude of the
wave as it emerges from the wedge, by simply putting an attenuation
factor in the different rays, consistent with the path lengths that they
have travelled.

The aim here is to show that determination of both the real and
imaginary parts of the refractive index is possible from simple
techniques like fringe counting and intensity pattern matching for
muliple beam interferometry. As a test case we consider the optical
properties of a material close to its absorbing frequency,i.e where the
imaginary part of the refactive index dominates. It is seen that
multiple reflections of single beam produce a flattened intensity
distribution from which it is diffcult to extract the real part of the
refractive index. . However, interferometry experiment with two incident
beams is suggested to determine the real and imaginary parts of the
refractive index. Both the beams are allowed to have multiple
reflections in the slab and the interference is allowed to take place
between all the emergent beams from the slab. This method gives good
results even when the absorption coefficient is high. The analysis of
the method begins with an analysis of the intensity distribution for
rays emerging from a wedge shaped thin film, when several beams are
incident on the wedge and are allowed to have multiple reflections in
the wedge.

We hope to use this method in surface profiling.

**2. Theory**

The theoretical expressions for the intensity are obtained from the
calculational methods given recently by the present author \[6\], with
the addition that for every emergent ray the amplitude has to be
calculated by incorporating an appropriate attenuation factor ,
consistent with the absorption coefficeint as also the path length of
the respective rays. We reproduce the main intensity expression here for
completeness and clarity in the subsequent discussions. .

We refer to figure 1 and consider a ray incident at A. For the first
emergent ray, the phase is given by

ψ(0)=2πρ/λ \[ μ~1~sin θ~1~ + μ~2~sin α/cos(α+θ^'^)\]
\...\...\...\...\...\...\.... (1)

and it emerges from the point

x(0,ρ)= ρ cos θ^'^/ cos(α+θ^'^)\...\...\...\...\..... (2)

The angle θ^'^ satisfies the Snell's law

μ~1~sinθ = μ~2~sinθ' (3)

The second beam travels an extra path CDE ≡ δl(ρ)

= 4ρ sinα cosα cosθ^'^/\[ cos 2α+ cos(2θ^'^ +
4α)\]\...\...\...\...\..... (4)

and emerges from

x(1,ρ) = ρ cosθ^'^/cos(3α+θ^'^)\...\...\...\...\...\...\...\...\.....
(5)

Similar calculations based on simple geometry of triangles show that the
p^th^ ray emerges

from

x(p, ρ) = ρcosθ^'^/cos \[(2p+1)α+ θ^'^\]\...\...\...\...\...\...\....
(6)

after travelling a total path L(p,ρ) in the wedge such that

δl(p,ρ) = L(p+1,ρ)-L(p,ρ) =

4ρ sinα cosα cosθ^'^/\[cos 2α+ cos(2θ^'^ + 4pα)\]\...\..... (7)

It is thus clear that for the pth emergent ray the total path length in
the wedge is found

over the entire zig -- zag path and is thus found from its sum.

~p~

L(p,ρ) = ∑ δl(k)

^k=1^

~p~

= δl(ρ) + ∫ δl(k, ρ) dk\...\...\...\...\..... (8)

^k=1^

where we are using an Euler -- Maclaurin summation formula, giving

L(p,ρ) = δl(ρ) + π(ρ cosθ^'^/α)ln\[(1+cos 2α+sin 2α tan(2pα+θ^'^)).
(1+cos 2α-sin 2α tan(2α+θ^'^))

/(1+cos 2α+sin 2α tan(2α+θ^'^)) (1+cos 2α-sin 2α
tan(2α+θ^'^))\]\...\...(9)

This sum given in (9) is a good approximation, whenever the terms in the
sum do not oscillate rapidly with p, i.e. when cosα \<\<1 and cosαp
\<\<1. Thus, every emergent beam arrives at p, with a phase,

ψ(p,ρ) = ψ(ρ,0)+(2π/λ) L(p,ρ)μ~2r~ -- (2π/λ) x(p,ρ) sinθ μ~1~

= (2π/λ) ρ χ(p,θ)\...\...\...\...\...\...\...\....(10)

in the limit of Fraunhoffer diffraction

Where μ~2r~ is the real part of the refractive index of medium 2. This
part gives rise to only the phase shift.The imaginary part of the
refractive index of the wedge gives rise to attenuation, so that the
amplitude of the p^th^ emergent ray is r^p^ e^-μ^~2i~^L(p,ρ)^ .We
further consider two rays incident at ρ = ρ~c1~ and ρ~c2~ respectively.
The waves from each of these two rays get divided on multiple
reflection.

We consider the incident beam to be a Gaussian, coherent one, with an
electric field distribution with ρ being

P(ρ) = (1/√π∆) exp( --(ρ-ρ~c~)^2^/∆^2^ )\...\...\...\...\...(11)

The net electric field contribution is, on adding over all ρ's, is
calculated as follows.

We designate as E~1~ and E~2~ the first emergent rays: They add up to
give a field,

~+∞~

E~1~ + E~2~ =∫ exp(i(ρ~1~-ρ~c1~+ ρ~c1~) (a+ib)) exp(-(ρ~1~-ρ~c1)~^2^ ∕
∆^2^ dρ~1~

^-∞^

~+∞~

\+ ∫ exp(i(ρ~1~-ρ~c1~+ρ~c1~) (a+ib)) exp(-(ρ~1~-ρ~c1~)^2^ / ∆~2~^2^
dρ~2~

^-∞^

\(12\)

In a similar way, the waves from the rest of the emergent rays, owing to
multiple reflections contribute,

~∞\ +∞~

E~3~+E~4~ = ∑ \[ exp(-βp) ∫ exp(i(ρ~1~-ρ~c1~+ ρ~c1~) (c+id))
exp(-(ρ~1~-ρ~c1)~^2^ ∕ ∆~1~^2^ dρ~1~

^p=1^ ^-∞^

~+∞~

\+ exp(-βp) ∫ exp(i(ρ~1~-ρ~c1~+ρ~c1~) (c+id)) exp(-(ρ~1~-ρ~c1~)^2^ /
∆~2~^2^ dρ~2~ \] (13)

^-∞^

Where,

.a =\[μ~1~ sinθ~1~+ μ~2~ sinα / cos(α+θ^'^) - μ~3~ cosθ^'^ sinθ /
cos(α+θ^'^)\] 2π/λ

.b = μ~2i~ sinα / cos(α+θ^'^)

.c = (2π/λ) \[μ~1~ sinθ~1~ + μ~2r~ sinα / cos(α+θ^'^)\] + μ~2~f(p) -
μ~3~f~1~(p)

d = μ~2i~ f(p) (14)

The total field is thus E~tot~ = E~1~+E~2~+E~3~+E~4~, which on
integration gives the complex field as,

E~tot~ = √π \[ exp(i(a-b)ρ~c1~) exp((a+ib)^2^ ∆~1~^2^ / 4) ∆~1~ +
exp(i(a-b)ρ~c2~) exp((a+ib)^2^ ∆~2~^2^ / 4) ∆~2~

~∞~

\+ ∑exp(-βp) exp((ic-d)ρ~c1~) exp((c+id)^2^ ∆~1~^2^ / 4) ∆~1~

^p=1^

~∞~

\+ ∑exp(-βp) exp((ic-d)ρ~c2~) exp((c+id)^2^ ∆~2~^2^ / 4) ∆~2~ \] (15)

^p=1^

as is expected considering the Gaussian nature of the integrals. The
intensity at any point

is given by

I(ρ,α,θ~1~;θ) = E~tot~(ρ,α,θ~1~;θ) ^2^ (16)

The Gaussian factor exp(-(2π∆χ(p,θ)/2λ)^2^) tends to center every term
in the sum around the values of θ, such that (2π∆χ(p,θ)/2λ)^2^ \<\<1, as
is expected due to diffraction for any finite ∆. Thus by making ∆
sufficiently small, these term may have considerable sweep over θ, as in
the present case , where we have chosen ∆≈1μ for λ = 6328 Angstroms. It
is to be understood that the Gaussian factor in (15) arises because we
have considered a Gaussian beam. For other kinds of beams, however, this
Gaussian factor has to be replaced by an appropriate factor consistent
with the amplitude and phase distribution in the incident beam. It can
be seen that the p=0 term is centered around Ψ(0)-(2π/λ) x~0~sinθ = 0,
which is exactly the geometrical optics formula for the emergent ray θ,
as given by the well known formula for a prism. However, due to finite
size of the Gaussian beam, there is a spread of intensity around this
central ray,-the central ray direction being given by the prism formula.
For different p values, the Gaussian factors are centered around χ(p,θ
)=0, i.e. give maxima in different θ directions, implying that the
different emergent rays from the wedge are not parallel to each other if
α be non zero.

The factors exp(iΨ(p,ρ~c~)) in the different terms in the sum are the
'phase terms' which give rise to interference terms in (14). The
interference results for thin films d ≤ 50μ and low wedge angle α ≤ 0.01
radians are now studied, with the aid of equation (14). It is to be
noted that the formula becomes inaccurate for α \> 0.03 radians as the
summation given by the Euler Mclaurin expression (8,9) then requires
more number of terms to be added,- a step that will be taken up in a
future communication.

**3. Results and discussion**

The issues involved here are a combination of both diffraction and
interference of light. The width ∆ of the beam defines the width of the
diffraction profile, being given by ∆θ≈λ∕∆ while the separation between
the fringes due to interference is nearly δθ≈λ∕d. Thus the number of
interference fringes that can be accommodated in the diffraction
envelope is n≈∆θ∕δθ=d/∆ ( apart from factors that depend on the angle of
incidence) ,so that it is essential that in order that the fringe count
method be effective n≥4. This puts the limitations on the application of
the method due to limitations of obtaining apertures with sufficiently
small ∆. In absence of the wedge, the interference should be exactly as
that of a Fabry-Perot, if the wedge angle be absent and if the light be
allowed to arrive at all possible angles,- i.e. if α and ∆ be both very
small

The effect of absorption on the intensity of the light is shown in
figure 2 and in Figure 3.In figure 2 we see that for a fixed wedge angle
and angle of incidence, intensity decreases as absorption increases, as
is expected. We also see that as we go from figure2a to figure2d (which
is in the order of increasing absorption) the number of fringes also
decrease, forcing us to use two incident beams for getting useful
information at high absorption. Figure 3 also shows the decrease of
intensity maximum as absorption increases.

The determination of the real and imaginary part of the refractive index
completely characterises the given system. The real part decides the
phase of the light beam as it travels in the medium and the imaginary
part determines the attenuation of the amplitude of the wave.

The role of the imaginary part of the refractive index is shown in
figure 3. Real part of refractive index is kept constant and the
imaginary part is varied over a wide range for different thicknesses of
the wedge. The plot of the wedge thickness verses log of intensity
maximum is a straight line for all values of the refractive index
(imaginary). These straight lines for a given wedge can be calibrated to
give an accurate value of the imaginary part of the refractive index.

Determination of the real part of the refractive index involves fringe
counting with the incidence of two beams at slightly8 different
distances from the apex of the wedge. This is because when a single beam
is used at high absorption values, only a broad peak is obtained and we
can not extract any information.

The computations in figure 4 and 5 were carried out using the following
relations. As mentioned earlier the computations are carried out in a
small range where the absorption is optimised.

μ~r~ = n~0~ + ( 4πe^2^/n~0~mω~s~^'^ ) ((Nf~s~)(ω~s~ - ω) / (4(ω~s~^'^ -
ω)^2^ + γ~s~^2^))

μ~i~ = 1//μ~r~ \[( 2πe^2^ / n~0~mω~s~^'^ ) ((Nf~s~)γ~s~ / 4(ω~s~^'^ -
ω)^2^ + γ~s~^2^)\]

where

μ~r~ = rel part of the refractive index

μ~i~ = imaginary part of the refractive index

ω~s~ = absorption frequency = 7.4\*10^15^ sec^-1^

γ~s~ = 3.3\*10^11^ sec^-1^

Nf~s~ = 9.2\*10^16^ cm^-3^

.e = electronic charge = 4.80325\*10^-10^ esu

.m = elecronic mass = 9.10956\*10^-28^ gms

In figures 4 and 5 which show a plot of ω vs the real part refractive
index and ω vs the fringe count after a reference point. In this case we
have used as fringe count as the number of intensity maxima that appear
after the reference point.. We note from figures 4 and 5 that the fringe
count verses the ω curve has a shape similar to the curve which gives
the variation of the real part of the refractive index with the
frequency ω of the incident light. This shows that fringe count helps us
to obtain an idea about the behaviour of real refractive index and can
get an approximate estimate of real refractive inex. An accurate
estimation of real refractive index would involve fitting the intensity
values to equation (16 ).

The work described here also has direct relevance to scattering by
absorbing flakes and fllows the analysis presented in reference \[7\].
It is to be noted that our analysis can be applied to flakes of fairly
large size so that the cross sectional area of the flake is greater than
the cross sectional area of the incident beam. This approximation allows
one to neglect the scattering at the edges. \[8,9\]. We plan to extend
the above analysis to incorporate the effct of the edges and the effect
of roughness on the intensity pattern of the transmitted light. As a
first approximation, the latter can be admitted by choosing a suitable
distribution and considering its autocorrelation. Such an averaging will
smear out the intensity distribution, given in equations (15) and (16).
However, observations conducted at multiple wavelengths, as suggested in
reference \[10\] are likely to reveal the parameters of the system.

## 

## 4. References

\[1\]. Castaneda,R. and Castenada,J.G.S.R., App.Optics **38** 6522(1999)

\[2\]. Vikram Chandra,S. and Martin Pechereky,J., Opt.Engg*.* **38**
1743(1999)

\[3\]. Malacara,D., "Optical Shop Testing" ( John Wiley and Sons Inc)
(1992)

\[4\]. Born,M. and Wolf,E., "Principles of Optics" ( Cambridge
University Press,Chapter 7, 1999)

\[5\]. Francon,M., "Optical Interferometry" (Academic Press, 1966)

\[6\]. Vani,V.C., Ind.Jour.Phys. **75B,** 433 (2001)

\[7\]. Waldemarsson,K.W.T. and Guatafson,B.A.S., "Fifth international
conference on light scattering by non spherical particles." ( Edited by
Videen,S., Fu.Q. and Chylek,P.,) (Army Research Lab Press, Adelphi,
Maryland, 2000).p34.

\[8\]. Kuryoumjian,R.G. and Pathak,P.H., Proc.IEEE.,**62,**1448 (1974).

\[9\]. Syed,H.H. and Volakis,J.L., Radio Science,**30,** 505 (1995).

\[10\] Chatterjee,S., Ind.J.Phys. **74B,** 363 (2000)

**Figure caption**

Figure 1. Sketch of the optics of the system. All calculations and
computations are based on this geometry.

Figure 2. Effect of absorption on the intensity of light. This figure
shows a plot of the scattering angle verses intensity. Figure shows four
plots giving different intensity profiles for different refractive
indicies.

a.  Refractive index = 1.0 + (1\*10^-5^ i).

b.  Refractive index = 1.0 + (1\*10^-4^ i)

c.  Refractive index = 1.0 + (1\*10^-3^ i)

d.  Refractive index = 1.0 + (1\*10^-2^ i)

Figure 3. Plot of the wedge thickness verses logarithm of the maximum
value of the intensity at a given value of the imaginary part of the
refractive index.. Imaginary part of the refractive index varies from
1\*10^-5^ to 10^-2^ and the real part is kept constant at 1.1.

Figure 4. Plot of omega verses the real part of the refractive index.
The relation between omega and refractive index is given in the text.

Figure 5. Plot of omega verses the fringe count after a reference point.
The relation between omega and refractive index is given in the text.