Download MICROWAVE OPTICS – THE MEASUREMENTS OF THE

MICROWAVE OPTICS – THE MEASUREMENTS OF THE WAVELENGTH OF THE
MICROWAVES BASED ON THE INTERFERENTIAL METHODS.
BASIC THEORY
Microwaves belong to the band of very short electromagnetic waves. The
wavelengths of the microwaves range from 30 centimeters down to single millimeters
(what corresponds to the frequencies of 109-1011Hz). These waves are used in
radars and other communication systems, as well as in the analyses of very fine
details of atomic and molecular structure, and are also generated by electronic
devices.
As everybody knows interference is one of the phenomena that demonstrate the
wave nature of the light but it is important not only in optics but manifested also in
acoustic and radio signals.
Generally speaking interference it is the phenomenon resulting from the meeting of
two or more waves (coexisting in the space and time), with an increase in intensity at
some points (where waves are in phase) and a decrease at others (where waves are
out of phase). It is important that the result of interference does have the appearance
only when the waves are coherent (i.e. their sources oscillate with the same
frequency and have a constant phase relationship). Otherwise, if the phase
difference of the sources changes erratically with time, even if they have the same
frequency, no stationary interference pattern is observed, and the sources are said to
be incoherent.
Let's consider two plane harmonic waves of the same polarizations, that oscillate with
the same angular frequency ω and amplitudes E01 and E02, and move in the +x
direction. The second wave (described by its amplitude E2) passes additional
distance in space ∆.
Propagation of the mentioned waves is given by following expressions:
E1=E01sin(ωt-kx)
E2=E02sin[ωt-k(x+∆)]
⎫
⎬ (1)
⎭
2π
is a wave number and λ is a wavelength (a distance advanced by the
λ
wave motion in the one period).
where: k =
If the wave does not propagate in vacuum (air) but in the medium, where refractive
index n≠1, its wavelength is decreasing to the value of λ0/n (where λ0 is the
wavelength in the air) and wave number is increasing to k0n. In this case the
difference between optical ways of the waves E1 and E2 is equal to n∆.
Let’s consider the simplest condition when both waves are traveling in the air (n=1).
In this case the optical way is equal to the geometrical one, optical ways difference
for the waves from eq. (1) comes to ∆, and eventually the phase difference between
two wave motions is equal to
(2)
2π
φ = k∆ =
∆.
λ
Considering interference of two waves from eq. (1), the resultant field is found from
the principle of superposition:
E=E1+E2=E01sin(ωt-kx)+E02sin[ωt-k(x+∆)]=E01sin(ωt-kx)+E02sin(ωt-kx-φ)
(3)
Detectors of electromagnetic radiation are sensitive for the intensity of the waves that
is defined as the energy incident per second per unit area normal to the direction of
propagation.
For analyzed type of waves it can be calculated using following formula:
T
1 2
2
(4)
I =< E > t = ∫ E dt
T0
where T is the period of oscillation.
In our case:
E2=(E1+E2)2=E012sin2(ωt-kx) + E022sin2(ωt-kx-φ) + 2E01E02sin(ωt-kx)sin(ωt-kx-φ)
By using the trigonometric identity: cos α − cos β = 2 sin
α+β
β−α
sin
we obtain that:
2
2
E2=E012sin2(ωt-kx) + E022sin2(ωt-kx-φ) + E01E02 { cosφ - cos[2(ωt-kx)-φ] }
T
(5)
(6)
T
1
1
1
Because
sin 2 (ωt + δ)dt = , and ∫ cos( ωt + γ )dt = 0
∫
T0
2
T0
we have finally that
2
2
E 01
E 02
I =< E > t =T =
+
+ E 01E 02 cos φ = I1 + I2 + 2 I1 I2 cos φ
(7)
2
2
where: I1 it is intensity of the first wave, I2 intensity of the second one, and the last
term of the equation above describes the result of mutual interference of the waves 1
and 2.
2
One can see that intensity falls between the values of ⎛⎜ I1 + I2 − 2 I1 I2 ⎞⎟ and
⎝
⎠
⎛⎜ I + I + 2 I I ⎞⎟ , depending on whether cosφ= -1 or +1 i.e. φ=(2m+1)π (waves are
1 2
⎝1 2
⎠
in phase) or φ=2mπ (waves are out of phase), where m is either a positive or a
negative integer.
In the first case we have maximum attenuation of the two wave motions, called as
destructive interference, and in the second case maximum reinforcement i.e.
constructive interference. That is,
⎧ 2mπ
constructi ve interferen ce
φ=⎨
⎩ (2m + 1)π destructiv e interferen ce
(8a)
using equation (2), we have
⎧ 2mπ
constructi ve interferen ce
k∆ = ⎨
⎩ (2m + 1)π destructiv e interferen ce
(8b)
which can be written as
⎧⎪ mλ
constructive interference
λ
∆=⎨
⎪⎩ (2m + 1) 2 destructiv e interference
(8c)
Therefore we can conclude from the last equations that when the optical ways
differences are: ∆=0, ±λ, ±2λ, ±3λ, ... the waves interfere with reinforcement, instead
when ∆=±λ/2, ±3λ/2, 5λ/2, ... the waves interfere destructively.
PRACTICAL APPLICATION OF THE INTERFERENCE PHENOMENON IN THE
MEASUREMENTS OF THE WAVELENGTHS:
Interferometers are the devices that use waves interference phenomenon for
determination of wavelength or for the precious measurements of the distances in
terms of the wavelength of used EM wave.
1. Michelson's Interferometer
The main principle of its functioning is shown in
Fig. 1.1, while the sketch of set-up used in
experiment is demonstrated in Fig. 1.2.
The beam of electromagnetic waves coming from
the source is partially transmitted and partially
reflected by the semi-transmitting (S-T) plate
placed in the middle part of described device.
Approximately half of the incident EM radiation,
transmitted by the S-T plate, goes to the mirror 1
and it is reflected back, and then finally reaches
the detector after being reflected by S-T plate.
The second part of the beam is reflected by S-T
plate, goes to the mirror 2 and after reflection,
Figure 1.1
travel trough S-T plate to reach the detector.
Eventually two waves, coming from the one
source but passing different ways (paths) in the space, recombine and interfere in the
plane of the detector head. Detector records the intensity of the resultant radiation
and exchange this information into electric signal that could be measured by
voltmeter (see Fig. 1.2).
Figure 1.2
By moving the mirror(s) (1
or/and 2), we obtain the
change of
the length of
interferometer 'arms'. It causes
the change of the path
difference between two rays.
Recalling general rule which
says
that
maximum
of
interference is obtained when
mentioned difference is equal
to
integer
multiple
of
wavelength one can prove that
successive maxima
of the
signal registered by detector
are observed when 2L=mλ
(see Fig 1.1).
It means that having the
position of the mirror for the
first and the mth maximum we
can find the wavelength of EM
waves
using
following
2
(L m − L 1 ) (9)
m −1
where Lm is the position of the mirror for the mth maximum and L1 for the first one.
equation: λ =
2. Fabry-Perot Interferometer
F-P interferometer is composed of two
perfectly parallel plates that transmit one
part of electromagnetic radiation but also
they have high ability of reflection. Rays
that get inside of cavity are multireflected –
both by the first and by the second plate.
Eventually, as the output, the group of
mutually parallel rays is obtained. They
interfere with each other and the result of
interference depends on the angle α of the
rays incidence on F-P interferometer and
on the distance d between two plates
Figure 2.1
forming cavity.
Let's analyze the situation shown in Figure
2.1. The path difference between wave 1-1' (passing trough the cavity without
reflection) and wave 2-2' (reflected twice inside of interferometer) is equal to
∆=OA+AB-B'R.
Taking that OA=AB=d/cosα , OB=2OD=2dtgα and B'R=B'Esinα=BOsinα we obtain
that:
2d
2d
(10)
∆=
− 2d sin αtgα =
1 − sin 2 α = 2d cos α
cos α
cos α
(
)
It means that by changing the position of the plate(s) i.e. by changing the distance d
between plates we change the optical ways difference ∆ i.e. we change the
conditions of the interference. Maxima of reinforcement take place for dm that fulfill
the following conditions:
∆=2dmcosα=mλ
(11)
Observing the following maxima registered by detector placed after interferometer we
are able to calculate the wavelength
of used electromagnetic radiation:
2
λ = cos α (dm+r − dm )
(12)
r
where dm is the width of cavity for
the mth maximum and dm+r for the
(m+r)th one.
Considering the simples case (see
Fig. 2.2), when the beam of
microwaves goes perpendicularly
to the plates (i.e. α=0) we observe
the maxima of the signal given by
detector for:
∆=2d=mλ
(13)
what means that having width of
cavity for the first and the mth
maximum we can find the
wavelength of incident EM waves
using following formula:
Figure 2.2
λ=
2
cos α (dm − d1 )
m −1
3. Diffraction Grating:
Another simple device for producing interference of light is
the one used by Thomas Young in his early, double-slit
experiment or diffraction grating. The latter consists of
greater number of parallel, identically spaced slits. The
distance between the middles of the neighboring slits is
called as the diffraction grating spacing d and it is a
parameter characterizing each grating (d=a+b, where a it
is a width of the slit and b it is a width of a matter between
two slits).
When the electromagnetic wave of the single wavelength
λ passes trough a diffraction grating the resultant radiation
forms the pattern of maxima and minima, depending on
the path difference between the waves coming from the
different slits.
Let's consider the rays coming from the neighboring slits
Figure 3.1
(14)
shown in Figure 3.1. In this case the path difference is equal to ∆=BC=d sinα.
Therefore the positions are given by following expression:
d sinαm = mλ
(15)
where: d it is diffraction grating
spacing, λ it is the wavelength of
used electromagnetic waves, α it is
the angle of diffraction and integer
m=0,1,2,3,… is referred to as the
order of observed maximum.
For m=0 we obtain the maximum
corresponding to the beam that is
not diffracted i.e. α=0 and for
m=1,2,3, ... there are maxima on
the left and right side that
correspond to the increasing value
of diffraction angle α at which
reinforcement of the signal is
observed.
All mentioned maxima are called
as the principle ones since the
waves from all slits are in phase.
Figure 3.2
Basing on equation (15), knowing
diffraction grating spacing d and by
measuring the angles of following
maxima, we may obtain the
wavelength λ.
EXPERIMENTAL PROCEDURE
1) Using Michelson interferometer (Fig. 1.2) and changing position of one of the
mirrors observe on an oscilloscope (or the voltmeter) connected to detector,
following reinforcements of the signal. Register the positions of as many
maxima as it is possible and calculate wavelength of examined waves using
formula (9).
2) Build F-P interferometer (Fig. 2.2) and increasing (or/and decreasing) the
width of cavity find the maxima of the signal registered by detector. Find great
number of them and calculate the wavelength of used microwaves from
equation (14).
3) Create the set-up with diffraction grating as it is shown in Fig. 3.2. Note that
during all measurements diffraction grating should be placed exactly
perpendicularly to incident beam of radiation. Measure the diffraction grating
spacing d. To minimize the error and to obtain higher precision it is better to
measure the distance between initial and ending slit and final result divide by
total number of slits.
Changing the angular position of the arm of the branch with detector find the
angles α corresponding to the following maxima of diffraction – both on the left
and the right side. From the diffraction grating equation (15) determinate
wavelength λ for diffraction grating spacing d calculated before.
In all cases 1) – 3) make quite a lot of measurements around maximum to find its
precious position.
CALCULATIONS AND DATA ANALYSES
Calculate the wavelength of examined microwaves using three different methods and
give the errors of obtain values. Compare obtained results. Write some conclusions
explaining which method gives the highest and the lowest precision, and why. Are
the results given by different methods similar? Are they in the same range? Are they
equal in the range of errors? Are the experimental results close to the theoretical
value? etc.