MICROWAVE OPTICS – the MEASUREMENTS of the WAVELENGTH of the MICROWAVES BASED on the INTERFERENTIAL METHODS. BASIC THEORY Mi

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) ⎫ (1) E2=E02sin[ωt-k(x+∆)] ⎬ ⎭ 2π where: k = is a wave number and λ is a wavelength (a distance advanced by the λ wave motion in the one period).

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 2π (2) 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: 1 T I =< E2 > = E2dt (4) t ∫ T 0 where T is the period of oscillation.

In our case:

2 2 2 2 2 2 E =(E1+E2) =E01 sin (ωt-kx) + E02 sin (ωt-kx-φ) + 2E01E02sin(ωt-kx)sin(ωt-kx-φ) (5)

α + β β − α By using the trigonometric identity: cosα − cosβ = 2sin sin we obtain that: 2 2

2 2 2 2 2 E =E01 sin (ωt-kx) + E02 sin (ωt-kx-φ) + E01E02 { cosφ - cos[2(ωt-kx)-φ] } (6)

1 T 1 1 T Because ∫ sin2 (ωt + δ)dt = , and ∫ cos(ωt + γ)dt = 0 T 0 2 T 0 we have finally that

E2 E2 I =< E2 > = 01 + 02 + E E cosφ = I + I + 2 I I cosφ (7) t=T 2 2 01 02 1 2 1 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.

One can see that intensity falls between the values of ⎜⎛I+ I − 2 I I ⎟⎞ and ⎝ 1 2 1 2 ⎠ ⎜⎛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 constructive interference φ = ⎨ (8a) ⎩ (2m +1)π destructive interference using equation (2), we have

⎧ 2mπ constructive interference k∆ = ⎨ (8b) ⎩ (2m +1)π destructive interference which can be written as

⎪⎧ mλ constructive interference ∆ = λ (8c) ⎨ (2m +1) destructive interference ⎩⎪ 2

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, travel trough S-T plate to reach the detector. Figure 1.1 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).

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 Figure 1.2 first and the mth maximum we can find the wavelength of EM waves using following 2 equation: λ = (L − L ) (9) m −1 m 1 th where Lm is the position of the mirror for the m maximum and L1 for the first one.

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 forming cavity. Figure 2.1 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 ∆ = − 2dsinαtgα = ()1− sin2 α = 2dcosα (10) cosα cosα 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 = cosα (d − d ) (12) r m+r m where dm is the width of cavity for th the m maximum and dm+r for the th (m+r) 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 2 λ = cosα (d − d ) (14) m −1 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. Figure 3.1 Let's consider the rays coming from the neighboring slits 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.

Basing on equation (15), knowing diffraction grating spacing d and by Figure 3.2 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.