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optical interferometry and its applications in absolute distance measurements

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### opticalinterferometryand its applications in absolute distance measurements

### Outlines

### Interferometry Concepts

### Interferometers configurations

### Absolute Distance Measurements

by: KHALED ALZAHRANI

Liverpool John Moores University

GERI

Interferometry Concepts

Popular inteferometric configurations

Absolute distance measurement (ADI)

Interference

Intensity

Visibility

Optical Path Length [OPL]

Optical Path Difference [OPD]

Coherence:

Spatial coherence

Temporal coherence

Optical Interfrometry is an optical measurement technique that provides extreme precise measurements of distance, displacement or shape and surface of objects.

It exploits the phenomenon of light waves interference .

Where under certain conditions a pattern of dark and light bars called interference fringes can be produced. Fringes can be analyzed to present accurate measurements in the range of nanometer.

The recent developments in laser, fiber optics and digital processing techniques have supported optical interferometry .

Applications ranging from the measurement of a molecule size to the diameters of stars.

General ConceptsLight waves

- For many centuries, light was considered a stream of particles .
- Light wave exhibits various behaviours which can not interpreted through the particles theory of light such as, refraction, diffraction and interference.
- in19th centurythe particles concept was replaced by the wave theory .
- light waves are transverse waves with two components; magnetic and electric field each one of them oscillating perpendicular to the other and to the propagation direction.
- The visible light is part of the electromagnetic spectrum it extends from 750nm for the red color to 380nm for the violet color.
- Light wave characteristics:

light speed in free space (c): C=300k (km/s)

C = λv

V = c/n

λn =λ/n

- Where: n is the refractive index of the medium in which the light travels.

λnis the wavelength in medium other than free space.

EM-wave propagation

Visible light spectrum

Refractive index

Interference

- Interference is a light phenomenon .It can be seen in everyday life. e.g.. colures of oil film floating on water.
- In electromagnetic waves , interference between two or more waves is just an addition or superposition process. It results in a new wave pattern .

Superposition of two waves

- When two waves with an equal amplitudes are superposed the output wave depends on the phase between the input waves.

Y = y1 + y2

Where: y1=A1 sin (wt + θ1 )

y2=A2 sin (wt + θ2)

- Since the energy in the light wave is intensity I ,which is proportional to the sum of square amplitudes A^2

where: A=A1^2+A2^2+2A1A2 cos (θ1 – θ2)

If A1=A2=A then:

A=2A^2+2A^2 cos (θ1 – θ2)

If y1&y2 in phase ,cos(0)=1 hence,

Y = 4A^2,it gives a bright fringe.

If y1&y2 out of phase by (π) ,cos (π)=-1 hence,

Y = 0 ,it gives a dark fringe

Optical Path Length [OPL]

- When light beam travels in space from one point to another, the path length is the geometric length d multiplied byn (the air refractive index) which is one:

OPL = d

- Light beam travels in different mediums will have different optical path, depending on the refractive index (n)of the medium or mediums.

OPL = n d

Optical Path Difference [OPD]

- If two beams with the same wavelength i.e same frequency, travel from two different points towards the same destination ,taking different paths there will be a difference in their optical path this difference is called the optical path difference [OPD].
- it is very important factor in determining fringes intensity.

OPD = mλ

- Here, If m=0 or any integer values there will be a bright fringe. Otherwise dark fringes (maximum darkness when )

OPD= (m-1/2) λ

Intensity of Interference fringes

- Intensity of interference fringes depends on the phase between the recombined waves i.e.
- Intensity I is the complex amplitude of the interferer waves A given as: I=│A│^2

I= lAl^2 = I1+I2+2(I1I2) cos (Δθ) ^1/2

- When Δθ = 0

I max = I1 + I2 +2(I1I2)^1/2

if I1=I2 then

I max=4I

- When Δθ = π

I min = I1 + I2 – 2(I1I2)^1/2

if I1=I2 then

I min=0

Visibility of Interference fringes

- Visibility determines the ability to resolve interference fringes. It depends on the coherence degree between the recombined light waves.
- It is defined as:

V = I max - I min / I max + I min

maximum if Imin= 0 , V= 1

When Imin = Imax , V= 0

[ 0 ≤V≤1 ].

Coherence

- Coherence of light wave is defined as the correlation between the electric field values at different locations or times. The coherent light source is able to produce a coherent waves able to interfere with each other.
- Ideal coherent source is a source with one wave length only ‘‘monochromatic’’ which does not exist in practice.
- Practically, there is no fully coherent light or fully incoherent light, but there are light sources with deferent coherence degree .

Spatial & Temporal Coherence

- Spatial coherence:

The degree of correlation between different points on the same wave front at the same time.

Spatial coherence is light source dependent, as the source size extends its spatial coherence degree deteriorate.

- Temporal coherence:

The correlation between the electric fields at the same point but at different times.

Temporal coherence proportionate to the wave train length. Monochromatic sources such as laser have a high degree of temporal coherence, because of the long wave trains.

- Coherence Length:ΔS = N λ.

where N is thewaves number contained in one wave train.

- Coherence time :Δt = ΔS / C

where C is the light speed in space .

Interferometers classifications:wave front division interferometers Amplitude division interferometer

Popular configurations:

Michelson interferometer

Twyman-Green interferometer

Mach-Zehnder interferometer

Fapry-Perot interferometer

Interferometer

- Interferometer:

Is an optical instrument that can produced two beams interference or multiple beam interference.

- wave front division interferometers:

Two light beams from the same wave front are made to interfere to produce an interference fringe pattern.

- Amplitude-division interferometers:

A light beam from one source point is divided into two beams using a beam splitter.

e.g. Michelson’s interferometer

Michelson interferometer

- Configuration:

Michelson interferometer consists of a coherent light source, a beam splitter BS a reference mirror ,a movable mirror and a screen .

- Applications:

There are many measurements that Michelson interferometer can be used for, absolute distance measurements, optical testing and measure gases refractive index.

- Work method:

The BS divides the incident beam into two parts one travel to the reference mirror and the other to the movable mirror .both parts are reflected back to BS recombined to form the interference fringes on the screen.

Twyman-Green interferometer

- Configuration:
- A modified configuration of Michelson interferometer ( rotatable mirror& a monochromatic point source)
- Applications:length measurements,optical testing e.g. lenses ,prisms, mirrors.
- Work method:

When the interferometer aligned properly, two images of the light source S from the two mirrors M1&M2 will coincide. The superposed waves are parallel and have a constant phase difference. On the serene a uniform illumination can be seen with a constant intensity depends on the path difference.

- Mirror imperfections test:

There will be an interference fringes due to the path difference between W2 and the reference plan wave W1

Mach-Zehnder interferometer

- Configuration:

consists of a light source, a detector, two mirrors to control the beams directions and two beam splitters to split and recombine the incident beam.

- Applications: refractive indexfluid flow ,heat transfer.
- Work method: BS1 divides the incident beam into 2 beams,mirrors M1&M2 reflect beams to BS2 . BS2 recombine the beams. interference fringes produced depending on the path difference .
- measure thickness at constant refractive index
- measure refractive index at constant thickness

Fabry-Perot Interferometer [FPI]

- Configuration:

consisting of two parallel high reflecting glass plates separated several millimeters , a focusing lens and a display screen.

- Advantages& disadvantages:
- high sensitivity to wave length changes. (used in laser to select wave length)
- High resolution fringes (used in optical spectroscopy)
- Applications:
- measure or control the light wave lengths e.g. in laser as a resonator to select a single wave length. optical spectroscopy.
- Work method:
- the beam falls on L1, part of the beam is transmitted to L2, other part is reflected .the transmitted part partially reflected back to L1. Then again reflected to the L2 which partially reflects and transmit each incident light. The transmitted lights from L2 falls on the Focusing lens. beams are focused on the screen at point P .these beams interfere constructively or destructively according to the phase difference between them .

.

Developments in laser techniques and digital image processing have made distance measurement by optical techniques very attractive at variety of applications in industrial fields e.g. tool calibration, aircraft industry and robotics.

Two measurement techniques:

Non-Coherent methods:

Triangulation techniques

Time-of-flight systems

Measurement accuracy larger than a 1mm

Coherent methods:

based on interferometry, enable high precision measurements of distances or displacements.

Classical interferometry: (i.e. one-wavelength)

commonly used for high-resolution displacement measurements. Resolution better than 100 nm .Drawback of this technique is the incremental manner of measuring, resulting from the counting of optical fringes.

ADI cannot be covered by classical interferometry since the range of non-ambiguity is limited to half the optical wavelength

multiple-wavelength interferometry:(MWI)

offers great flexibility in sensitivity by an appropriate choice of the different wavelengths

Example: conceder two optical wavelength λ1&λ2 with PD=L .the phases φ1 and φ2 corresponding to the wavelengths λ1 and λ2

∆φ1 =(2π/ λ1) 2L & ∆φ2=(2π/ λ2) 2L

∆φ12= (∆φ1- ∆φ2) = 2π/[1/ λ1 - 1/ λ2] 2L = [2π/ λs]2L

λs= [λ1λ2/(λ1- λ2)]this synthetic wavelength is much longer than λ1 or λ2.

The range of non-ambiguity of the phase difference Δφ12, which is also known as the synthetic phase, is therefore increased compared to the range of non-ambiguity of classical interferometry. Moreover, the sensitivity of the measurement is reduced.

Practical example

- Iteration1
- Estimate distance manually e.g. L=235 mm
- Estimate error range and ambiguity length.
- e.g. error = ± 2 mm , 233mm≤L ≤ 237mm
- λS > 2* error range, to be say 5mm
- Adjust tunable laser source at arbitrary λ 1 such as 682 nm and grab image-1
- Calculate λ 2 : λ 2 = λ 1 / [ (λ 1/ λS ) +1] = 681.908 nm
- Adjust tunable laser source at λ 1 = 681.908 nm and grab image-2
- δØ calculated using FTA (it represents a fraction of a fringe)
- Divide ambiguity range by 2 so λ s =2.5mm

Practical example

- Iteration2
- Adjust laser at λ 1 = 681.908 nm then Calculate λ 2= 681.522nm grab image-3 .
- calculate δØ between image 2&3.
- Iteration3
- image-4 is for wavelengths 681.15 nm.
- calculate δØ between image3&4.
- This way ambiguity decreased and error decreased by 2 hence better accuracy. It is possible to converge in fewer steps if the confidence factor is higher.

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