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Optical Profilometry and Vibration Amplitude
Measurement with Multicore Fibers
M. Naci Inci
Physics Department, Bogazici University
Layout
• Fourier Transform Profilometry (FTP)
• Vibration Amplitude Study with FT Analysis
Optical Profilometry

It employes the wave nature of light to determine
shape and dimensions of objects. It uses
structured light patterns that are generated
through optical interference.

A structured light pattern based on a two-beam
optical interference
Advantageous of the Optical Profilometry
 Applicable

in real-time
Non-invasive
 Applicable
to large areas

Hight resolution and high sensitivity

Computer compatibility
Applications





Industrial otomation
Robotic vison
Quality control
Biomedical applications
CAD/CAM modelling
Optical Profilometry:
It is a measurement method based on
the wave nature of light, which uses
optical interference fringes of the laser beam
How do we obtain a structured light pattern?

(A double-slit Young experiment)
Why Fourier Transform Profilometry (FTP)?

FTP’s main advantage is that it uses only a
single image to extract profile of an object. In
other techniques, 3 or 4 images are required.
Methodology:
Aim: To obtain a direct
relationship between the
object’s surface
topography (z(x,y)) and
the phase () of the
structured light pattern
Two-beam interference fringe pattern analysis

Light intensity distribution over the surface in concern is
(1)

For the Fourier fringe analysis, Eq.1 can be written as
(2)

The FT of I(x,y) at the CCD camera is
(3)
A
C
C*
u0
A
Fringe analysis
C*
C
u0






C (or C*) is isolated and then translated to the origin by u0 amount.
A(u, v) and C*(u+u0, v) are eliminated by bandpass filtres
Inverse of FT is applied to determine the complex fn. c(x,y)
Phase of the structured light pattern is determined as
Phase-unwrapping is applied to correct 2π phase jumps
Surface topography and phase of the fringes are related as
Optical fiber
>
Interference with two fibers
Interference with four fibers
Mutual coherence is required between fibre ams
to obtain interference pattern
Single source with a 2x2 fiber coupler
Fibre arms are difficuilt to aline properly.
Vibration, temperature, polarization, etc. result in
a poor fringe visibility
Alignment is even more difficuilt with 4 fibers
Interferece with a two-core optical fibre
Interferece with
a four-core optical fibre
Four-core fiber
Manufactured by Hesfibel, Kayseri, Turkey
(www.hesfibel.com)



x cos   z ( x, y) sin    2 cos 2  y  
I x, y   2 I 0 2  2 cos 2
 f

 f 




x cos   z ( x, y) sin   y   cos 2  x cos   z ( x, y) sin   y 
cos 2
 f

 f

I ( x, y )  ax, y   cx, y  exp i2u0 x   c * x, y  exp  i2u0 x  
d x, y  exp i2u0 y   d * x, y  exp  i2u0 y  
ex, y  exp i2 (u0 x  u0 y   e * x, y  exp  i2 (u0 x  u0 y  
f x, y  exp i2 (u0 x  u0 y   f * x, y  exp  i2 (u0 x  u0 y 

u 0 
f
FT of I(x,y)
I (u, v)  A(u, v)  C (u  u 0 , v)  C * (u  u 0 , v) 
D(u, v  u 0 )  D * (u, v  u 0 ) 
E (u  u 0 , v  u 0 )  E * (u  u 0 , v  u 0 ) 
F (u  u 0 , v  u 0 )  F * (u  u 0 , v  u 0 )

FFT of the light pattern
I (u, v)  A(u, v)  C (u  u 0 , v)  C * (u  u 0 , v) 
D(u, v  u 0 )  D * (u, v  u 0 ) 
E (u  u 0 , v  u 0 )  E * (u  u 0 , v  u 0 ) 
F (u  u 0 , v  u 0 )  F * (u  u 0 , v  u 0 )
Phase
Surface and phase are related as
Experimental Setup
(a) Triangular shape object; (b) projected
fringe pattern; (c) reconstructed surface of the
object
(a) Sculptured head object and the outlined
area shows the analysed surface; (b)
projected fringe pattern; (c)reconstructed
surface of the object
(a) An object made from sand and the outlined
area shows the analysed surface; (b)
projected fringe pattern; (c)reconstructed
surface of the object
(a) Projected fringe pattern of a flat plate with a 2 mm step. The
area in the upper right-hand corner is 2mm higher than the rest
of the plate; (b) 2D Fourier spectra of the analyzed pattern (c)
Reconstructed surface
K Bulut, MN Inci, Optics & Laser Technology (in press)
A board marker
K Bulut, MN Inci, Optics & Laser Technology (in press)
4
3
2,5
2
1,5
1
Surface Height (mm)
3,5
Measured
0,5
Circle, r =14,4 mm
-6,50
-4,88
-3,25
-1,63
0,00
1,63
3,25
4,88
0
6,50
y (mm)
Comparison between a cross section of the reconstructed surface
with a circle of radius 14.4 mm. The RMS error is 0.4 mm.
Vibration Amplitude Measurements
If the object vibrates sinusoidal with an angular frequency ,
then the out-of-plane displacement of the object surface at (x, y) is given by
zx, y, t   z 0 x, y   V x, y cos t
V(x, y): local amplitude of vibration
A
B



x cos   z 0 ( x, y ) sin   V x, y cos t sin  
I ( x, y, t )  2 I 0 2  2 cos 2
 f



 
 2 cos 2
y
 f 


x cos   y  z 0 ( x, y ) sin   V x, y cos t sin  
 cos 2
 f




x cos   y  z 0 ( x, y) sin   V x, y  cos t sin   
 cos 2
 f

I  x, y, t   p  r exp i  x  x    r  x, y   t  x, y, t 
 r * exp  i  x x    r  x, y   t x, y, t 
 r exp i y  y 




 r * exp  i y  y 
 m exp i  x x    y  y    r x, y   t  x, y, t 




 m * exp  i  x  x    y  y    r x, y   t  x, y, t 
 n exp i  x x    y  y    r  x, y   t  x, y, t 
 n * exp  i  x  x    y  y    r  x, y   t x, y, t 






x cos  ;
 y  y   2
y
f
f

 r x, y   2
z 0 x, y sin 
f

t x, y, t   2 V x, y  cos t sin  ; i   1
f
 x x   2
Since the frame rate of the CCD camera is much lower than
the vibration angular frequency ω, the light pattern captured
is proportional to the time average of I(x, y ,t) over one period:
I  x, y , t 
T
t
1
  I  x, y, t dt
T 0
 p  r '  x, y  exp i 2u 0 x   (r ' ) *  x, y  exp  i 2u 0 x 




 m'  x, y  exp i 2u x  2u y 
 (m' ) * x, y  exp  i 2u x  2u y 
 n'  x, y  exp i 2u x  2u y 
 (n' ) *  x, y  exp  i 2u x  2u y 
 r exp i 2u 0' y  r * exp  i 2u 0' y
'
0
0
'
0
0
'
0
0
0
'
0
2

V ( x, y )
f
T
1
r ' x, y   r exp i r x, y    exp  it  x, y, t dt
T 0
 r exp i r x, y   J 0  
u0 
 cos 

; u 0' 
f
f
T

2
C and D are processed to obtain
J 0   
Vibration amplitude is obtained from  
r ' x, y 
r x, y 
2
V ( x, y )
f
4 different vibration amplitudes studied
ST Yilmaz, U Ozugurel, K Bulut, MN Inci, Optics Communications (to be published)
Conclusion
Multicore fiber based optical profilometry and vibration
amplitude measuremets are promising.
However, a larger fiber core seperation will
improve the resolution of the optical method
Acknowledgement
Karahan Bulut, Tunç Yılmaz, Umut Özuğurel