Abstract

A theoretical analysis of the fringe pattern produced by a capillary tube interferometer is presented, which is expected to be two-beam interference, and a computer program to simulate the interference fringe pattern is established. By comparing the simulated fringe pattern and the experimental fringe pattern, the refractive index of the liquid can be given when the two fringes coincide best. The results of this method are close to those of the Abbe refractometer.

© 2006 Optical Society of America

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References

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  1. H. El-Ghandoor, E. Hegazi, I. Nasser, and G. M. Behery, "Measuring the refractive index of crude oil using a capillary tube interferometer," Opt. Laser Technol. 35, 361-367 (2003).
    [CrossRef]
  2. H. El-Ghandoor, I. Nasser, M. A. Abd-El Rahman, and R. Hassan, "Theoretical model for the transverse interference pattern of GRIN optical fiber using a laser sheet of light," Opt. Laser Technol. 32, 281-286 (2000).
    [CrossRef]
  3. H. El-Ghandoor, I. Nasser, A. Abdel-Hady, and A. Al-Shukri, "A comparation study of spliced optical fibers," Opt. Lasers Eng. 41, 277-287 (2004).
    [CrossRef]
  4. A. A. Hamza, M. A. Mabrouk, W. A. Ramadan, and H. H. Wahba, "Core index determination of a thick fibre using lens-fibre interference (LFI) technique," Opt. Lasers Eng. 42, 121-130 (2004).
    [CrossRef]
  5. W. A. Ramadan, "On-line lens-fiber interference method for testing a thick fiber," J. Opt. A 2, 234-238 (2000).
    [CrossRef]
  6. M. Fukuta, T. Yanagisawa, S. Miyamura, and Y. Ogi, "Concentration measurement of refrigerant/refrigeration oil mixture by refractive index," Int. J. Refrig. 27, 346-352 (2004).
    [CrossRef]
  7. S. A. Khodier, "Refractive index of standard oils as a function of wavelength and temperature," Opt. Laser Technol. 34, 125-128 (2002).
    [CrossRef]
  8. N. Barakat, H. A. El-Hennawi, E. Abd. El-Ghafar, H. El-Ghandoor, R. Hassan, and F. El-Diasty, "Three-dimensional refractive index profile of a GRIN optical waveguide using multiple beam interference fringes," Opt. Commun. 191, 39-47 (2001).
    [CrossRef]
  9. A. A. Hamza, M. A. Mabrouk, W. A. Ramadan, and M. A. Shams-Eldin, "Determination of GRIN optical fibre parameters from transverse interferograms considering the refraction of the incident ray by the fibre," Opt. Commun. 200, 131-138 (2001).
    [CrossRef]
  10. M. A. El-Morsy, T. Yatagai, A. A. Hamza, M. A. Mabrouk, and T. Z. N. Sokkar, "Automatic refractive index profiling of fibers by phase analysis method using Fourier transform," Opt. Lasers Eng. 38, 509-525 (2002).
    [CrossRef]
  11. M. A. El-Morsy, K. Harada, M. Itoh, and T. Yatagai, "A subfringe integration method for multiple-beam Fizeau fringe analysis," Opt. Laser Technol. 35, 223-232 (2003).
    [CrossRef]

2004 (3)

H. El-Ghandoor, I. Nasser, A. Abdel-Hady, and A. Al-Shukri, "A comparation study of spliced optical fibers," Opt. Lasers Eng. 41, 277-287 (2004).
[CrossRef]

A. A. Hamza, M. A. Mabrouk, W. A. Ramadan, and H. H. Wahba, "Core index determination of a thick fibre using lens-fibre interference (LFI) technique," Opt. Lasers Eng. 42, 121-130 (2004).
[CrossRef]

M. Fukuta, T. Yanagisawa, S. Miyamura, and Y. Ogi, "Concentration measurement of refrigerant/refrigeration oil mixture by refractive index," Int. J. Refrig. 27, 346-352 (2004).
[CrossRef]

2003 (2)

H. El-Ghandoor, E. Hegazi, I. Nasser, and G. M. Behery, "Measuring the refractive index of crude oil using a capillary tube interferometer," Opt. Laser Technol. 35, 361-367 (2003).
[CrossRef]

M. A. El-Morsy, K. Harada, M. Itoh, and T. Yatagai, "A subfringe integration method for multiple-beam Fizeau fringe analysis," Opt. Laser Technol. 35, 223-232 (2003).
[CrossRef]

2002 (2)

M. A. El-Morsy, T. Yatagai, A. A. Hamza, M. A. Mabrouk, and T. Z. N. Sokkar, "Automatic refractive index profiling of fibers by phase analysis method using Fourier transform," Opt. Lasers Eng. 38, 509-525 (2002).
[CrossRef]

S. A. Khodier, "Refractive index of standard oils as a function of wavelength and temperature," Opt. Laser Technol. 34, 125-128 (2002).
[CrossRef]

2001 (2)

N. Barakat, H. A. El-Hennawi, E. Abd. El-Ghafar, H. El-Ghandoor, R. Hassan, and F. El-Diasty, "Three-dimensional refractive index profile of a GRIN optical waveguide using multiple beam interference fringes," Opt. Commun. 191, 39-47 (2001).
[CrossRef]

A. A. Hamza, M. A. Mabrouk, W. A. Ramadan, and M. A. Shams-Eldin, "Determination of GRIN optical fibre parameters from transverse interferograms considering the refraction of the incident ray by the fibre," Opt. Commun. 200, 131-138 (2001).
[CrossRef]

2000 (2)

H. El-Ghandoor, I. Nasser, M. A. Abd-El Rahman, and R. Hassan, "Theoretical model for the transverse interference pattern of GRIN optical fiber using a laser sheet of light," Opt. Laser Technol. 32, 281-286 (2000).
[CrossRef]

W. A. Ramadan, "On-line lens-fiber interference method for testing a thick fiber," J. Opt. A 2, 234-238 (2000).
[CrossRef]

Abd-El Rahman, M. A.

H. El-Ghandoor, I. Nasser, M. A. Abd-El Rahman, and R. Hassan, "Theoretical model for the transverse interference pattern of GRIN optical fiber using a laser sheet of light," Opt. Laser Technol. 32, 281-286 (2000).
[CrossRef]

Abdel-Hady, A.

H. El-Ghandoor, I. Nasser, A. Abdel-Hady, and A. Al-Shukri, "A comparation study of spliced optical fibers," Opt. Lasers Eng. 41, 277-287 (2004).
[CrossRef]

Al-Shukri, A.

H. El-Ghandoor, I. Nasser, A. Abdel-Hady, and A. Al-Shukri, "A comparation study of spliced optical fibers," Opt. Lasers Eng. 41, 277-287 (2004).
[CrossRef]

Barakat, N.

N. Barakat, H. A. El-Hennawi, E. Abd. El-Ghafar, H. El-Ghandoor, R. Hassan, and F. El-Diasty, "Three-dimensional refractive index profile of a GRIN optical waveguide using multiple beam interference fringes," Opt. Commun. 191, 39-47 (2001).
[CrossRef]

Behery, G. M.

H. El-Ghandoor, E. Hegazi, I. Nasser, and G. M. Behery, "Measuring the refractive index of crude oil using a capillary tube interferometer," Opt. Laser Technol. 35, 361-367 (2003).
[CrossRef]

El-Diasty, F.

N. Barakat, H. A. El-Hennawi, E. Abd. El-Ghafar, H. El-Ghandoor, R. Hassan, and F. El-Diasty, "Three-dimensional refractive index profile of a GRIN optical waveguide using multiple beam interference fringes," Opt. Commun. 191, 39-47 (2001).
[CrossRef]

El-Ghafar, E. Abd.

N. Barakat, H. A. El-Hennawi, E. Abd. El-Ghafar, H. El-Ghandoor, R. Hassan, and F. El-Diasty, "Three-dimensional refractive index profile of a GRIN optical waveguide using multiple beam interference fringes," Opt. Commun. 191, 39-47 (2001).
[CrossRef]

El-Ghandoor, H.

H. El-Ghandoor, I. Nasser, A. Abdel-Hady, and A. Al-Shukri, "A comparation study of spliced optical fibers," Opt. Lasers Eng. 41, 277-287 (2004).
[CrossRef]

H. El-Ghandoor, E. Hegazi, I. Nasser, and G. M. Behery, "Measuring the refractive index of crude oil using a capillary tube interferometer," Opt. Laser Technol. 35, 361-367 (2003).
[CrossRef]

N. Barakat, H. A. El-Hennawi, E. Abd. El-Ghafar, H. El-Ghandoor, R. Hassan, and F. El-Diasty, "Three-dimensional refractive index profile of a GRIN optical waveguide using multiple beam interference fringes," Opt. Commun. 191, 39-47 (2001).
[CrossRef]

H. El-Ghandoor, I. Nasser, M. A. Abd-El Rahman, and R. Hassan, "Theoretical model for the transverse interference pattern of GRIN optical fiber using a laser sheet of light," Opt. Laser Technol. 32, 281-286 (2000).
[CrossRef]

El-Hennawi, H. A.

N. Barakat, H. A. El-Hennawi, E. Abd. El-Ghafar, H. El-Ghandoor, R. Hassan, and F. El-Diasty, "Three-dimensional refractive index profile of a GRIN optical waveguide using multiple beam interference fringes," Opt. Commun. 191, 39-47 (2001).
[CrossRef]

El-Morsy, M. A.

M. A. El-Morsy, K. Harada, M. Itoh, and T. Yatagai, "A subfringe integration method for multiple-beam Fizeau fringe analysis," Opt. Laser Technol. 35, 223-232 (2003).
[CrossRef]

M. A. El-Morsy, T. Yatagai, A. A. Hamza, M. A. Mabrouk, and T. Z. N. Sokkar, "Automatic refractive index profiling of fibers by phase analysis method using Fourier transform," Opt. Lasers Eng. 38, 509-525 (2002).
[CrossRef]

Fukuta, M.

M. Fukuta, T. Yanagisawa, S. Miyamura, and Y. Ogi, "Concentration measurement of refrigerant/refrigeration oil mixture by refractive index," Int. J. Refrig. 27, 346-352 (2004).
[CrossRef]

Hamza, A. A.

A. A. Hamza, M. A. Mabrouk, W. A. Ramadan, and H. H. Wahba, "Core index determination of a thick fibre using lens-fibre interference (LFI) technique," Opt. Lasers Eng. 42, 121-130 (2004).
[CrossRef]

M. A. El-Morsy, T. Yatagai, A. A. Hamza, M. A. Mabrouk, and T. Z. N. Sokkar, "Automatic refractive index profiling of fibers by phase analysis method using Fourier transform," Opt. Lasers Eng. 38, 509-525 (2002).
[CrossRef]

A. A. Hamza, M. A. Mabrouk, W. A. Ramadan, and M. A. Shams-Eldin, "Determination of GRIN optical fibre parameters from transverse interferograms considering the refraction of the incident ray by the fibre," Opt. Commun. 200, 131-138 (2001).
[CrossRef]

Harada, K.

M. A. El-Morsy, K. Harada, M. Itoh, and T. Yatagai, "A subfringe integration method for multiple-beam Fizeau fringe analysis," Opt. Laser Technol. 35, 223-232 (2003).
[CrossRef]

Hassan, R.

N. Barakat, H. A. El-Hennawi, E. Abd. El-Ghafar, H. El-Ghandoor, R. Hassan, and F. El-Diasty, "Three-dimensional refractive index profile of a GRIN optical waveguide using multiple beam interference fringes," Opt. Commun. 191, 39-47 (2001).
[CrossRef]

H. El-Ghandoor, I. Nasser, M. A. Abd-El Rahman, and R. Hassan, "Theoretical model for the transverse interference pattern of GRIN optical fiber using a laser sheet of light," Opt. Laser Technol. 32, 281-286 (2000).
[CrossRef]

Hegazi, E.

H. El-Ghandoor, E. Hegazi, I. Nasser, and G. M. Behery, "Measuring the refractive index of crude oil using a capillary tube interferometer," Opt. Laser Technol. 35, 361-367 (2003).
[CrossRef]

Itoh, M.

M. A. El-Morsy, K. Harada, M. Itoh, and T. Yatagai, "A subfringe integration method for multiple-beam Fizeau fringe analysis," Opt. Laser Technol. 35, 223-232 (2003).
[CrossRef]

Khodier, S. A.

S. A. Khodier, "Refractive index of standard oils as a function of wavelength and temperature," Opt. Laser Technol. 34, 125-128 (2002).
[CrossRef]

Mabrouk, M. A.

A. A. Hamza, M. A. Mabrouk, W. A. Ramadan, and H. H. Wahba, "Core index determination of a thick fibre using lens-fibre interference (LFI) technique," Opt. Lasers Eng. 42, 121-130 (2004).
[CrossRef]

M. A. El-Morsy, T. Yatagai, A. A. Hamza, M. A. Mabrouk, and T. Z. N. Sokkar, "Automatic refractive index profiling of fibers by phase analysis method using Fourier transform," Opt. Lasers Eng. 38, 509-525 (2002).
[CrossRef]

A. A. Hamza, M. A. Mabrouk, W. A. Ramadan, and M. A. Shams-Eldin, "Determination of GRIN optical fibre parameters from transverse interferograms considering the refraction of the incident ray by the fibre," Opt. Commun. 200, 131-138 (2001).
[CrossRef]

Miyamura, S.

M. Fukuta, T. Yanagisawa, S. Miyamura, and Y. Ogi, "Concentration measurement of refrigerant/refrigeration oil mixture by refractive index," Int. J. Refrig. 27, 346-352 (2004).
[CrossRef]

Nasser, I.

H. El-Ghandoor, I. Nasser, A. Abdel-Hady, and A. Al-Shukri, "A comparation study of spliced optical fibers," Opt. Lasers Eng. 41, 277-287 (2004).
[CrossRef]

H. El-Ghandoor, E. Hegazi, I. Nasser, and G. M. Behery, "Measuring the refractive index of crude oil using a capillary tube interferometer," Opt. Laser Technol. 35, 361-367 (2003).
[CrossRef]

H. El-Ghandoor, I. Nasser, M. A. Abd-El Rahman, and R. Hassan, "Theoretical model for the transverse interference pattern of GRIN optical fiber using a laser sheet of light," Opt. Laser Technol. 32, 281-286 (2000).
[CrossRef]

Ogi, Y.

M. Fukuta, T. Yanagisawa, S. Miyamura, and Y. Ogi, "Concentration measurement of refrigerant/refrigeration oil mixture by refractive index," Int. J. Refrig. 27, 346-352 (2004).
[CrossRef]

Ramadan, W. A.

A. A. Hamza, M. A. Mabrouk, W. A. Ramadan, and H. H. Wahba, "Core index determination of a thick fibre using lens-fibre interference (LFI) technique," Opt. Lasers Eng. 42, 121-130 (2004).
[CrossRef]

A. A. Hamza, M. A. Mabrouk, W. A. Ramadan, and M. A. Shams-Eldin, "Determination of GRIN optical fibre parameters from transverse interferograms considering the refraction of the incident ray by the fibre," Opt. Commun. 200, 131-138 (2001).
[CrossRef]

W. A. Ramadan, "On-line lens-fiber interference method for testing a thick fiber," J. Opt. A 2, 234-238 (2000).
[CrossRef]

Shams-Eldin, M. A.

A. A. Hamza, M. A. Mabrouk, W. A. Ramadan, and M. A. Shams-Eldin, "Determination of GRIN optical fibre parameters from transverse interferograms considering the refraction of the incident ray by the fibre," Opt. Commun. 200, 131-138 (2001).
[CrossRef]

Sokkar, T. Z. N.

M. A. El-Morsy, T. Yatagai, A. A. Hamza, M. A. Mabrouk, and T. Z. N. Sokkar, "Automatic refractive index profiling of fibers by phase analysis method using Fourier transform," Opt. Lasers Eng. 38, 509-525 (2002).
[CrossRef]

Wahba, H. H.

A. A. Hamza, M. A. Mabrouk, W. A. Ramadan, and H. H. Wahba, "Core index determination of a thick fibre using lens-fibre interference (LFI) technique," Opt. Lasers Eng. 42, 121-130 (2004).
[CrossRef]

Yanagisawa, T.

M. Fukuta, T. Yanagisawa, S. Miyamura, and Y. Ogi, "Concentration measurement of refrigerant/refrigeration oil mixture by refractive index," Int. J. Refrig. 27, 346-352 (2004).
[CrossRef]

Yatagai, T.

M. A. El-Morsy, K. Harada, M. Itoh, and T. Yatagai, "A subfringe integration method for multiple-beam Fizeau fringe analysis," Opt. Laser Technol. 35, 223-232 (2003).
[CrossRef]

M. A. El-Morsy, T. Yatagai, A. A. Hamza, M. A. Mabrouk, and T. Z. N. Sokkar, "Automatic refractive index profiling of fibers by phase analysis method using Fourier transform," Opt. Lasers Eng. 38, 509-525 (2002).
[CrossRef]

Int. J. Refrig. (1)

M. Fukuta, T. Yanagisawa, S. Miyamura, and Y. Ogi, "Concentration measurement of refrigerant/refrigeration oil mixture by refractive index," Int. J. Refrig. 27, 346-352 (2004).
[CrossRef]

J. Opt. A (1)

W. A. Ramadan, "On-line lens-fiber interference method for testing a thick fiber," J. Opt. A 2, 234-238 (2000).
[CrossRef]

Opt. Commun. (2)

N. Barakat, H. A. El-Hennawi, E. Abd. El-Ghafar, H. El-Ghandoor, R. Hassan, and F. El-Diasty, "Three-dimensional refractive index profile of a GRIN optical waveguide using multiple beam interference fringes," Opt. Commun. 191, 39-47 (2001).
[CrossRef]

A. A. Hamza, M. A. Mabrouk, W. A. Ramadan, and M. A. Shams-Eldin, "Determination of GRIN optical fibre parameters from transverse interferograms considering the refraction of the incident ray by the fibre," Opt. Commun. 200, 131-138 (2001).
[CrossRef]

Opt. Laser Technol. (4)

S. A. Khodier, "Refractive index of standard oils as a function of wavelength and temperature," Opt. Laser Technol. 34, 125-128 (2002).
[CrossRef]

H. El-Ghandoor, E. Hegazi, I. Nasser, and G. M. Behery, "Measuring the refractive index of crude oil using a capillary tube interferometer," Opt. Laser Technol. 35, 361-367 (2003).
[CrossRef]

H. El-Ghandoor, I. Nasser, M. A. Abd-El Rahman, and R. Hassan, "Theoretical model for the transverse interference pattern of GRIN optical fiber using a laser sheet of light," Opt. Laser Technol. 32, 281-286 (2000).
[CrossRef]

M. A. El-Morsy, K. Harada, M. Itoh, and T. Yatagai, "A subfringe integration method for multiple-beam Fizeau fringe analysis," Opt. Laser Technol. 35, 223-232 (2003).
[CrossRef]

Opt. Lasers Eng. (3)

H. El-Ghandoor, I. Nasser, A. Abdel-Hady, and A. Al-Shukri, "A comparation study of spliced optical fibers," Opt. Lasers Eng. 41, 277-287 (2004).
[CrossRef]

A. A. Hamza, M. A. Mabrouk, W. A. Ramadan, and H. H. Wahba, "Core index determination of a thick fibre using lens-fibre interference (LFI) technique," Opt. Lasers Eng. 42, 121-130 (2004).
[CrossRef]

M. A. El-Morsy, T. Yatagai, A. A. Hamza, M. A. Mabrouk, and T. Z. N. Sokkar, "Automatic refractive index profiling of fibers by phase analysis method using Fourier transform," Opt. Lasers Eng. 38, 509-525 (2002).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Ray tracing of selected rays impinging on the upper semisphere of the capillary tube. The locations of three possible fringes are shown on the screen at right (Fig. 3 in Ref. 1).

Fig. 2
Fig. 2

Ray tracing inside the optical fiber and in the surrounding medium where A ( X A , Y A ) , B ( X B , Y B ) , D ( X D , Y D ) , F ( X F , Y F ) , and H ( X H , Y H ) (Fig. 2 in Ref. 2).

Fig. 3
Fig. 3

Experimental setup to produce the interference pattern.

Fig. 4
Fig. 4

Schematic diagram represents the path of the incident beam passing through the capillary tube.

Fig. 5
Fig. 5

Ray-tracing diagram from cylindrical lens to the screen. Rays 1, 2, and 3 are typical rays.

Fig. 6
Fig. 6

OPL as a function of the ray position on the cylindrical lens.

Fig. 7
Fig. 7

Ray position on the screen as a function of the position on the cylindrical lens.

Fig. 8
Fig. 8

Experimental and theoretical estimated interference pattern. (a) Experimental pattern, (b) experimental pattern after digital processing, (c) estimated pattern, (d) intensity changes with the experimental fringe position on the screen, (e) intensity changes with the simulated fringe position on the screen.

Tables (1)

Tables Icon

Table 1 Refractive Indices of Abbe Refractometer and Capillary Tube Interferometer Methods a

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

OPL = MAn a + ABn cy + BCn a + 2 CDn g +   DEn + FGn a ,
MA = R cy - R cy 2 - y 2 ,
x A = [ R cy 2 - y A 2 + ( R + l 1 + d cy - R cy ) ] ,
y A = y ,
x B = ( R + l 1 ) ,
y B = k AB ( x B - x A ) + y A ,
x C = - k BC b 1 - R 2 + k BC 2 R 2 - b 1 2 1 + k BC 2 ,
y C = k BC x C + b 1 .
x D = - k CD - r 2 + k CD 2 r 2 - b 2 2 1 + k CD 2 ,
y D = k CD x D + b 2 .
x E = - k DE b 3 + R in 2 + k DE 2 R in 2 - b 3 2 1 + k DE 2 ,
y E = k DE x E + b 3 ,
FG = ( l 2 - x F ) cos ( φ F - α F ) ,
s = ( l 2 - x F ) tan ( - φ F + α F ) + y F ,
x H = - k CF ( y C - k CF x C ) 1 + k CF 2 ,
y H = y C - k CF x C 1 + k CF 2 .
δ = 2 π λ Δ ,
I = 4 A 2 cos 2 ( δ 2 ) ,

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