Abstract

A mathematical model is derived for the focusing method which relates refractive-index and deflection function errors to errors in the measured light intensity. The model predicts that refractive-index measurement noise increases linearly from zero at the outside diameter of the fiber to a maximum value at the center of the core. Measurement errors for the focusing method determined by experiment are in close agreement with predictions obtained using the theoretical model.

© 1985 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. L. Chu, “Nondestructive Measurement of Index Profile of an Optical Fibre Preform,” Electron. Lett. 24, 736 (1977).
    [CrossRef]
  2. D. Peri, P. L. Chu, T. W. Whitbread, “Direct Display of the Deflection Function of Optical Fiber Preforms,” Appl. Opt. 21, 809 (1982).
    [CrossRef] [PubMed]
  3. P. L. Chu, T. W. Whitbread, “Nondestructive Determination of Refractive Index Profile of an Optical Fiber: Fast Fourier Transform Method,” Appl. Opt. 18, 1117 (1979).
    [CrossRef] [PubMed]
  4. C. Saekeang, P. L. Chu, T. W. Whitbread, “Nondestructive Measurement of Refractive-Index Profile and Cross-sectional Geometry of Optical Fiber Preforms,” Appl. Opt. 19, 2025 (1980).
    [CrossRef] [PubMed]
  5. D. Marcuse, “Refractive Index Determination by the Focusing Method,” Appl. Opt. 18, 9 (1979).
    [CrossRef] [PubMed]
  6. H. M. Presby, D. Marcuse, W. G. French, “Refractive Index Profiling of Single Mode Optical Fibres and Preforms,” Appl. Opt. 18, 4006 (1979).
    [CrossRef] [PubMed]
  7. P. L. Chu, T. W. Whitbread, P. Y. P. Cheu, “Noise Analysis of Refractive Index Profile Measurements of Optical Fibre and Preform,” Appl. Sci. Res. 41, 289 (1984).
    [CrossRef]
  8. D. Marcuse, H. M. Presby, “Focusing Method for Nondestructive Measurement of Optical Fiber Index Profiles,” Appl. Opt. 18, 14 (1979).
    [CrossRef] [PubMed]
  9. A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1984).

1984 (1)

P. L. Chu, T. W. Whitbread, P. Y. P. Cheu, “Noise Analysis of Refractive Index Profile Measurements of Optical Fibre and Preform,” Appl. Sci. Res. 41, 289 (1984).
[CrossRef]

1982 (1)

1980 (1)

1979 (4)

1977 (1)

P. L. Chu, “Nondestructive Measurement of Index Profile of an Optical Fibre Preform,” Electron. Lett. 24, 736 (1977).
[CrossRef]

Cheu, P. Y. P.

P. L. Chu, T. W. Whitbread, P. Y. P. Cheu, “Noise Analysis of Refractive Index Profile Measurements of Optical Fibre and Preform,” Appl. Sci. Res. 41, 289 (1984).
[CrossRef]

Chu, P. L.

French, W. G.

Marcuse, D.

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1984).

Peri, D.

Presby, H. M.

Saekeang, C.

Whitbread, T. W.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Single-mode fiber profile measurement by the focusing method. Deflection function geometry.

Fig. 2
Fig. 2

Single-mode fiber profile measurement by the focusing method. Deflection function plots with integration drift both present and removed. Drift exaggerated for illustration.

Fig. 3
Fig. 3

Single-mode fiber profile measurement by the focusing method. Broken line—error predicted by Eq. (17). Solid line—error determined from twenty profile measurements.

Fig. 4
Fig. 4

Single-mode fiber profile measurement by the focusing method. Broken line—error predicted by Eq. (18). Solid line—error determined from twenty profile measurements.

Equations (36)

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

n ( r ) = n 0 + n 0 π r b Φ ( y ) · d y ( y 2 r 2 ) 1 / 2 ,
y = 0 t 1 P 1 ( U ) · d u = 0 t 2 P 2 ( u ) · d u .
Φ tan Φ = t 2 t 1 L ,
Δ n ( r ) = n 0 π r b Δ Φ ( y ) · d y ( y 2 r 2 ) 1 / 2 .
var [ n ( r ) ] = n 0 2 π 2 r b r b R ϕ ( x , y ) · d x · d y ( x 2 r 2 ) 1 / 2 ( y 2 r 2 ) 1 / 2 ,
y = 0 t 1 + Δ t 1 [ P 1 ( u ) + Δ P 1 ( u ) ] · d u = 0 t 2 + Δ t 2 [ P 2 ( u ) + Δ P 2 ( u ) ] · d u ,
Δ t 1 = 0 t 1 Δ P 1 ( u ) · d u P 1 ( t 1 ) ,
Δ t 2 = 0 t 2 Δ P 2 ( u ) · d u P 2 ( t 2 ) .
Δ Φ Δ t 2 Δ t 1 L ,
Δ Φ ( y ) 1 L 0 y [ Δ P 1 ( u ) P 1 ( y ) Δ P 2 ( u ) P 2 ( y ) ] · d u .
Δ Φ ( b ) 1 L 0 b [ Δ P 1 ( u ) P 1 ( b ) Δ P 2 ( u ) P 2 ( b ) ] · d u .
Δ Φ ( y ) = 0 y [ p 1 ( u ) p 2 ( u ) ] · d u y b 0 b [ p 1 ( u ) p 2 ( u ) ] · d u ,
p 1 ( u ) = Δ P 1 ( u ) L P 1 ( y ) and p 2 ( u ) = Δ P 2 ( u ) L P 2 ( y ) .
N ( y ) = 0 y [ p 1 ( u ) p 2 ( u ) ] · d u .
Δ Φ ( y ) = N ( y ) y b N ( b )
R ϕ ( x , y ) = E { [ N ( x ) x b N ( b ) ] [ N ( y ) y b N ( b ) ] } = R N ( x , y ) x b R N ( b , y ) y b R N ( x , b ) + x y b 2 R N ( b , b ) ,
R N ( x , y ) = K σ 2 y + O [ K 2 ] x > y = K σ 2 x + O [ K 2 ] y > x ,
R ϕ ( x , y ) = K σ 2 y [ 1 x / b ] x > y , R ϕ ( x , y ) = K σ 2 x [ 1 y / b ] y > x ,
S D [ Φ ] = [ R ϕ ( x , x ) ] 1 / 2 = σ ( K b ) 1 / 2 ( x / b ) 1 / 2 [ 1 x / b ] 1 / 2 .
S D [ n ( r ) ] = ( K B ) 1 / 2 · ( σ n 0 / π ) [ 1 r / b ] .
S D [ n ( 0 ) ] = ( K b ) 1 / 2 · ( σ n 0 / π ) .
σ = σ p L { 1 + var [ P ( y ) ] } 1 / 2 1 . 05 σ p L ,
R N ( x , y ) = 0 x 0 y E { [ p 1 ( u ) p 2 ( u ) ] · [ p 1 ( υ ) p 2 ( υ ) ] } · d u · d υ = 2 0 x 0 y R ( u υ ) · d u · d υ .
R N ( x , y ) = 2 [ 0 y ( y z ) · R ( z ) · d z + 0 x y y · R ( z ) · d z + x y x ( x z ) · R ( z ) · d z ] x > y , R N ( x , y ) = 2 [ 0 x ( x z ) · R ( z ) · d z + 0 y x x · R ( z ) · d z + y x y ( y z ) · R ( z ) · d z ] y > x .
R ( u ) = σ 2 exp ( | u | / ρ ) ,
R n ( x , y ) 2 σ 2 = 2 ρ y ρ 2 { 1 exp ( y / ρ ) exp ( x / ρ ) + exp [ ( x y ) / ρ ] } x > y , R n ( x , y ) 2 σ 2 = 2 ρ x ρ 2 { 1 exp ( y / ρ ) exp ( x / ρ ) + exp [ ( y x ) / ρ ] | y > x .
p r ( u ) = n = n = + p ( n ) · h ( u n X ) ,
h ( u ) = ( 1 | u | / X ) | u | < X , h ( u ) = 0 | u | > X ,
R ( u , υ ) = 2 · σ 2 n = n = h ( u n X ) · h ( υ n X ) ,
R n ( x , y ) = 2 · σ 2 n = n = + 0 x h ( u n X ) · d u 0 y h ( υ n X ) · d υ .
I ( n , x ) = 0 x h ( u n X ) · d u ,
I ( n , x ) = 0 x / X < ( n 1 ) = [ x ( n 1 ) X ] 2 2 X ( n 1 ) < x / X < n , = X [ ( n + 1 ) X x ] 2 2 X n < x / X < ( n + 1 ) = X ( n + 1 ) < x / X .
R n ( x , y ) = 2 · σ 2 X n = 0 n = M I ( n , x ) + Δ ,
R n ( x , y ) 2 σ 2 = X y + O [ X 2 ] x > y , R n ( x , y ) 2 σ 2 = X x + O [ X 2 ] y > x ,
var [ n ( r ) ] K ( σ n 0 π ) 2 = x = r b d x ( x 2 r 2 ) 1 / 2 y = r x y ( 1 x / b ) · d y ( y 2 r 2 ) 1 / 2 + x = r b d x ( x 2 r 2 ) 1 / 2 y = x b x ( 1 y / b ) · d y ( y 2 r 2 ) 1 / 2 .
S D [ ( n ( r ) ] = ( K b ) 1 / 2 ( σ n 0 / π ) · ( 1 r / b ) .

Metrics