Abstract

A fiber diameter variation measurement system is described which is capable of measuring transparent fibers with 0.02% diameter resolution and 6-μm axial resolution at a measurement rate of 1 kHz and with a working distance of >100 mm. The principles of its operation are discussed in detail, and experimental confirmation of its performance is reported. A theoretical calculation of the optimum obtainable diameter resolution for a given set of experimental parameters is also presented.

© 1985 Optical Society of America

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References

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  1. M. M. Fejer, J. L. Nightingale, G. A. Magel, R. L. Byer, “Laser Heated Miniature Pedestal Growth Apparatus for Single Crystal Optical Fibers,” Rev. Sci. Instrum. 55, 1791 (1984).
    [CrossRef]
  2. J. Stone, C. A. Burrus, “Self Contained LED-Pumped Single Crystal Nd:YAG Fiber Lasers,” Fiber Integrated Opt. 2, 19 (1979).
    [CrossRef]
  3. T. J. Bridges, J. S. Hasiak, A. R. Strand, “Single Crystal AgBr Infrared Optical Fibers,” Opt. Lett. 5, 85 (1980).
    [CrossRef] [PubMed]
  4. G. D. Boyd, L. A. Coldren, R. N. Thurston, “Acoustic Clad Fiber Delay Lines,” IEEE Trans. Sonics Ultrason. SU-24, 246 (1977).
    [CrossRef]
  5. C. A. Burrus, J. Stone, “Single Crystal Fiber Optical Devices: A Nd:YAG Fiber Laser,” Appl. Phys. Lett. 26, 318 (1975).
    [CrossRef]
  6. L. S. Watkins, “Scattering from Side-Illuminated Clad Class Fibers for Determination of Fiber Parameters,” J. Opt. Soc. Am. 64, 767 (1974).
    [CrossRef]
  7. D. H. Smithgall, L. S. Watkins, R. E. Frazee, “High-Speed Noncontact Fiber-Diameter Measurement Using Forward Light Scattering,” Appl. Opt. 16, 2395 (1977).
    [CrossRef] [PubMed]
  8. M. A. G. Abushagur, N. George, “Measurement of Optical Fiber Diameter Using the Fast Fourier Transform,” Appl. Opt. 19, 2031 (1980).
    [CrossRef] [PubMed]
  9. A similar analysis for determination of fiber location in silicone coatings appears in D. H. Smithgall, “Light Scattering Model for the Determination of Fiber Location in Silicone Coatings,” Appl. Opt. 21, 1326 (1982).
    [CrossRef] [PubMed]

1984

M. M. Fejer, J. L. Nightingale, G. A. Magel, R. L. Byer, “Laser Heated Miniature Pedestal Growth Apparatus for Single Crystal Optical Fibers,” Rev. Sci. Instrum. 55, 1791 (1984).
[CrossRef]

1982

1980

1979

J. Stone, C. A. Burrus, “Self Contained LED-Pumped Single Crystal Nd:YAG Fiber Lasers,” Fiber Integrated Opt. 2, 19 (1979).
[CrossRef]

1977

G. D. Boyd, L. A. Coldren, R. N. Thurston, “Acoustic Clad Fiber Delay Lines,” IEEE Trans. Sonics Ultrason. SU-24, 246 (1977).
[CrossRef]

D. H. Smithgall, L. S. Watkins, R. E. Frazee, “High-Speed Noncontact Fiber-Diameter Measurement Using Forward Light Scattering,” Appl. Opt. 16, 2395 (1977).
[CrossRef] [PubMed]

1975

C. A. Burrus, J. Stone, “Single Crystal Fiber Optical Devices: A Nd:YAG Fiber Laser,” Appl. Phys. Lett. 26, 318 (1975).
[CrossRef]

1974

Abushagur, M. A. G.

Boyd, G. D.

G. D. Boyd, L. A. Coldren, R. N. Thurston, “Acoustic Clad Fiber Delay Lines,” IEEE Trans. Sonics Ultrason. SU-24, 246 (1977).
[CrossRef]

Bridges, T. J.

Burrus, C. A.

J. Stone, C. A. Burrus, “Self Contained LED-Pumped Single Crystal Nd:YAG Fiber Lasers,” Fiber Integrated Opt. 2, 19 (1979).
[CrossRef]

C. A. Burrus, J. Stone, “Single Crystal Fiber Optical Devices: A Nd:YAG Fiber Laser,” Appl. Phys. Lett. 26, 318 (1975).
[CrossRef]

Byer, R. L.

M. M. Fejer, J. L. Nightingale, G. A. Magel, R. L. Byer, “Laser Heated Miniature Pedestal Growth Apparatus for Single Crystal Optical Fibers,” Rev. Sci. Instrum. 55, 1791 (1984).
[CrossRef]

Coldren, L. A.

G. D. Boyd, L. A. Coldren, R. N. Thurston, “Acoustic Clad Fiber Delay Lines,” IEEE Trans. Sonics Ultrason. SU-24, 246 (1977).
[CrossRef]

Fejer, M. M.

M. M. Fejer, J. L. Nightingale, G. A. Magel, R. L. Byer, “Laser Heated Miniature Pedestal Growth Apparatus for Single Crystal Optical Fibers,” Rev. Sci. Instrum. 55, 1791 (1984).
[CrossRef]

Frazee, R. E.

George, N.

Hasiak, J. S.

Magel, G. A.

M. M. Fejer, J. L. Nightingale, G. A. Magel, R. L. Byer, “Laser Heated Miniature Pedestal Growth Apparatus for Single Crystal Optical Fibers,” Rev. Sci. Instrum. 55, 1791 (1984).
[CrossRef]

Nightingale, J. L.

M. M. Fejer, J. L. Nightingale, G. A. Magel, R. L. Byer, “Laser Heated Miniature Pedestal Growth Apparatus for Single Crystal Optical Fibers,” Rev. Sci. Instrum. 55, 1791 (1984).
[CrossRef]

Smithgall, D. H.

Stone, J.

J. Stone, C. A. Burrus, “Self Contained LED-Pumped Single Crystal Nd:YAG Fiber Lasers,” Fiber Integrated Opt. 2, 19 (1979).
[CrossRef]

C. A. Burrus, J. Stone, “Single Crystal Fiber Optical Devices: A Nd:YAG Fiber Laser,” Appl. Phys. Lett. 26, 318 (1975).
[CrossRef]

Strand, A. R.

Thurston, R. N.

G. D. Boyd, L. A. Coldren, R. N. Thurston, “Acoustic Clad Fiber Delay Lines,” IEEE Trans. Sonics Ultrason. SU-24, 246 (1977).
[CrossRef]

Watkins, L. S.

Appl. Opt.

Appl. Phys. Lett.

C. A. Burrus, J. Stone, “Single Crystal Fiber Optical Devices: A Nd:YAG Fiber Laser,” Appl. Phys. Lett. 26, 318 (1975).
[CrossRef]

Fiber Integrated Opt.

J. Stone, C. A. Burrus, “Self Contained LED-Pumped Single Crystal Nd:YAG Fiber Lasers,” Fiber Integrated Opt. 2, 19 (1979).
[CrossRef]

IEEE Trans. Sonics Ultrason.

G. D. Boyd, L. A. Coldren, R. N. Thurston, “Acoustic Clad Fiber Delay Lines,” IEEE Trans. Sonics Ultrason. SU-24, 246 (1977).
[CrossRef]

J. Opt. Soc. Am.

Opt. Lett.

Rev. Sci. Instrum.

M. M. Fejer, J. L. Nightingale, G. A. Magel, R. L. Byer, “Laser Heated Miniature Pedestal Growth Apparatus for Single Crystal Optical Fibers,” Rev. Sci. Instrum. 55, 1791 (1984).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Schematic of the fiber diameter measurement approach. An incident plane wave of wavelength λ scatters off a fiber. The resulting interference pattern is projected by a lens of focal length f onto a photodiode array with element spacing s and element width d.

Fig. 2
Fig. 2

Geometry of reflected and transmitted rays that interfere to form the fringe pattern. The path difference between AB and CD is given by Λ(θ) in Eq. (2) when θt = θr = θ. Angles and lengths indicated are used in the Appendix to calculate scattering amplitudes a1(θ) and a2(θ) referred to in Eq. (1).

Fig. 3
Fig. 3

Plot of sensitivity function g(n,θ) defined by Eqs. (6) and (7). Solid lines show g(n,θ) vs θ for various values of n.

Fig. 4
Fig. 4

Schematic diagram of the fiber diameter measurement optical system.

Fig. 5
Fig. 5

Schematic diagram of the fiber diameter measurement electronic system.

Fig. 6
Fig. 6

Typical electronic signals: (a) output of a section of the photodiode array; (b) window signal bracketing fringe to be tracked; (c) output of boxcar integrator equal to the average value of the fringe selected by the window signal.

Fig. 7
Fig. 7

Scattering angle for highest resolution for several values of n vs log, where is a dimensionless parameter defined by Eq. (19) characterizing the relative importance of nonuniform photodiode responsivity and noise. Note that the optimum angle is not a function of fiber diameter.

Fig. 8
Fig. 8

Logarithm of normalized optimized resolution ( Δ ρ / ρ ) ¯ opt [defined by Eq. (20)] for several values of n vs log, [ is defined by Eq. (19).]

Fig. 9
Fig. 9

Diameter vs length of a 350-μm diam glass fiber. The solid line was measured using the fringe tracking apparatus; the closed circles were measured using an optical microscope.

Fig. 10
Fig. 10

Diameter change vs length for a 50-μm diam ruby fiber grown under open-loop conditions by the fiber growth apparatus described in Ref. 1.

Equations (39)

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P ( θ ) = a 1 ( θ ) + a 2 ( θ ) cos k Λ ( θ ) ,
Λ ( θ ) = 2 ρ [ sin 1 2 θ + ( n 2 + 1 2 n cos 1 2 θ ) 1 / 2 ] + λ / 4 ,
k Λ ( θ j ) = 2 π ( j + j 0 ) ,
Δ x = θ j ρ f Δ ρ .
Δ ρ min = ( s f ) | ( θ j ρ ) | 1 .
θ j ρ = 2 ρ sin 1 2 θ j + ( n 2 + 1 2 n cos 1 2 θ j ) 1 / 2 cos 1 2 θ j + n ( n 2 + 1 2 n cos 1 2 θ j ) 1 / 2 sin 1 2 θ j .
θ j ρ = 1 ρ g 1 ( n , θ j ) ,
( Δ ρ ρ ) min = s f g ( n , θ j ) .
a 1 ( θ ) = a 2 ( θ ) = P 0 λ 2 π 5 / 2 w 0 θ 2 ,
Λ ( θ ) = 2 ρ θ ( λ / 2 ) .
g ( n , θ ) = g ( θ ) 1 / θ .
V m = d f R m τ P ( θ m ) + υ m ,
δ V m = V m + 1 V m
δ V m = ( d f ) τ [ P ( θ m + 1 ) R m + 1 P ( θ m ) R m ] + ( υ m + 1 υ m ) .
δ V m = ( d f ) τ [ s f R m d P d θ | θ m + ( R m + 1 R m ) P ( θ m ) ] + ( υ m + 1 υ m ) .
δ V = U + E ,
U = s d f 2 τ R m a 2 ( θ m ) k d Λ d θ | θ m , E = d f τ ( R m + 1 R m ) a 1 ( θ m ) + ( υ m + 1 υ m ) .
U ( f opt , θ ) = E ( f opt , θ ) .
0 = [ a ¯ 2 h ( θ ) / g 2 ] ( Δ ρ / ρ ) 2 + ( a ¯ 1 / 2 π g ) ( δ R / R ) ( λ / ρ ) ( Δ ρ / ρ ) + ( 2 2 π ) 1 ( E 0 / P 0 τ ) ( s / d ) ( w 0 / ρ ) ( λ / ρ ) ,
h ( θ ) = 1 ρ Λ θ = cos ( 1 2 θ ) + n sin ( 1 2 θ ) ( n 2 + 1 2 n cos 1 2 θ ) 1 / 2 .
ε = ( 2 π 3 ) 1 / 4 ( δ R R ) ( P 0 τ E 0 d s λ w 0 ) 1 / 2 ,
( Δ ρ / ρ ) ¯ opt = ( Δ ρ / ρ ) opt ( 2 2 π P 0 τ E 0 d s ρ w 0 ρ λ ) 1 / 2
( Δ ρ ρ ) opt = π 2 ( E 0 P 0 τ w 0 ρ s d ) 1 / 2 .
P ( θ ) = P r ( θ ) + P t ( θ ) + 2 P r ( θ ) P t ( θ ) cos k Λ ( θ ) ,
P r ( θ ) = I R ( θ ) | d x r ( θ ) d θ r | ,
P t ( θ ) = I T ( θ ) | d x t ( θ ) d θ t | ,
d ϕ d x r = 1 ρ sin ϕ .
d θ r d x r | θ r = θ = 2 ρ sin 1 2 θ r .
θ t = 2 ( β 0 β 1 ) ,
β 1 = sin 1 ( x t / n ρ ) ,
β 0 = sin 1 ( x t / ρ ) ,
d θ t d x t | θ t = θ = 2 ρ ( 1 cos β 0 1 n cos β 1 ) .
R ( θ ) = ( sin 1 2 θ n cos γ sin 1 2 θ + n cos γ ) 2 ,
T ( θ ) = [ 4 n cos β 0 cos β 1 ( cos β 0 + n cos β 1 ) 2 ] 2 ,
P ( θ ) = P 0 ( 2 π ) 1 / 2 ( ρ w 0 ) [ a ¯ 1 ( θ ) + a ¯ 2 ( θ ) cos k Λ ( θ ) ] ,
a ¯ 1 ( θ ) = 1 2 [ R ( θ ) sin ( 1 2 θ ) + T ( θ ) ( 1 cos β 0 1 n cos β 1 ) 1 ] ;
a ¯ 2 ( θ ) = [ R ( θ ) T ( θ ) sin ( 1 2 θ ) ( 1 cos β 0 1 n cos β 1 ) 1 ] 1 / 2 .
a 1 ( θ ) = P 0 ( 2 π ) 1 / 2 ( ρ w 0 ) a ¯ 1 ( θ ) ,
a 2 ( θ ) = P 0 ( 2 π ) 1 / 2 ( ρ w 0 ) a ¯ 2 ( θ ) .

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