Abstract

A simple nondestructive method is described for determining the spot size w0 (width of the fundamental mode)—one of the most important quantities to characterize monomode optical fibers. The theory involves the definition of effective values of core radius and normalized frequency. Thereby, graded-index fibers can be replaced by equivalent step-index fibers. The experiment consists in illuminating the fiber perpendicularly to its axis and determining the position of the first minimum in the diffraction pattern. In this way, just one measurement yields w0 regardless of the index profile and for any operating wavelength, provided the cutoff wavelength is known.

© 1979 Optical Society of America

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References

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  1. W. A. Gambling, H. Matsumura, Opt. Quantum Electron. 10, 31 (1978).
    [CrossRef]
  2. D. Marcuse, J. Opt. Soc. Am. 68, 103 (1978).
    [CrossRef]
  3. D. Marcuse, Bell Syst. Tech. J. 56, 703 (1977).
  4. J. Albrecht, Ph.D. Thesis, University Bochum, Germany (1977).
  5. W. A. Gambling, D. N. Payne, H. Matsumura, R. B. Dyott, Microwaves Opt. Acoust. 1, 13 (1976).
    [CrossRef]
  6. T. Okoshi, K. Hotate, Appl. Opt. 15, 2756 (1976).
    [CrossRef] [PubMed]
  7. K. Iga, Y. Kokubun, Appl. Opt. 17, 1972 (1978).
    [CrossRef] [PubMed]
  8. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  9. E. Brinkmeyer, Appl. Opt. 16, 2801 (1977).
    [CrossRef]
  10. F. Oberhettinger, Tables of Bessel Transforms (Springer, New York, 1972), p. 33, No. 4.5.
  11. M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 480, Eq. (11.1.1).
  12. S. H. Wemple, D. A. Pinnow, T. C. Rich, R. E. Jaeger, L. G. Van Uitert, J. Appl. Phys. 44, 5432 (1973).
    [CrossRef]
  13. D. Gloge, Appl. Opt. 10, 2252 (1971).
    [CrossRef] [PubMed]
  14. The index of refraction liquids in steps of 0.002 were supplied by R. P. Cargille Laboratories, Inc., Cedar Grove, N.J. 07009.
  15. K. Hotate, T. Okoshi, Electron. Lett. 14, 246 (1978).
    [CrossRef]

1978

W. A. Gambling, H. Matsumura, Opt. Quantum Electron. 10, 31 (1978).
[CrossRef]

D. Marcuse, J. Opt. Soc. Am. 68, 103 (1978).
[CrossRef]

K. Iga, Y. Kokubun, Appl. Opt. 17, 1972 (1978).
[CrossRef] [PubMed]

K. Hotate, T. Okoshi, Electron. Lett. 14, 246 (1978).
[CrossRef]

1977

E. Brinkmeyer, Appl. Opt. 16, 2801 (1977).
[CrossRef]

D. Marcuse, Bell Syst. Tech. J. 56, 703 (1977).

1976

W. A. Gambling, D. N. Payne, H. Matsumura, R. B. Dyott, Microwaves Opt. Acoust. 1, 13 (1976).
[CrossRef]

T. Okoshi, K. Hotate, Appl. Opt. 15, 2756 (1976).
[CrossRef] [PubMed]

1973

S. H. Wemple, D. A. Pinnow, T. C. Rich, R. E. Jaeger, L. G. Van Uitert, J. Appl. Phys. 44, 5432 (1973).
[CrossRef]

1971

Abramovitz, M.

M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 480, Eq. (11.1.1).

Albrecht, J.

J. Albrecht, Ph.D. Thesis, University Bochum, Germany (1977).

Brinkmeyer, E.

Dyott, R. B.

W. A. Gambling, D. N. Payne, H. Matsumura, R. B. Dyott, Microwaves Opt. Acoust. 1, 13 (1976).
[CrossRef]

Gambling, W. A.

W. A. Gambling, H. Matsumura, Opt. Quantum Electron. 10, 31 (1978).
[CrossRef]

W. A. Gambling, D. N. Payne, H. Matsumura, R. B. Dyott, Microwaves Opt. Acoust. 1, 13 (1976).
[CrossRef]

Gloge, D.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Hotate, K.

K. Hotate, T. Okoshi, Electron. Lett. 14, 246 (1978).
[CrossRef]

T. Okoshi, K. Hotate, Appl. Opt. 15, 2756 (1976).
[CrossRef] [PubMed]

Iga, K.

Jaeger, R. E.

S. H. Wemple, D. A. Pinnow, T. C. Rich, R. E. Jaeger, L. G. Van Uitert, J. Appl. Phys. 44, 5432 (1973).
[CrossRef]

Kokubun, Y.

Marcuse, D.

D. Marcuse, J. Opt. Soc. Am. 68, 103 (1978).
[CrossRef]

D. Marcuse, Bell Syst. Tech. J. 56, 703 (1977).

Matsumura, H.

W. A. Gambling, H. Matsumura, Opt. Quantum Electron. 10, 31 (1978).
[CrossRef]

W. A. Gambling, D. N. Payne, H. Matsumura, R. B. Dyott, Microwaves Opt. Acoust. 1, 13 (1976).
[CrossRef]

Oberhettinger, F.

F. Oberhettinger, Tables of Bessel Transforms (Springer, New York, 1972), p. 33, No. 4.5.

Okoshi, T.

K. Hotate, T. Okoshi, Electron. Lett. 14, 246 (1978).
[CrossRef]

T. Okoshi, K. Hotate, Appl. Opt. 15, 2756 (1976).
[CrossRef] [PubMed]

Payne, D. N.

W. A. Gambling, D. N. Payne, H. Matsumura, R. B. Dyott, Microwaves Opt. Acoust. 1, 13 (1976).
[CrossRef]

Pinnow, D. A.

S. H. Wemple, D. A. Pinnow, T. C. Rich, R. E. Jaeger, L. G. Van Uitert, J. Appl. Phys. 44, 5432 (1973).
[CrossRef]

Rich, T. C.

S. H. Wemple, D. A. Pinnow, T. C. Rich, R. E. Jaeger, L. G. Van Uitert, J. Appl. Phys. 44, 5432 (1973).
[CrossRef]

Stegun, I. A.

M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 480, Eq. (11.1.1).

Van Uitert, L. G.

S. H. Wemple, D. A. Pinnow, T. C. Rich, R. E. Jaeger, L. G. Van Uitert, J. Appl. Phys. 44, 5432 (1973).
[CrossRef]

Wemple, S. H.

S. H. Wemple, D. A. Pinnow, T. C. Rich, R. E. Jaeger, L. G. Van Uitert, J. Appl. Phys. 44, 5432 (1973).
[CrossRef]

Appl. Opt.

Bell Syst. Tech. J.

D. Marcuse, Bell Syst. Tech. J. 56, 703 (1977).

Electron. Lett.

K. Hotate, T. Okoshi, Electron. Lett. 14, 246 (1978).
[CrossRef]

J. Appl. Phys.

S. H. Wemple, D. A. Pinnow, T. C. Rich, R. E. Jaeger, L. G. Van Uitert, J. Appl. Phys. 44, 5432 (1973).
[CrossRef]

J. Opt. Soc. Am.

Microwaves Opt. Acoust.

W. A. Gambling, D. N. Payne, H. Matsumura, R. B. Dyott, Microwaves Opt. Acoust. 1, 13 (1976).
[CrossRef]

Opt. Quantum Electron.

W. A. Gambling, H. Matsumura, Opt. Quantum Electron. 10, 31 (1978).
[CrossRef]

Other

J. Albrecht, Ph.D. Thesis, University Bochum, Germany (1977).

F. Oberhettinger, Tables of Bessel Transforms (Springer, New York, 1972), p. 33, No. 4.5.

M. Abramovitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), p. 480, Eq. (11.1.1).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

The index of refraction liquids in steps of 0.002 were supplied by R. P. Cargille Laboratories, Inc., Cedar Grove, N.J. 07009.

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Figures (4)

Fig. 1
Fig. 1

Schematic representation of the disturbance of a laser beam normally incident on a fiber which is immersed into index-matching liquid.

Fig. 2
Fig. 2

Intensity distribution in the diffraction pattern caused by a monomode fiber for three values of the profile parameter α.

Fig. 3
Fig. 3

Normalized spot size w0/aeff as a function of the effective normalized frequency Veff for some values of the profile parameter α. The graphs were calculated from Ref. 1.

Fig. 4
Fig. 4

Same plot as in Fig. 3 but calculated from Refs. 2 and 3.

Tables (4)

Tables Icon

Table I Comparison of the Numerical Values of the Quantity pmin with Those Calculated from the Empirical Formula Eq. (11) for Various Profile Parameters α

Tables Icon

Table II Single-Mode Limit as a Function of the Profile Exponent α

Tables Icon

Table III Comparison of the Normalized Near-Field Width with the Corresponding Normalized Spot Size for Step-index Fibers

Tables Icon

Table IV Comparison of the Normalized Half-intensity Angle ka sinθh of the Far Field for Step-Index Fibers with the Term 1.177/(w0/a) Emerging from the Gaussian Approximation of the Near Field

Equations (18)

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n ( r ) = n 0 [ 1 - Δ ( r / a ) α ] for r < a = n 0 ( 1 - Δ ) = n c l for r > a .
U 0 ( x ) = exp [ j ϕ ( x ) ] ,
Δ n ( x , z ) = n ( x , z ) - n cl .
U 0 , w ( x ) = { 0 for x a w 1 elsewhere .
F ( f x ) = F [ U 0 ( x ) ] = - U 0 ( x ) exp ( - j 2 π f x x ) d x ,
F ( f x ) = F [ ϕ ( x ) ] .
F ( f x ) = B [ k Δ n ( r ) ] = 2 π k Δ n ( r ) r J 0 ( 2 π r f x ) d r .
F ( f x ) = 2 π k n 0 Δ 0 a [ 1 - ( r / a ) α ] r J 0 ( 2 π r f x ) d r = 2 π k n 0 Δ α 2 [ 0 1 q J 0 ( p q ) d q - 0 1 q α + 1 J 0 ( p q ) d q ] = 2 π k n 0 Δ a 2 [ I 1 ( p ) - I 2 ( p ) ] ,
I 1 ( p ) = J 1 ( p ) / p , I 2 ( p ) = Γ ( α 2 + 1 ) p Γ ( - α 2 ) · k = 0 ( 2 k + 1 ) Γ ( - α 2 + k ) Γ ( α 2 + 2 + k ) J 2 k + 1 ( p ) ,
F ( f x ) = 2 π k n 0 Δ a 2 α ( 2 + α ) p k = 0 ( 2 k + 1 ) c k J 2 k + 1 ( p ) ,
c 0 = 1 , c 1 = 1 / [ 2 + ( α / 2 ) ] , c m = c m - 1 m - 1 - ( α / 2 ) m + 1 + ( α / 2 )             for m 2.
p min ( α ) = p min ( ) ( 2.5 + α 0.5 + α ) 1 / 2 ,
a eff = a ( 0.5 + α 2.5 + α ) 1 / 2 = p min ( ) 2 π f x , min = 3.832 λ 2 π sin θ min ,
w 0 a eff = w 0 a ( 2.5 + α 0.5 + α ) 1 / 2
V eff = V · V co ( ) V co ( α ) = V · 2.405 V co ( α ) ,
w 0 a eff = { 0.283 + 3.422 V eff - 6.807 V eff 2 + 7.996 V eff 3 0.65 + 1.619 V eff 1.5 + 2.879 V eff 6 .
V eff = V · 2405 V co ( α ) 2 π λ a eff n 0 ( 2 Δ ) 1 / 2 = V · ( 0.5 + α 2.5 + α ) 1 / 2 .
Δ ϕ c l = 2 k Δ n mis a c l - 2 k Δ n mis [ a c l 2 - ( b 2 ) 2 ] 1 / 2 = 2 k Δ n mis a cl { 1 - [ 1 - ( b 2 a cl ) 2 ] 1 / 2 } ;

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